Local stereology

LOCAL STEREOLOGY

ADVANCED SERIES ON STATISTICAL SCIENCE & APPLIED PROBABILITY Editor: Ole E. Barndorff-Nielsen

Published Vol. 1: Random Walks of Infinitely Many Particles by P. Revesz Vol. 4: Principles of Statistical Inference from a Neo-Fisherian Perspective by L Pace and A. Salvan Vol. 5: Local Stereology by Eva B. Vedel Jensen

Forthcoming Vol. 2: Ruin Probability by S. Asmussen Vol. 3: Essentials of Stochastic Finance by A. Shiryaev

Advanced Series on Statistical Science & Applied Probability

LOCAL STEREOLOGY

Eva B. Vedel Jensen University ofAarhus

World Scientific Singapore • New Jersey 'London • Hong Kong

Vol.5

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LOCAL STEREOLOGY Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-2454-0

Printed in Singapore.

Preface The aim of this book is to give a unified exposition of local stereological methods. It is my belief that such a book containing the mathematical and statistical foundations of local stereology is needed since these methods are now in world-wide use in the microscopical study of biological tissue. In many ways this is a very personal book, for a large part based on my own research, and I hope that some of my enthusiasm and joy in working in this area will be acknowledged by the reader. Local stereology has gradually developed during the last fifteen years, as a product of practical needs from the biologists and new advances in the microscopical observation techniques and in the theoretical foundations of local stereology, integral geometry and geometric probability. Three colleagues have had a profound influence on my engagement in local stereology. Roger Miles wrote a fundamental series of papers about stereology and integral geometry in the seventies and the eightties, partly together with Pamela Davy. His paper Some new integral geometric formulae, with stochastic applications, published in Journal of Applied Probability in 1979, became especially important since it contained the tools for constructing local estimators of volume, as we simultaneously realized in 1983. The needs from the users of stereology have been realized through my close collaboration with Hans Jørgen G. Gundersen since 1977. I have during the years tried to cope with his never ending series of questions and their solutions occupy a fair part of the book. The mathematical fundament of local stereology has been built together with my former Ph.D. student Kién Kieu. The book is written for researchers, teachers and graduate students in mathematical statistics and probability. In order to reach a broader audience, the book is not only for specialists in stereology, integral geometry and geometric measure theory. In particular, Chapter 1 is an elementary introduction to stereology. Local stereology involves, however, advanced mathematical tools which is an important part of the book too. Glancing through the book, the reader will realize that this is a book about geometric sampling. Chapter 1 contains an exposition of classical global stereology, including Cavalien's method of volume estimation, estimation via ratios, number estimation using the disector, length and surface area estimation under isotropy, vertical sections. The rest of the book is devoted to local stereology. The mathematical tool for developing local stereological methods is the coarea formula, involving v

vi

PREFACE

generalized Jacobians. This transformation formula is presented in Chapter 2 together with many simple geometric examples and also more complicated examples needed in the following chapters. Chapter 3 is a study of the rotation invariant measure on linear subspaces in Rn. Geometric decompositions of the q—fold product of Lebesgue measure are treated in Chapter 4 and generalized to decompositions of Hausdorff measures in Chapter 5. Local slice formulae and some projection formulae are discussed in Chapter 6. Practical aspects of local methods, including their implementation in the microscopical study of biological tissue and some biological applications are presented in Chapter 7 while Chapter 8 discusses the so-called modelbased approach, a somewhat misleading terminology since the models considered are of a general non-parametric type. I want to thank Ole E. Barndorff-Nielsen for originally suggesting to write this book and for following my work with interest. In the academic year 95/96, I lectured on the subjects of the book at the Department of Theoretical Statistics, Institute of Mathematics, University of Aarhus. I am grateful to the 12 students patiently listening to the lectures, reading the lecture notes, discussing the unclear points, doing the exercises, etc. Especially thanks to my Ph.D. student Jørgen Nielsen and to Kristian Stegenborg Larsen and Niels Væver Petersen. I also want to thank Jørgen Hoffmann-Jørgensen for fruitful discussions about the coarea formula and related matters. Comments on earlier drafts by Morten Bech, Hans Jørgen G. Gundersen and Rick Vitale are also gratefully acknowledged. The illustrations have been made with utmost skill and care by Jakob Goldbach, using the programs Adobe Illustrator 6.0 and STRATA STUDIO Pro VERSION 1.75+. Trine Tandrup and Anette Larsen have provided the photographs of biological tissue. Last but not least I want to thank my husband Kaj and my daughter Jane for patiently listening to all my stories during the writing process and for giving me the opportunity to write this book.

Aarhus January 1997

Eva B. Vedel Jensen

Contents Preface

v

List of notation

xi

1 Introduction to stereology 1.1 Sampling theory 1.2 Stereological estimation of number 1.3 Stereological estimation of volume 1.4 Length and surface area estimation under isotropy 1.5 Local stereology 1.6 Exercises 1.7 Bibliographical notes

1 1 3 11 18 27 29 33

2 The coarea formula 2.1 Hausdorff measures 2.2 The coarea formula 2.3 The special case d = n 2.4 Hausdorff measures on affine subspaces 2.5 Polar decomposition of Lebesgue measure 2.6 Translative decompositions of Hausdorff measures 2.7 A transformation result for surface area measure 2.8 Simplices 2.9 Exercises 2.10 Bibliographical notes

35 35 37 42 44 45 46 49 52 56 61

3 Rotation invariant measures on C™ 3.1 Construction of rotation invariant measures on C™ 3.2 Crofton's formula 3.3 A result on projections 3.4 Pairs of subspaces 3.5 Random subspaces 3.6 Random grids 3.7 Exercises 3.8 Bibliographical notes

63 63 72 75 78 82 83 88 93

vii

viii

CONTENTS

4 The classical Blaschke-Petkantschin formula 4.1 A local estimator of planar area 4.2 Decompositions involving lines in Rn 4.3 Proof of the classical Blaschke-Petkantschin formula 4.4 Local estimators of volume 4.5 Local integral geometric formulae for powers of volume 4.6 Exercises 4.7 Bibliographical notes

95 95 97 100 105 112 114 119

5 The generalized Blaschke-Petkantschin formula 5.1 A local estimator of length in R2 5.2 Prerequisites concerning G—factors 5.3 Decomposition of a single Hausdorff measure 5.4 Decomposition of a product of Hausdorff measures 5.5 Local estimators of d—dimensional Hausdorff measure 5.6 An alternative estimator of surface area \1^~1{X) 5.7 Exercises 5.8 Bibliographical notes

121 121 123 126 132 138 142 146 153

6 Local slice formulae 6.1 A local estimator of number in R3 6.2 A local slice formula for d—dimensional Hausdorff measure 6.3 Local slice estimators of d—dimensional Hausdorff measure 6.4 The special case d — n 6.5 The case 0 < d < n 6.6 Some further developments for n = 3 6.7 Exercises 6.8 Bibliographical notes

155 155 157 159 160 164 167 170 173

7 Design and implementation of local stereological experiments 7.1 Optical sectioning 7.2 Implementation of local designs 7.3 Local stereological estimators in use 7.4 Particle aggregates 7.5 Applications of local stereological methods 7.6 Systematic sampling along an axis 7.7 The circular case 7.8 Exercises 7.9 Bibliographical notes

175 175 175 178 181 183 187 192 195 199

CONTENTS

ix

8 The model-based approach 8.1 Point processes in Rn 8.2 Marked point processes in Rn 8.3 A few results from invariant measure theory 8.4 Estimation of the if-function of the reference point process 8.5 Estimation of moments in the mark distribution 8.6 Exercises 8.7 Bibliographical notes

201 201 206 208 208 212 219 221

9 Perspectives and future trends 9.1 Mathematical and statistical aspects 9.2 Affine version of local stereology 9.3 Curvatures and other parameters 9.4 Future trends

223 223 224 226 227

Appendix Invariant measure theory

229

References

233

Subject index

245

This page is intentionally left blank

List of notation Sets Rn

n-dimensional Euclidean space. The i'th coordinate of x G Rn is denoted xi or {x)i ^R1

R O

The origin of Rn

i?+

Positive real numbers

[a, b] (a, b) [a, 6) Bn(x,r) 5n~!

Closed, open and half-closed-open intervals, respectively, between aeRn and b G Rn, e.g. [a, b) = {aa+ (1 - a)b : 0 < a < 1} Open ball in i? n with centre x e Rn and radius r The unit sphere in i? n

5J-1

{u G S™'1 : o;n > 0}

£n

The unit ball in Rn

Lp

p—subspace; p—dimensional linear subspace of Rn

Cp

The set of p—subspaces of Rn

£ntr\

The set of p—subspaces of Rn, containing a fixed r—subspace Lr, say. Note that ££ (0) = ££

Tp

p—slice of thickness 2t, i.e. a set of the form Lp + Bn(0, t)

n

Tp

The set of p-slices of Rn of thickness 2t

TJ7,

The set of p-slices Lp + Bn(0, t) for which L p D L r , where L r is a fixed r—subspace. Note that T 7 ^ = 7^,n

Gg

^—grid, i.e. grid of parallel g-dimensional affine subspaces

QV:

A set of q—grids in Rn, identical up to rotations and translations

Convex set

A set X C Rn with the property [x,y] C X for all x, y G X

xi

xii Starshaped set

LIST OF NOTATION A set X C Rn is said to be star-shaped relative to x G X, if [x, y] C X for all y G X

Constants cdn

The volume (Lebesgue measure) of the unit ball Bn in Rn, = 7r^ n r(l + \n)~x, cf. Exercise 2.7

^

The surface area of the unit sphere Sn'1 Exercise 2.7

c(n,p)

=

in Rn, = 2 ( 7 r ) H \ ± n ) - 1 , cf.

^-i-^-r+i

c(n,0)

=1

5/;.

The /c'th Bernoulli number, A; = 0 , 1 , . . .

Inner product and norm {x,y)

\\x\\

= YsxiViixiy

e

Rn

=(x)x)1/2,xeRn

Concepts relating to linear subspaces span

s p a n { x i , . . . , Xd} is the linear subspace of Rn, spanned by xi,..-,Xd £ ^ n , i-e- a*l vectors of the form a\x\ -\ h a^x^, where ai G R, i — 1 , . . . , d

L1-

The orthogonal complement of the linear subspace L of i ? n , x G L1- if and only if (x, y) = 0 for all y G L

dimL

Dimension of L

L ©M

Orthogonal sum of the linear subspaces L and M of Rn, i.e. L ø M i s the sum of L and M , and L and M are orthogonal

Le M

Orthogonal difference of L and M , i.e. L e M = L n ( L n M)" 1

7T^

Orthogonal projection onto L. Recall that if a\,..., orthogonal basis of L, then

TTUJ

ITLX

= J2 n'tf ad

Orthogonal projection onto span{a;},u; G Rn

a^ is an

LIST OF NOTATION

xiii

Matrices In

n x n identity matrix

det(A)

Determinant of the matrix A

T

A

Transposed matrix of A

SO(n)

The group of rotations in Rn, special orthogonal transformations

SO(n, Lr)

The subgroup of SO{n) consisting of rotations, keeping the r—subspace Lr fixed

Set operations X +y

= X + {y}, X C Rn, y G Rn

X

={-x

X\Y

Set difference, = {z e Rn : z e X, z <£Y},X,Y

diamX

= sup{||x - x'\\ : x € X, x' £ X}, X C Rn

convX

Convex hull, the smallest convex set containing X C Rn

dX

Boundary of X C Rn

starX

The part of X C i? n which can be seen directly from O, = {x eX : [0,x] C X }

:x eX},X

CRn C Rn

Measures #n

The Borel a-algebra in Rn

BQ

The set of bounded Borel sets

A®B

The product of the a—algebras A and B

Mr

The measure \i lifted by the mapping T, i.e. // r (A) = /i(T -1 (,4))

\d

d—dimensional Hausdorff measure in Rn

dxd

= \dn{dx)

\n

Lebesgue measure in Rn, = A™

y

Volume in R3 or generally in Rn, Lebesgue measure

S

Surface area in R3 or Rn, = A™"1

L

Length in Æ2, R3 or Æn, = A*

H

Number of elements in •, =Xn

xiv

LIST OF NOTATION

/xn, x

Rotation invariant measure on £ V x

dL

=Vnp{r)(dLp) Rotation invariant measure on T7„\

P(r) n v( v p(r)

p{r)

dT

=^(r)(dTp)

Ar)

j]q

Measure on Q™, invariant under rotations and translations

n

dG q

= tf(dGq)

Functions gi or {g)i

The i'th coordinate function of g : Rn —► Rk

g=c

The function g is constant equal to c

ker#

The kernel of g : Rn -» Rk, i.e. kerp =

1{-}

Indicator function

[•]

Integer part

r(-)

The gamma function

£(-, •)

The Beta function

F(-, •, •; •)

The hypergeometric function

Pk{-)

The fc'th Bernoulli polynomial, k — 0 , 1 , . . .

Vp

V p O i , . . . , xp) = p!A£(conv{0, x\,...,

g'l(0)

ar^})

Differential calculus Df(x)

k x n matrix of partial derivatives of the differentiable function / : Rn -+ Rk at a; G i? n

/'(x)

= D/(x) for n = A; = 1

Tan[X, x]

Tangent space of I at x G I

Jf(x\X)

Generalized Jacobian

Statistics E(X)

Mean of the random variable X

Var(X)

Variance of the random variable X

Cov(X, Y)

Covariance between the two random variables X and Y

2

N((i, a )

The normal distribution on R with mean /i and variance a2

LIST OF NOTATION

xv

b(n,p)

The binomial distribution with number parameter n and probability parameter p

X2(f)

The x2~distribution with / degrees of freedom

B(a, (3)

The Beta distribution with parameters a and ft

~

The notation X ~ b(n,p) is used in the case where the random variable X is binomial distributed, say

Fa£

Distribution function of the Beta distribution with parameters (a/2,6/2)

h (\

The quantity JvVjx, Lr) is the probability that x G Rn belongs to an isotropic p—slice of thickness 2t, containing Lr

a.s.

Almost surely convergence

2^

Convergence in distribution

o(-)

Small o function, / = o(g) if f/g —► 0 in some limit

O(-)

Big O function, f = O(g) if f/g is bounded in some limit

Chapter 1

Introduction to stereology This chapter is an elementary introduction to stereology and at the same time, a first course in geometric probability for those readers not familiar with this field. Although stereological methods are applicable for general spatial structures, the applications mentioned in this and the coming chapters are mainly taken from the microscopical study of biological tissues. Indeed, many of the methods described here have evolved with such applications in mind. Modern stereology can be regarded as spatial sampling theory. We start by introducing the few necessary concepts from classical sampling theory.

1.1 Sampling theory Let us consider a finite population V of N objects. We represent the objects by the integers from 1 to TV such that V =

{1,...,N}.

A characteristic y(i) G R is associated to each object i G V. For instance, if the objects are line-segments, y(i) may be the length of the i'th line-segment. The parameter of interest is the population total

The estimate will be based on a sample from the population which is a non-empty subset of the population. The set of all possible samples is denoted S. A sampling design is a probability function p(-) defined on S, i.e. p(s) > 0 for all s G S and ^ P p ( s ) = 1. ses 1

2

1. INTRODUCTION TO STEREOLOGY

A well-known sampling design is simple random sampling of n objects, n < AT, defined by

*.>=M") 'fw=» 10 otherwise, where |-| is the number of elements in •. The sampling designs we will consider in the following sections are however not of this well-known type, the major difference is that the designs we are going to consider have a geometric flavour. Using the concept of a sampling design, we can define what we mean by a random sample. Definition 1.1. A random sample is a discrete random variable S which takes values in S and has a distribution determined by P(S = s) = p(s),s G S, where p is a sampling design. □ An important concept in sampling theory is sampling probabilities of first, second and higher order. The i'th sampling probability p(i) of first order is p(i) =

P(ieS),teV,

and the (i, j)'th sampling probability of second order is P(ij)

= P(i e Sj e S),i ePj

eP.

Note that p(i,i) = p(i) for all i G P. An estimator of the population total, well-known from survey sampling theory, is the Horvitz-Thompson estimator, cf. Horvitz & Thompson (1952), Cochran (1977), Brewer & Hanif (1983), Krishnaiah & Rao (1988). The estimator is given by

Note that we implicit assume that p(i) > 0 for all i G P. This is of course a very mild assumption. The Horvitz-Thompson estimator is an unbiased estimator of y-p, as indicated in Proposition 1.2 below. The proof of the proposition is left as Exercise 1.1. Proposition 1.2. Suppose that p(i) > 0 for all i £ P and let

1.2 STEREOLOGICAL ESTIMATION OF NUMBER

3

Then,

E(yv) = w-

D Sampling with unequal sampling probabilities has been introduced in the survey sampling literature in order to reduce variability. A situation of special interest occurs when the sampling probabilities are proportional to the characteristics p(i) = cy(i),i <EV, for some c ^ 0. Then,

yv = -\s\. C

If in particular the sample size is constant, the Horvitz-Thompson estimator has variance equal to 0. The Horvitz-Thompson estimator may however have a large variance if the sampling probabilities p(i) and the characteristics y(i) are unrelated. One famous and humorous example is given in the anecdote about Basu' elephants, cf. Basu (1971, p. 212-213). In the sections to follow, we will see that the classical stereological estimators can be regarded as Horvitz-Thompson estimators associated with geometric sampling designs fulfilling the assumption of proportionality stated above.

1.2 Stereological estimation of number Let us suppose that an open and bounded subset X of R3 contains particles Z i , . . . , Z/v, and that the object is to estimate their number N. To fix ideas, you may imagine that X is a human brain and that the particles are a particular type of biological cells, e.g. neurons. The problem of estimating the number N is of the type described in Section 1.1. The objects are here the particles and the characteristic associated with each object is identically 1, such that iev At this stage, no specific assumptions are made about the shape of the particles, they are simply compact subsets of R3. One plane sampling design. Let us start by discussing the one plane sampling design where a random sample of particles are selected as those hit by a random plane section L
4

L INTRODUCTION TO STEREOLOGY

Figure 1.1, The set X sectioned by Li.

The basic difficulty in constructing a steieological estimator of N from obsewaticMS in such a plane section through X is that the number of particles seen in a planar section does not depend only on N, but also on the size and the shape of the particles. To see this» let us consider a random plane section through X with ixed orientation., but with uniform random position relative to X. To be more precise, we consider a random plane L2 distributed as L2(o) + UUJ9 where L2(o) is a fixed plane through the origin O, u) is a unit normal vector of L2(o) and U is uniform random in the hitting set {«€i?:A'n(L2(0)+iL)^}. cf. Figure 1.2. Note that X n (I2(0) + ^ ' ) # Ö <=> 'Ma,' € TT^X, where w^ is the projection onto the line through () spanned by w. Therefore, the density of U is ( l/L(Ku,X) X0

if uweiTuX otherwise,

where L denotes length (1-dimensional Lebesgue measure).

1 1 SmSKEOLOGICAL ESTIMATION' OF "NUMBER--

5

Figure 1.2. The section plane shown in Figure 1.1 is distributed as L-1:{J] — l'~ where L^Q) and u arc ixed», and U is uniform random.

By the one plane sampling design, we simply sample all particles hit by the plane, i.e. S = {i eV : Zi^.L-2 ^ 0 } . The sampling probability of the /Th particle becomes pij) = p(i e 5) = P{ZinL>> # 0 )

I = 1 , . . . , :V. cf. Exercise 1.2. The set of particles hit by the plane is therefore a size-biased sample. Under the assumption that L^^Zi) > 0 for all i, we can construct the Horvitz™ Thompson estimator of N,

ies In practice, L(irwX) can easily be measured directly, cf. Figure 1,1» while L(7rwZi) cannot be determined from a single plane section. Under the assumption that all particles ha¥e the same» known and simple shape like spherical shape, the estimator can be simplified, cf. Wicksell (1925) for the first reference. Craz-Orive (1980a)

6

1. INTRODUCTION TO STEREOLOGY

suggested to determine the projection lengths using local serial sectioning» thereby avoiding specific shape assumptions. A more convenient procedure was however suggested by Sterio (1984). Di'sector sampling design. A breakthrough in the number estimation came in 1984. when Sterio (1984) suggested to sample particles by means of two planes a short, but known distance h apart, a so-called disector (di=two). The idea is to sample all particles hit by one of the planes» the reference plane, but not by the other» the look-up plame, cf. Figure 1.3.

Figure 1.3. A disector consists of a reference plane and a look-up plane, a distance h apart.

The reference plane is denoted by i#2 and is distributed as before, the look-up plane is L-2 — JtUJ.

The disector sample is S = {i € V : Zi H L2 ± 0. Z, P (L-2 - M - 0}. Under weak assumptions about particle shape, the sampling probabilities can be calculated» as shown in the proposition below. As we will see» the sampling probability does not depend on the particle considered. In essence, we sample from the population of particle caps instead of whole particles» so all particles are 'made into* the same height.

1.2 STEREOLOGICAL ESTIMATION OF NUMBER

7

Proposition 1.3. Suppose that for all i G V, the projection ir^Zi is a line-segment of length at least h, i.e. KujZi = {uu : a,i < u < bi}, where a^bi G R,bi — ai > h,i = 1 , . . . , N. Then the sampling probabilities of the disector design are the same for all objects, viz. p(i)

= p(i eS) = h/LfaX^ie

V.

Proof. Recall that L 2 has the same distribution as L2(o) + Uu. For any particle Z we have Z n (L2(o) + Uu) ^ 0 & Uu G iruZ. Therefore, we get P(i eS) =

P{UUJ

= P(Uu

G irUJZl, {U - h)u 0 TTLUZ1) G ITuZi^TTuZi

= LfaZiXfruZi +

+

hu))

hu^/L^^X)

( }

= h/L(ir„X),

where we at (*) have used the assumption of the proposition.

□

The Horvitz-Thompson estimator of the particle number N therefore becomes

In order to determine \S\, it must be possible for all i G V and for all u G R to identify the set {Zi D (L2(o) + uu), Zi D (L2(o) + 0 - /i)^)} generated by a disector with position u. This requires either prior knowledge about the particles or extra information collected from e.g. moving optical planes. In the latter case, the assumption of Proposition 1.3 concerning a minimal 'height' of the particles is not necessary, cf. Chapter 7. Systematic disector sampling design. Usually, it is not enough to collect data from a single disector. Instead, a systematic set of disectors can be used. The reference planes of the disectors constitute a series of equidistant, parallel planes with distance A>h between neighbour planes. The series is {L2(0)+ (U + iA)w}°°=-00,

8

1. INTRODUCTION TO STEREOLOGY

cf. Figure 1,4, where I' is now a uniform random variable in the interval [0,A), say. If we let Siu) = {ieV:Zin

(L 2(0) + 1^) ^ ø.Z, n (L 2(0) + (u - h)u) = 0}

be the set of particles sampled by a disector with position m, thee the sample generated by the systematic set of disectors is

Figure 1.4. A systematic set of disectors.

In the proposition below» the sampling probabilities are derived. Proposition 1.4. Suppose that for all i € V, the projection ir^Zi is an interval of length, at least h, cf. Proposition 1.3. Then the sampling probabilities of the systematic disector design are />(/) = P(ie

S) = h/Xi

GT.

Proof. Because of the assumption of the proposition» we have for all u S(u + ji A) fl S(u +

JL> A)

= 0. for all ji # J2-

1.2 STEREOLOGICAL ESTIMATION OF NUMBER

9

(A particle is sampled in at most one disector.) If 1{-} denotes indicator function, we therefore get oo

P(ieS)=

£

P(isS(U

+ jA))

j=-oo

= f; Ji{ieS{u + jA)}% ;=-oo

0

^'+1>A

oo

=i E

/

J=-oo

i{ies(«)}d«

j A

oo

= i

M { i G 5(u)}
(

=^/A,

where we at (*) also have used the assumption of the proposition.

□

The Horvitz-Thompson estimator of TV based on a systematic disector design therefore is A

A

°° .7=—oo

Subsampling in the section planes. In many biological applications, the number of particles hit by a planar section is so large that it is out of question to count all particles hit by a reference section, and not by the corresponding look-up section. Subsampling of the section by means of windows is therefore needed. The problem now is whether a particle hitting the boundary of the window should be counted or not. In the literature, a number of ways to cope with this edge effect problem exist, see Miles (1974), Gundersen (1977, 1978), Jensen & Sundberg (1986b) and references therein. Here, we will only discuss one of the methods, which is frequently used. The idea behind the method can be presented as follows: imagine the plane section covered by windows which are non-overlapping congruent quadrats. Order the windows by e.g. the lexicographic ordering. Count a particle in a window if this window is the first one where the particle appears. Then, each particle is counted in exactly one window. Note that a particle which is connected in space does not necessarily have a connected particle section. For simplicity, it is often assumed that any section through a particle is connected, but this assumption is not necessary for the method to work.

10

1. MTRODUCTfOM TO STEREOLOGY

The important thing is to be able to identify» by means of optical sectioning» for instance, the different connected parts in the particle section which originate from the same particle. See also Chapter 7. Now» let us discuss how we can use this idea to get an estimate of the number of particles in a planar section based on subsamplieg in windows. Let us co¥er the planar section with N windows, say» cf. Figure 1.5. Let us denote this population of N windows by V. To each window i, we associate the number y(i) of particles which are counted in this window. The total number of particles in the planar section can then be represented by the population total

ie'P Let us now take a random sample S of windows such that window i has sampling probability p(i)ji € V. The Horvitz-Thompson estimator of the number of particles in the planar section becomes

ies Very often, the sampling probability does not depend on /. i.e. /){/) = p. / e V, say.

Figure 1.5. Subsampling of particles in a planar section. In Figure 1.5, the planar' section is co¥ered by N = 36 windows. A systematic sample of windows has been selected as those lightly hatched. The sampling proba­ bility is p = 1/9. The dark particles are counted in one of the selected windows when the lexicographic ordering is used.

1.3 STEREOLOGICAL ESTIMATION OF VOLUME

1.3 Stereologieal estimation of volume Let us suppose that we want to estimate the volume, i.e. 3-dimensional Lebesgue measure, V(Z) of a subset Z of X. Both X and Z are assumed to be open and bounded subsets of Ä 3 . This estimation problem has numerous applications in. biology. One example is the case where X is a kidney and Z is some substructure of the kidney the content of which may change during diseases. The simplest possible design is the Spatial point grid design. In practice» a spatial point grid is generated by making a series of parallel plane sections through X and then» placing a planar point grid on each section, cf. Figure 1.6. The collection of planar point grids constitutes the spatial point grid.

Figure 1.6. The spatial point grid design. Formally, we will dehne a uniform spatial point grid, using a series {£>/}'^_0 of bounded, space-filling, non-overlapping and congruent subsets of lf\ In particular, the /.Vs satisfy

Dhr\Dh =

^h±j2.

Since the D,\s are congruent we can find UJ € R:i such that Dj = for all j ,

DQ

+ Uj = {u +

UJ

: u G D0}.

12

1. INTRODUCTION TO STEREOLOGY

Definition 1.5. A uniform spatial point grid is a set of random points in R3 distributed as Go(£/) = {£/ + ^ : j = 0 , l , . . . } , where U is uniform random in Do-

□

Below, we write Go as short for GQ(U), when convenient. The definition of a uniform spatial point grid is illustrated by the 2-dimensional analogue shown in Figure 1.7.

Figure 1.7. The uniform planar point grid. The random variable U is uniform random in the region DQ.

In practice, V(Z) is estimated, using the spatial point grid, by

V(Z) =

V(D0)\ZnG0\.

As shown in the proposition below, this estimator can be regarded as a HorvitzThompson estimator. Proposition 1.6. Let G0 be a uniform spatial point grid and let Z = U ^ Z » be any decomposition of Z, such that Z{ n Z2■ = 0,z / j . Suppose that the decomposition is chosen so fine that for all u G DQ \ZinGo(u)\e{0,lhi

= l,.-.,N.

(1.1)

1.3 STEREOLOGICAL ESTIMATION OF VOLUME

13

Let V = { 1 , . . . , N} and associate to object i the characteristic y(i) = V(Zi), i G V. Let S be the random sample given by S =

{ieV:ZinGo^Qi}.

Then, the estimator V(Z) is the Horvitz-Thompson estimator associated with this sampling design. □ Note that one possible decomposition satisfying (1.1) is the finite number of non­ empty sets among { Z n D j } ^ . Proof of Proposition 1.6. First note that V(Z) can be expressed as the population total

V(Z) = £V(Z;) = 5>(z) = yp. lev

lev

The next step is to determine the sampling probabilities. We find P(i eS) = P(Zt n G 0 jt 0) {1

=]E\ZtnG0\ OO

= E^2i{u

+

UjeZi}

j=0 OO

= Y,P(u + ujeZi) 3=0

J

-°Do 1

°° f J

= =

Dj

VW) 11{u e Zl}du V(Zi)/V(D0).

The Horvitz-Thompson estimator of V(Z) therefore becomes

v(z) = Y/y-j\ = v(Do)\s\.

14

1. INTRODUCTION TO STEREOLOGY

The last step is to show that | 5 | = \Z n G0\. We get N

2= 1

N

i=l

= \s\.

n The spatial point grid design cannot be used in all cases of practical interest. Very often, the 'smallness' of the structure of interest Z suggests a cascade of different sampling levels at different magnifications, whereby the object phase of one level becomes the reference phase of the next level, cf. Cruz-Orive and Weibel (1981), X = Y0DY1D---DYN

= Z.

The parameter of interest V(Z) is then estimated via the equation

The first quantity V(YQ)= V(X) is estimated using the spatial point grid design, or simply by fluid displacement, while the ratios are estimated using different types of sampling designs depending on the magnification. At intermediate levels, light microscopy may be used corresponding to a magnification of the order of 102 to 103, while at higher levels, electron microscopy may be necessary giving magnifications from 103 to 105. Bounded point grid design. We will now discuss the bounded point grid design which is used in cases where high magnifications are needed. Let us assume that the object is to estimate the ratio

VV = V(Z)/V(Y), where Y = Y/v-i is an appropriately chosen reference space containing Z — Y/v. Both Y and Z are assumed to be open and bounded subsets of Æ3. The set X is cut into small blocks and a section of negligible extent compared to X is made through each one of a random sample of blocks. Only sections hitting Y are analysed. Equivalently, we consider small sections hitting Y as the one shown in Figure 1.8. The sections are equipped with a finite set of points.

1.3 STEREOLOCilCAL ESTIMATION OF VOLUME

15

Figure 1.8. The bounded point grid design. The .set Y is here a cell population while the set 2 is the darker regions inside the cells.

Formally» we will define a uniform bounded point grid hitting Y, using a set of n points m T\) =

{itoii-.-iUu,,}.

In the applications, we ha¥e in mind, the points will lie in the same plane but this is not necessary for the theoretical considerations that follow. Definition 1.7. A uniform bounded point grid hitting Y is a set of random points in Ä 3 with the same distribution as

C\)(L:) = r -f r 0 = { r + uoi

r + no,,}.

where U is uniform random in the hitting set {'/ € RA : Y n (u ~t- 7o) ^ 0}«

□

As before, we write (/;. instead of GVT: when convenient, Let us find the density of U. Since Y

" ..<; - IG = 0 c- a rL y +

f{h

where Jit = { n € J?'1 : -// e 7{)} = {-»ui, • • •, -^0n},

16

1. INTRODUCTION TO STEREOLOGY

we have 10

otherwise.

Note that 0
+ f0) < o c ,

because Y is open and bounded. Using a bounded point grid design, the ratio Vy — V(Z)/V(Y) estimated by

is usually

vv = |znG 0 |/|ynGo|. As we will now see, this estimator can be regarded as the ratio between two HorvitzThompson estimators of V(Z) and V(Y), respectively. In the proposition below, we start by constructing a Horvitz-Thompson estimator of V(Z). Proposition 1.8. Let Go be a uniform bounded point grid hitting Y and let

z = uf=1zu ZiH Zj — 0, for i / j , be any decomposition of Z, satisfying for all u G R3 l ^ n G o W l G{0,l},« = l,...,iV.

(1.2)

Let V = {1,. ..,N} and associate to object i the characteristic y(i) = V(Z{),i G V. Let S be the random sample given by S={ieV:ZlnG0^®}. Then, the estimator

v(Z) = V(y

+

To)

\znG0\

is the Horvitz-Thompson estimator of V(Z), associated with this sampling design. Proof. Because of condition (1.2), we get n

P(i € S) = P(Zi n G0 # 0) = J2 p(u + UOJ e Zi). 3=1

1.3 STEREOLOGICAL ESTIMATION OF VOLUME

17

Since Zi C 7 , we have u + uoj eZi=>ueY

+ f0=> pu(u) > 0,

and accordingly, U V(y +f)

1{U + U0J €Zi} R3 _ V(Zj - UQJ)

°

~ V(Y + t0)

v(Y + t0y The sampling probability therefore becomes nV

P(i e s) =

&l .

V(Y + T0) The Horvitz-Thompson estimator of V(Z) is then of the form 7tsp{l)

£lnV(Zi)/V(Y

+ t0)

n

Finally because of (1.2) we have \S\ = \Z D G0\-

' '' D

Using Proposition 1.8 once more with Z = Y, we also get

v(Y) = V
+

t

n

ti\Ynöo\, '

'

and therefore V(Z)/V(Y)

= \ZH G0\/\Y n Go| = Vv.

The estimator in common use can thus be regarded as a ratio between two HorvitzThompson estimators. Usually, observations from a number of point grids are combined, as shown in the proposition below. Proposition 1.9. Let GQI, • • •, Gom,... be a sequence of independent and identically distributed uniform bounded point grids hitting Y. Let m

m

Znö

Vv = Y/\ t=l

oi\/J2\Ynöoi\i=l

18

1. INTRODUCTION TO STEREOLOGY

Then, Vy " '

m—>-oo

a i l d

Vy 2

V^(Vv-Vv) -

2

yv(0,^(VV2—))>

where for a single uniform bounded point grid Go hitting Y, we have fiy = E\Y nGo\,fjLz = E\Z

nG0\

o\ = Var|F n G 0 |, a* = Var|Z n G 0 |

ay2 = Cov(|ynGo|,|znG 0 |).

□ The proof of the Proposition 1.9 is left as an exercise. Regression type estimators of Vy based on the data

(|ynGoi|,|^nGo,-|),2 = i , . . . , m have been discussed in Cruz-Orive (1980b) and Jensen & Sundberg (1986a).

1.4 Length and surface area estimation under isotropy The principle of estimating length and surface area under isotropy goes back to Buffon. Here is a formulation of his original problem, cf. Miles & Serra (1978), Je suppose que dans une chambre, dont le parquet est simplement divisé par des joints paralléles, on jette en Vair une baguette, & que Vun des joueurs parie que la baguette ne croisera aucune des paralleles du parquet, & que Vautre au contraire parie que la baguette croisera quelques-unes de ces paralleles; on demande lefort de ces deux joueurs. On peujouer cejeu sur un damier avec une aiguille ä coudre ou une épingle sant téte. Buffon found that the needle hits one of the boundary lines with probability 2//(?rA), provided that the length I of the needle is smaller than the width A of the boards. In Buffon's formulation, the needle is random while the line system is fixed. Exactly the same result is obtained if the situation is reversed, i.e. the needle is fixed while the line system is random, cf. Exercise 1.7.

1,4 LENGTH AND SURFACE AREA ESTIMATION

19

Buff oil's ideas are nowadays applied in the stereological estimation of curve length in 3-dimensional space. In biological applications, the curves may be Mood Yessels» muscle fibres» fibre structures inside cells» etc. We suppose that our set X contains a spatial curve Z. The objective is to estimate its length L(Z). For the moment, it suffices to have an intuitive idea of what is meant • by the length of a spatial curve. In Chapter 2» this concept is formally defined, using a Hausdorff measure. The 3-dimensional analogue of Buffoe's design is the Plane grid design. In contrast to earlier sampling designs» this design involves a randomization in orientation as well as position. The analogue of a uniform position becomes an isotropic direction. In order to define this concept, let S\ be the unit hemisphere in Ä 3 and let fi be the uniform measure on S+ with total mass p(S+) = 2w. This measure will be defined formally in Chapter 2 as a Hausdorff measure. An isotropic direction is now a random direction O € S+ with constant density l/2n~ with respect to fi.

Figure 1.9. Buff on's needle problem.

In some cases» it will be convenient to use polar coordinates instead of Cartesian coordinates. Let (6, ) be the polar coordinates of a point ~ e >i on the hemisphere, i.e. u) = (sin9cos#,sin0sin#,cos0)Je [0,?r/2),oG

[0,2TT).

20

1. INTRODUCTION TO STEREOLOGY

We have the following transformation result for any non-negative measurable function g on S+ TT/2 2TT

I g(uj)ii(du:) = / S2+

/ p(sinØcosø,sinØsinø,cosØ)sinØdødØ.

(1.3)

o o

This result will be shown in Chapter 2» together with other transformation results of this type» see also Figure 1.10.

Figure 1.10. The area of the infinitesimal element is sinØctørfØ. We will estimate the length of the curve Z, using a systematic set of parallel planes with isotropic direction. Definition 1.10. A uniform and isotropic plane grid is a set of random parallel planes with the same distribution as

G 2 (r.O) = {L2(n)^(r

+ jA)n:j

= o.±i.r2....}.

where 1-2(0) is a plane through O with unit normal chosen as an isotropic direction O on S+ and U is independent of O and uniform, in an interval of length A. D Uniform and isotropic grids are treated in a more general framework in Chapter 3. Below» we use the short notation G2 for G2(U,Q)9 when convenient. In practice, if G-> is a uniform and isotropic plane grid, then L(Z) is estimated by I(Z) = 2A|ZnG2|, cf. Figure 1.11. As we will see below, L(Z) can in fact be regarded as a HorvitzThompson estimator in case Z is a polygonal curve.

1.4 LENGTH AND SURFACE AREA ESTIMATION

21

Figure 1.11. The plane grid design.

Proposition 1.11. Let GL> be a uniform and isotrepic plane grid and let Z = U ^ Z » , where Z% is a line-segment of length less than A. Let V = { I , . . . , N} and associate to object i the characteristic y(i) = L{Z{), i € P . Let S be the random sample given by

5 = {/€P:ZinG2#0}. Thee, the estimator I ( Z ) = 2A|Z~=GV is the Horvitz-Thompson estimator of L(Z), associated with this sampling design. Proof. Let us find the sampling probabilities. Conditionally on c^» we get

P(i € S\Q = uO = P(Zi n G2 + #|0 = a;) = Since LfanZi)/L(Zi)

L Zi

^ r.

is uniform in [0,1], cf. Exercise 1.8, we find» if p(w) is the

22

1. INTRODUCTION TO STEREOLOGY

density of fi with respect to \x, P(i eS)=

f P(i£

S\Q = w)p(w)n(dto)

si _ f L^Zj)

~~ J si

n(du)

Ä

2^T

_ L(Zj) f L^Zj) /i(duj) A J L{Z%) 2TT s%

0

L(Zj) = 2A ' The Horvitz-Thompson estimator is therefore

LIZ) = Y^-

= 2A|5| = 2A\Z n G2\.

At (*) we have used that L(Zi) < A for all i e V.

□

The variance of the estimator L(Z) may be large because only one orientation is represented by the planes. One such example occurs when the spatial curve has a preferred direction, as muscle fibres have. In such cases, alternative designs are necessary, cf. e.g. Cruz-Orive et al. (1985) and Gokhale (1990). In case high magnification is needed for observing Z, planar sections of negligible extent compared to X are used. The parameter L(Z) is estimated as described in Section 1.3 via a ratio LV =

L(Z)/V(Y),

where Y is an appropriately chosen reference space containing Z. We assume that Y is an open and bounded subset of R3. One example of a design used in this situation is the Circular window design. In order to define the circular window design we let To (a;), a; G 5^., be circular discs of the same size, centred at the origin O, such that TO(CJ) has normal LJ, UO G S\. We use A3 as the notation for Lebesgue measure in R3. Definition 1.12. A uniform and isotropic circular window hitting Y is a random circular disc with the same distribution as G2(£/,fi) = C/ + T0(ft),

1.4 LENGTH AND SURFACE AREA ESTIMATION

23

where the density of [U. f>) G RA x S | with respect to A3 x /1 is concentrated on the hitting set {* >/, ^) € i? 3 x S% : F n (11 + r 0 U\)) # 0 } and is of the form [ 0 otherwise.

D The definition of a uniform and isotropic circular window G2 hitting Y is illustrated in Figure 1.12. One may think of the circular window as the ield of vision in the microscope.

Figure 1.12. The circular window design. Here» >" is as in Figure 1.8 while Z is the union of the spatial curves inside the cells. The marginal density of 0 becomes p{u!) = / p(u*u:)du =

/

cdu

Y~To(u)

= rr(r + 7b(u/)), and therefore s2

24

1. INTRODUCTION TO STEREOLOGY

The conditional density of U given £1 = u is then

^=w=W&j)- , o r " e y + f o M Therefore, the conditional distribution of U given Q, — UJ is uniform in the hitting set Y + f0{uj). The number of intersections between G2 and Z can be used to construct an estimator of L(Z). Proposition 1.13. Let G2 be a uniform and isotropic circular window hitting Y and suppose that Z = ufL-^Zi, where Z{ is a line-segment. Let V — { l , . . . , i V } and associate to object i the characteristic y(i) — L(Zi),i G V. Let 5 be the random sample given by

S=

{ieV:Zlnö2^Q}.

Then, if A (To) denotes the area of To, the estimator TrcA(Io)

'

is the Horvitz-Thompson estimator of L{Z), associated with this sampling design. Proof. It suffices to find the sampling probabilities. We get P(i G S) = P(ZinG2^0) =

l{Zi n (u+T0(cj)) ^ 0}cdu/j,(du>) 3

SI R

= c I V(Zi +

f0(u))ij,(du)

si = c [ L(TruZi)A(T0)ti(dio)

S%

= KcL(Zi)A{TQ). The last equality follows from the fact that L(-KQ,Zi)/L{Zi) is uniform random in [0,1], when Q is an isotropic direction. □

1.4 LENGTH AND SURFACE AREA ESTIMATION In order to estimate Ly = L(Z)/V(Y) to C?2- Then,

Y(Y) =

we suppose that n points Go are attached

^—\YnG0\

can be regarded as a Horvitz-Thompson estimator of V(Y). estimated by I

_ L(Z) _ r

25

2n

Accordingly, Ly is

\ZnG2:

" r(r) ~-4(r 0 )|ynG„r

As in the previous section» we can also consider a sequence of independent and identically distributed uniform and isotropic circular windows hitting Y, G'21i • • • ?G2TO» . . .

and construct a consistent and asymptotic normally distributed estimator of Ly. The dual estimation problem is estimation of the area of a spatial surface Z, This can be done by using a spatial line grid. We will not go into details with this estimation problem since the principles have already been presented. Here, a uniform and isotropic line grid is used.

Figure 1.13. The spatial line grid design.

All the lines in the grid are parallel to Li(O), which is a random line through the origin O with an isotropic direction 0» cf. Figure 1.13. The position of the line

26

1. INTRODUCTION TO STEREOLOGY

grid is determined by a uniform planar.point grid in the plane Li(il)

. The estimate

of surface area is then Ä(Z) =

2A(Do)\ZnChl

where Do is indicated on Figure 1.13. It is often difficult to make sections with an isotropic direction (normal ¥ector) through a biological specimen. In fact» sections at a particular orientation may be preferred because such sections reveal interesting information. Typical examples are muscle tissue where longitudinal sections are often taken; and skin» where perpendic­ ular sections are used, cf. Baddeley et al. (1986). Longitudinal and perpendicular sections are examples of so-called vertical sections» i.e. sections containing a fixed axis (in the skin example» it is the normal to the skin surface). It is therefore ¥ery important from a practical point of ¥iew that it is possible to generate for instance lines with isotropic directions ¥ia ¥ertical sections. This problem was sol¥ed by Baddeley (1983, 1984). Consider a random vertical plane» i.e. a random plane ^2(#) containing the vertical axis and with a uniform rotation # € [0,1") around the vertical axis, cf. Figure 1.14. Generate a random line through O in this plane, L i ( 6 , $ ) , with an angle (-) to the vertical axis having the density p(0) = ^sinØ, 0 €[0,7T). Then» it is easy to show, using the transformation, result (1.3), that I i ( B . # ) can be regarded as a random line in Ä 3 through O with an isotropic direction.

Figure 1.14. Generation of an isotropic line via a vertical plane.

1.5 LOCAL STEREOLOGY

27

1.5 Local stereology Local stereology is a branch of stereology which has been developed since the beginning of the eighties. The target is here quantitative parameters of structures which can be regarded as neighbourhoods of points, called reference points. An important example is the case where the structure is a biological cell and the reference point is the cell nucleus or some identifiable part of the nucleus such as the nucleolus. The cell is thus regarded as a neighbourhood of its nucleus, cf. Figure 1.15. Local stereology is a collection of sampling designs based on sections through the reference points which have been developed with structures of the above mentioned type in mind. It is possible to determine sampling probabilities associated with such designs and thereby to construct estimates of parameters such as cell volume and cell surface area. Other parameters which can be estimated by local techniques are the length of fibres inside cells and the number of cells of another type sitting around the cell in question. Below, we give a simple example of a sampling design in local stereology.

Figure 1.15. The local set-up. A typical cell size is 5-50 fim. Example 1.14. Let Z be a subset of R2 consisting of N points Z =

{ZI,...,ZN}.

Suppose that we want to estimate the number N. As previously, we can think of the points as objects in a population V — { 1 , . . . , Af} and associate to object i the characteristic y(i) = 1, i G V. The population total is then N. We will estimate N by using an isotropic band through a reference point. Let the reference point be the origin O in the plane. Then, an isotropic band T\ of width 2£

28

1. INTRODUCTION TO STEREOLOGY

through O is determined by its mid-line L\ which makes a uniform angle 6 G [0,7r) with a fixed axis, cf. Figure 1.16. The sample consists of the points hit by the isotropic band, S = {i G V : Zi G Ti}. Let us find the sampling probabilities. In case ||^|| < t, p(i) — 1. Now, suppose that ll^ll > t. Without loss of generality let us assume that Zi is positioned on the positive part of the y—axis. Then, if ai = arcsin (t/lkdl) w e n a v e p(i) =

P(zieT1)

= P ( - - a{ < 6 < - +

ai)

_ 2ai 7T

Figure 1.16. The band Ti is determined by the mid-line Lx which makes a uniform angle 0 G [0,7r) with a fixed axis. The Horvitz-Thompson estimator of TV becomes

*-£$ ies

!»-'■

= p{l

>

ies

Note that we always have p(i) > 0. The estimator TV is bounded. To see this, let

Vt = {ieV:

11*11 < t},

1.6 EXERCISES

29

dmax = max||zi|| and am[n — arcsin(t/d m a x ). Then

N = \snvt\+ Yl

P^1

iesyPt <\SnVt\

+

^n\S\Pt\.

In particular, TV has always finite variance.

□

In the earlier sections of this chapter, we have considered sampling designs for which p(i) = cy(i),i e V fo> = - | S | c The local design described above is an example of a design of the type p(i) = c(i)y(i),i eV ies

v

'

The quantity c(i) depends on distance measurements which can be measured in an actual application. More details about practical aspects will appear in Chapter 7.

1.6 Exercises Exercise 1.1. Prove Proposition 1.2. Show also that the variance can be rewritten as \r (- \

ST1-P(i) lev

HK J

/-x2 ,

v^ ievjev,i^j

P{hJ)~ P{i)p(j) r\ ( -\ HK JHKJJ

Exercise 1.2. Derive the sampling probabilities p(i) = P(i eS) = L(iruZi)/L(iruX),i

G V,

associated with the one plane sampling design. Exercise 1.3. This exercise concerns the variance of the estimator ^ L ^ X ) | 5 | h of N based on the disector sampling design. 1. Let Lij = L(7TwZi fl irwZj\[(TTwZi + huj) U (TTWZJ + hu)]),

30

1. INTRODUCTION TO STEREOLOGY ij

= 1,.. .,7V. Show that

Var(tf) = : É ^ £ > , ■ - * ' . Hint. It is a good idea to start by finding the second-order sampling probabilities.

2. Show that Var(7V)>^^7V-iV2. 3. Let us suppose that the particles are positioned such that the minimum of the variance is attained, i.e. - N2.

Var(iV) = ^ p - N Show under this assumption that | 5 | ~ b(l,p)

w tn

i

parameter p =

Nh/L(7ri0X).

Exercise 1.4. In Figure 1.5, a planar section with N — 15 particles is shown. The Horvitz-Thompson estimator is in this example of the form TV = 9 ^ 2 / ( 2 ) , ies where S is the systematic sample of windows and y(i) is the number of particles counted in window i, i.e. the number of particles first seen in window i. Find the distribution of N and show that EN = 15. Exercise 1.5. This exercise concerns the 1-dimensional analogue of the spatial point grid design. We consider an open and bounded subset Z of the real line R. The object is to estimate the length (1-dimensional Lebesgue measure) of Z, using a uniform point grid with distance A between neighbour points, say. This point grid is distributed as {U + i A } ^ z _ 0 0 , where U is uniform random in the interval [0,A). The estimator of L(Z) becomes 00

L(Z) = A ^2 !{U + j=-oo

1. Show directly that EL(Z)

= L(Z).

jAeZ}.

1.6 EXERCISES

31

2. Suppose that Z is a line-segment of length L(Z) — kA + 6, where A; is a nonnegative integer and 0 < 6 < A. Show that the distribution of L(Z) in this case is given by

P(L(Z) = A(fc + l)) = A P(L(Z) = Afc) = 1 - -^. Also show that the variance of the estimator is VarL(Z) = (A - 6)6. Notice that the variance does not depend on the length of the interval. Is this reasonable? 3. Show for an open and bounded set ZC R that oo

VarL(Z) = A ^

L(Z)2

L(Z n (Z + jA)) -

j=-oo

Exercise 1.6. Prove Proposition 1.9, using the standard strong law of large numbers and the central limit theorem. Exercise 1.7. Buffons's needle problem. Let Z be a line-segment in the plane R2 of length I. Let be a random set of parallel lines with distance A between neighbour lines. Here, Li(fi) is the line through the origin O with unit normal vector £1 chosen as a uniform point on the unit semicircle, and U is independent of £2 and uniform in an interval of length A. The distribution of Q, is the same as that of (cos $, sin ), where 3> is uniform in the interval [0,7r). 1. Show under the assumption I < A that, conditionally on Q, P(Z is hit by a line|fi = u) =

L

^

Z

\

Hint. Recall that Z n (Li(u) + (u + jA)uu) / 0 ^> (u + jA)cj e TTujZ. 2. Show that L(TTQZ)/1 has the same distribution as |cos$|, where $ is uniform random in [0,7r). Hint. Without loss of generality Z can be assumed to be parallel to the x—axis. 3. Show that the unconditional hitting probability becomes 2/ P(Z is hit by a line) = —-. 7rA

32

1. INTRODUCTION TO STEREOLOGY

Exercise 1.8. Let Z be a line-segment in R3 and let ft G 5+ be an isotropic direction. Show that the random variable L(TT^Z)/L(Z) is uniform in the interval [0,1]. Hint. Without loss of generality, it can be assumed that Z is parallel to the z—axis. In that case L(7r n Z)/L(Z) = cos/(fi), where /(ft) G [0, f ] is the angle between ft and the z-axis. transformation result (1.3), that /(ft) has density

Show, using the

Exercise 1.9. Let Z be a line-segment in Æ3 and let G 2 = G2(f/,ft) = {L2(ft) + ([/ + j A ) f t : j = 0 , ± l , ± 2 , . . . } be a uniform and isotropic plane grid, as defined in Definition 1.10. Finally, let L(Z) = 2A\ZnG2\. 1. Show that the conditional mean value and variance of L(Z) are E(L(Z)\Q = uj) = 2L(7rUJZ) Var(L(Z)|ft = co) = 4(A - «„)«„, where ^ is defined by L^^Z) = k^A + 6U, where A^ is a non-negative integer and 0 < du < A. Hint. Exercise 1.5. 2. Show that the unconditional mean value and variance are E(L(Z)) = L{Z) Var(L(Z)) - \h{Zf

+ 4£((A -

Sn)Sa).

Hint. It is useful to use the identity Var(L(Z)) = Var£(L(Z)|Q) + £Var(L(Z)|fi). 3. Show that \\mN^{L{Z))=l-L{Z)\ The variance does not tend to 0, although the number of planes hitting Z is increasing to infinity. The reason for this is of course that under the limiting procedure only one direction is represented.

1.7 BIBLIOGRAPHICAL NOTES

33

1.7 Bibliographical notes Stereological methods can be applied in biology, medicine, metallography, min­ eralogy and many other fields. For this reason, the same stereological method has often been invented independently in different fields. It was first in 1962 that the International Society for Stereology was founded and a forum was created for mutual exchange among scientists with interests in stereology. The society is responsible for organizing international congresses (every fourth year) and publishes Journal of Microscopy (together with The Royal Microscopical Society) and Acta Stereologica. Since 1994, Advances in Applied Probability has included the subsection Stochas­ tic Geometry and Statistical Applications (SGSA) where papers on more theoretical aspects of stereology have been published. Workshops on stochastic geometry, stere­ ology and image analysis have been held every second year since 1981. Abstracts from the workshop in 1995 can be found in Jensen (1996). A solid theoretical framework for stereology has been laid down by Roger Miles and Pamela Davy in a series of papers from the seventies and onwards, cf. Davy & Miles (1977), Miles & Davy (1976, 1977) and Miles (1978a, b). See also Coleman (1979). These papers have had a profound impact on the further development of stereological methods. Another series of interesting papers from the eighties has been written by Luis Cruz-Orive and coauthors, cf. Cruz-Orive & Myking (1981), Cruz-Orive & Weibel (1981) and Cruz-Orive (1982, 1983a). This series of papers is more close to biological applications. Analyses of stereological data are included. Reviews of stereology can be found in Weil (1983), Jensen et al. (1985), CruzOrive (1987b), Stoyan (1990) and Baddeley (1993). The monographs Weibel (1979, 1980) and the more recent papers by Gundersen et al. (1988a, b) present stereology to biologists and other users. Baddeley (1993) has recently emphasized that modern stereology can be regarded as spatial sampling theory. In this chapter, I have introduced stereology from this viewpoint and described a stereological method as a geometric sampling design and an associated Horvitz-Thompson estimator. As also noted by Baddeley (1993), this viewpoint does not lead to refined methods of estimating the variance of the estimators. The main problem is that the second-order sampling probabilities may well be zero. As can be seen from the exercises at the end of this chapter, the variance depends in a complicated way on the spatial arrangement of the structure in question. The disector design discussed in Section 1.2 is a sampling design which uses local 3-dimensional information and represents a major breakthrough. It does not only solve the problem of estimating particle number but provide a statistical satisfactory solution to the classical sphere size problem, stated in Wicksell (1925), cf. e.g. Jensen & Gundersen (1987b). A nice review of earlier solutions to the sphere size problem

34

1. INTRODUCTION TO STEREOLOGY

has been written by Cruz-Orive (1983b), see also Jensen (1984) and Coleman (1989). Another very useful design closely related to the disector design is the fractionator design, cf. Gundersen (1986). The variance of the estimator of volume based on a spatial point grid design, cf. Section 1.3, can be approximated, using the theory of regionalized variables, cf. Matheron (1965, 71). Recently, these methods have been reconsidered and further developed, cf. Souchet (1995), Kiéu (1997) and Kiéu et al. (1997). See also Chapter 7. Practical considerations concerning the efficiency of stereological procedures can be found in Gundersen & Østerby (1981). An interesting biological example of volume estimation using magnetic resonance imaging is described in Roberts et al. (1993). The invention of vertical designs by Baddeley (1983, 1984) has had a major importance in practice. The sin-weighted lines illustrated in Figure 1.14 can be replaced by a cycloid test system, cf. Baddeley et al. (1986). Vertical designs have also been developed in local stereology. A dual model-based approach has earlier been studied, cf. Mathieu et al. (1983) and Cruz-Orive et al. (1985), using methods from directional statistics. A vertical design for estimating curve length has been developed by Gokhale (1990). See also Gokhale (1992, 1993) and Batra et al. (1995).

Chapter 2

The coarea formula The coarea formula is a transformation formula for integrals with respect to Hausdorff measures. In this chapter we will present this formula and derive a number of applications which are important for the development of local stereological methods.

2.1 Hausdorff measures Let ujd = 7T2^r(l + \d)~l be the d—dimensional Lebesgue measure of the ball in R with radius 1. In particular, LOQ — 1, u\ — 2, U2 = TT and ^3 = |7r. Let A C Rn. For any e > 0 we then let d

\dn(A,e) = i n f { £ w d ( ^ ^ )

d

: A C U ^ d i a m A , < e,Vj},

(2.1)

3

where diamAj = sup{||x — x'\\ : x £ Aj,xf € Aj}. The number of sets in a covering {Aj} of A may be finite or infinite. The sum in the definition of A^(A, e) may accordingly be finite or infinite. Note that 6i<e2^\dn(A,61)>\dn(A,e2).

(2.2)

This is easy to see because the set of coverings appearing in (2.1) is decreasing when e decreases. We now define the d—dimensional Hausdorff measure A^ in Rn by \dn{A)=\\m\dn(A,e). ejO

Note that because of (2.2), we also have A ^ 4 ) = supA^4,e). €>0

35

36

2. THE COAREA FORMULA

As mentioned earlier, the Hausdorff measures have been introduced to measure d—dimensional volume in Rn. In particular, A° is counting measure, cf. Exercise 2.1, and AJJ = An, the Lebesgue measure in Rn, cf. Hoffmann-Jørgensen (1995b, p. 6-7). Note that

ua{

-) = 2s

3

diam

^ i ' for d = 1

J

and, in case A is a spatial curve, the sum of the diameters of the covering sets is converging, when e tends to zero, to what is usually called the length of the curve, cf. Figure 2.1 and Hoffmann-Jørgensen (1995b, p. 6-7). In Exercise 2.2, it is shown that \ln([a,b)) =

\\a-bla,beRn.

Figure 2.1. The sum of the diameters of the circles converges to the length of the curve when e goes to zero.

Furthermore, as we will show later in this chapter, if A is a subset of a d-dimensional affine subspace of Rn, then X^(A) is the ^-dimensional Lebesgue measure defined on this affine subspace of the set A, cf. Figure 2.2. The Hausdorff measures also satisfy the following relation for a G Ä\{0}, a e Rn and A C Rn \dn(aA + a) = cf. Exercise 2.3.

\a\d\dn(A),

' 2.2 ;,THB "CÖÄRIA 'TCRftRJLA

37

Figure 2.2. If A is a subset of a ^-dimensional affine subspace, then Xrl,iA) is the J-dimensional Lebesgue measure defined on this affine subspace of the set A.

2.2 The coarea formula The coarea formula IEYOIYCS transformations between differentiable manifolds. Intuitively» X C Rn is a d—dimensional manifold if X locally looks like R*. Formally, X is said to be a d—dimensional manifold, if there exists for all x £ X a neighbourhood U of x and a homeomorphism ^ : f." — R
Figure 2.3. A ^-dimensional differentiable manifold in Rn.

38

2. THE COAREA FORMULA

The manifold X is said to be a differentiable manifold if, intuitively speaking, X is smooth. Formally, we require that if (U,(p) and (V,^) are two coordinate neighbourhoods of, say» x and y, respectively» such that U n V ^ 0, thee the two mappings (po^r1 and tp o ip""1 are diffeomorphisms, cf. e.g. Boothby (1975). The unit sphere Sn^1 is an (n — t)—dimensional differentiable manifold. If A" is a (/-dimensional differentiable manifold» then there exists for each x £ X a »/-dimensional linear suhspace of /?", denoted Tan|X,x] and called the tangent space of A* at .r £ A". Let H\p) be a coordinate neighbourhood of x. Then, the tangent space can be determined as Tan [A*. ./•; — s]>an{ai,..., atj} where «,• £ It" are given by

/— 1 (L and (^ _ 1 )j is the j ' t h coordinate function of s~[.j ----- 1 //. For a '/--dimensional differentiable manifold A. we denote A.'.'AA the length. surface area or volume of X, depending on whether d --- L // — 1 or //. Below, we present the version of the coarea formula appropriate tor our purposes. We use the notation dxd as a short notation for X^(dx). Furthermore, vectors are represented as column vectors and Är is the transposed matrix of A. fn Figure 2.4, the situation in Theorem 2.1 is illustrated.

Figure 2,4. In Theorem 2.1» the mapping / is defined on an open subset D of Rn, Here» we have for simplicity chosen I) ■■■- />'".

2.2 THE COAREA FORMULA

39

Theorem 2.1. Let D C Rn be an open set and let y C i?fc be a differentiable manifold of dimension p. Let / : Z) —» y be a differentiable mapping. Furthermore, let I C D be a differentiable manifold of dimension d, where d > p. Then, there exists a function Jf(-;X) : X —> i?+ U {0}, called the Jacobian, such that for any non-negative measurable function # defined on X we have y g{x)Jf(x- X)dxd = J X

Let Df(x)

g(x)dxd-?dyF.

j

Y

Xnf-i(y)

be the k x n matrix of partial derivatives

and let keiDf(x) = {y e Rn : Df(x)(y) Jf(x]X) > 0 iff the dimension of

= O} be the kernel of Df(x).

Tan[X, x] n (kerD/(x) n Tan[X, x])"1

Then, (2.3)

is equal to p. If the subspace (2.3) has dimension p, then the Jacobian can be calculated as Jf(x;X)

y/det{Df(x;X)Df{x;X)T},

=

where Df(x; X) is the following p x k matrix r Tn

I1] Df(x;X)=

•

\Df(x)T

\eT and e i , . . . , ep is an orthonormal basis of the subspace (2.3). In particular, if g{x) = h(f(x)) J h(f(x))Jf(x; x

we get the following transformation result

X)dxd = y h(y)Xdn-P(X n Z " 1 ^ ) ) ^ y

Example 2.2. Let £>

=

□

JR

2

X = (0,TT) x (0,2TT) C L>

y = 5 2 c R3

(2.4)

40

2. THE COAREA FORMULA

and consider the mapping / : R2 - S2 (#, (/)) —> (sin 9 cos 0, sin 9 sin 0, cos 9) Here, n = 2, fc = 3, d = 2 and p = 2. The matrix af partial derivatives can be written as [cos 9 cos 0 — sin 9 sin ø 1 Df(0,4>)= cosØsinø sin 9 cos <j) .

[ -sin(9

0

J

The rank of Df(0,4>) is 2 and keiDf(9, ) = {O}. Therefore, Tan[X, (0, 0)] n (ker£>/((0, ø)) n Tan[X, (0, 0)]) X = Æ2n

({OjnR2)1-

= R2. The orthonormal basis becomes

ei=

G)' e 2 = (0

and Z?/((0,0); X) = I2Df{0,
Df(6,4>f.

The Jacobian can now be calculated as Jf((9, ); X) = yJdet{Df(0,4>)TDf(e,<j>)}

=

V d 6 t { [j sin%]>

= sinØ. Using the coarea formula with a function of the form g{9,4>) = h(f(0, )), we get the following transformation result, cf. (2.4), / / /i(sm 0 cos , sin 0 sin 0, cos 0) sin ØdødØ = oo s2

h(y)dy2.

In particular, we get the transformation result (1.3) of Chapter 1 by restricting attention to functions vanishing on S2\S+. □ Example 2.3. Let D = X = R2\{0} Y = S1 CR2

C R2

2.2 THE COAREA FORMULA

41

and consider the mapping / : R2\{0} x->x/\\x\\.

S1

Note that the mapping / has a simple geometric interpretation as the mapping associating to a point x G R2\{0} the direction of the ray starting at O and passing through x, cf. Figure 2.5.

Figure 2.5. The mapping / from Example 2.3. The matrix of partial derivatives of / is

Df(x) = ^ \

^

IWJ

1

~~ iN 7 r s p a n ^ J ~' where ix

s x±

denotes the orthogonal projection onto spanjx} . Accordingly, kevDf(x) — span{x},

and Tan[X, x] n (keiDf(x) n Tan[X, x]r = R2 n (span{a:} O Ä 2 )- 1 =span{x} has dimension 1 for all x e R2\{0}. described in Theorem 2.1.

We can therefore calculate the Jacobian as

42

2. THE COAREA FORMULA Let e = ||x|| _ 1 (-X2,xi) T be a unit vector spanning (keiDf(x)) Df{x;R2\{0})

= eTDf(xf

= (Df(x)(e)f

. It follows that

= \\x\\~leT

and Jf(x;R2\{0})

= W - y d e t { e T e } = »iir1.

We then have the following transformation result, cf. Theorem 2.1, j g(x)\\x\\-ldx2

= f

2

1

R

S

f

g{x)dx1duj\

f-i(u)

At the left-hand side, we have replaced R2\{0} with R2. Note that f~l(u) is the ray starting in O with direction u.

D

2.3 The special case d = n In this section, we present three corollaries of Theorem 2.1 which are concerned with the case where the dimension of the set X appearing in Theorem 2.1 is n. Corollary 2.4. Let the situation be as in Theorem 2.1. Assume in addition that d = n, p = k and that the rank of Df(x) is k. Then, the Jacobian can be calculated as

Jf(x;X) =

^det{Df(x)Df(xf}.

Proof. Since d > p we have under the assumptions of the corollary that n > k. Furthermore, if fi denotes the z'th coordinate function of / , i — 1 , . . . , k, we get since Tan[X, x] — Rn Tan[X, x] n (keiDf(x)

n Tan[X, x})1-

= (kerD/(x)) ± = span{D/i(x),...,

Dfk(x)}.

Since Df(x) has full rank k, the vectors Dfi(x),..., Dfk(x) are linearly independent and the subspace spanned by these vectors has dimension p = k. We can therefore use the procedure of Theorem 2.1 to calculate the Jacobian. Let e i , . . . , e& be an orthonormal basis of span{D/i ( £ ) , . . . , £ / / , ( > ) } .

2.3 A SPECIAL CASE

43

We then have r T-]

\l I Df(x) = A\ • ,

where A is a A; x fc invertible matrix. Therefore, £>/(*; X ) D / 0 r ; X ) T r T ~i

r T i

1

p1

r1 =

[ei,...,ejfe]i4T>l

"

Lei J

•

[ei,...,efc]

14 J

It follows that

Jf{x;X) =

^ået{ATA}

=

jdet{AAT}

[eil = [detM

•

[ei,...,e fe ]j4 T }] 1 / 2

lek J

=

y/det{Df(x)Df(xf}. D

Corollary 2.5. Let the situation be as in Theorem 2.1. Assume in addition that d = n, p = n and that the rank of Df(x) is n. Then, Jf(x; X) = V / det{D/(x) T D/(x)}. Proof. Since p < k we have under the assumptions of the corollary that n < k. Since Df(x) is of rank n, the columns of Df{x) are n linearly independent vectors in Rk. Accordingly, keiDf(x)

= {O} =» (kerD/^))" 1 = i T

44

2. THE COAREA FORMULA

Therefore, if In is the n x n identity matrix,

= Df(x)T

Df(x;X) = InDf(xf and the result follows immediately.

□

Note that Example 2.2 is of the type described in Corollary 2.5 above. Corollary 2.6. Let the situation be as in Theorem 2.1. Assume in addition that d = p = k = n and that Df{x) has rank k = n. Then,

Jf(x]X)

= \det{Df(x)}\.

□ Corollary 2.6 gives the usual way of calculating the Jacobian.

2.4 Hausdorff measures on affine subspaces As mentioned in Section 2.1, if A is a subset of a d—dimensional affine subspace of Rn, then X^(A) is the d—dimensional Lebesgue measure defined on this affine subspace of the set A. In the proposition below, this result is proved, using the coarea formula, see also Figure 2.2. Proposition 2.7. Let Fj be a d—dimensional affine subspace of Rn Fd = span{ai,...,ad} + a, where a\,..., a^ is a set of d orthonormal vectors in Rn. A' C Zi^ be the unique set such that

Let A C F^ and let

d A =

{^XiUi

+ a : (xi,...,xd)

€ A'}.

2= 1

Then, A^(A) = \d(Af), where Arf is ordinary Lebesgue measure in Rd. Proof. Let / be the 1-1 mapping

f-.&^Fd d

(xi,...,xd)

- > ^Xidi 2=1

+ a.

2.5 POLAR DECOMPOSITION OF LEBESGUE MEASURE. Note that A — f(Af).

45

The matrix of partial derivatives of / is DfUi

rf/) = [ a i , . . . , a j .

In particular, Df(X{... .. xj) does not depend on ( x i , . . . , rr^). We can now use the coarea formula with D = X = Rd and Y = Fj. According to Corollary 2,5, •//((■n

xd); Rd) = \/dex{Df(jL^...<xd)TDf{x1,...,xd)}

= 1.

Since / is 1—1, the version of the coarea formula given in (2.4) now becomes

I h(f{x))dxd = I h(y)dyd. By using h(y) = l{y e A}, we find Xd(Af) = Xi(A).

D

2.5 Polar decomposition of Lebesgne measure This section concerns the generalisation to Rn of Example 2.3. This result will be important in the coming chapters» especially in Chapter 4. For later use, we formulate the result in Proposition 2.8 below. The mapping / of the proposition is illustrated in Figure 2.6.

Figure 2.6.- The mapping / from Proposition 2.8. Proposition 2.8. Let f : Rn\{0}

-

.r^x/Wxl

S"" 1

46

2. THE COAREA FORMULA

Then, for any non-negative function g on Rn, f g{x)\\x\f{n~1)dxn

=

f

f

g{x)dx1dujn-\

Proof. We want to find the Jacobian of the mapping / . Note that although d — n we cannot use Corollary 2.4 or 2.5. The matrix of partial derivatives of / is given by

Df(x)(y) = \\x\f\y

{

-

-^x)

=M ^ ^ ^ y

G R\

cf. Exercise 2.8, and therefore, kerD/(x) = span{x}. Let e i , . . . , en-i be an orthonormal basis of span{x} . Then, r

Df(x;Rn\{0}) =

T

1

■

r

D/(a:)T = W - 1

T ~\

•

T T len-l J len-l J = [Df(x)(e1),...,Df(x)(en-1)]T

r ei l

= INI _ 1

• T

len-l J and r

Jf(x; Rn\{0}) = Wxr^-^deti

T

i

■ [elt..., e^]})1'2 T len-l J

=

M^^. D

2.6 Translative decompositions of Hausdorff measures It is well-known that ordinary Lebesgue measure in Rn is a product measure of n copies of Lebesgue measure in R. Furthermore, if Lq is a
Xn(X) = J \UXn(Lq + y))dyn-
2.6 IRANSLATIVE DECOMPOSITIONS OF HAUSDORFF MEASURES

47

This result is no longer true if Lebesgue measures are substituted by Hausdorff measures. It turns out that the local geometry of X relative to Lq becomes important» viz. the 'orientation* of the d—dimensional linear subspaces Tan[X,a:],a: € X, relative to Lq. The definition below turns out to be useful in this connection. Note that for any d—dimensional subspace L& and q—dimensional subspace Lqy we have dim(Lrf fl (Lq H Ld)±)
Figure 2.7. Translative decomposition of Lebesgue measure.

Definition 2.9. Let 0 < d. q < n be noe-eegative integers» satisfying d + q > n. Let Lj and Lt} be d— and 7—dimensional linear subspaces of Rn. If q < n and dim(I rf n (Lq n Ld)L)=n - f,

(2.5)

we let .4 = [7r /j: L«i,...,7r£±a n _ 9 ], where a\,...»an^q is an orthoeormal basis of X L(i n (Lq n L(/) and TLL is the orthogonal projection onto L~. Then» we deine ( 1 G(Llh Lq) = < d e t { ^ r . l } 1 ; ' 2 v0

if q - n if q < n and (2.5) is Milled otherwise.

D

48

2. THE COAREA FORMULA

The quantity G(Li, Lq) has a simple geometric interpretation for n = 2 and 3» cf. Exercises 2.10 and 2.14 and Figure 2.S. In the ex-ample presented in Figure 2.8, X is a smooth spatial curve and G(Tan[X,x],L q ) is equal to sin«, where a is the angle between the tangent line Tan[X,s] and the plane Lq.

Figure 2.8. The G-factor G(Ta,n[X,x],Lq) is equal to sin«.

Using Definition 2.9 we ha¥e the following translative decomposition result for Haesdorif measures. Proposition 2.10. Let X C Rn be a ^/-dimensional differentiable manifold and let 0 < rf, n. Then, for any nonnegative measurable function g defined on X, I f/(x)G(Taa[X,xlLq)d.vd •v

= /

/

g(x)(Lr(^ll+(idf^.

L$Xf\(Lq+y)

Proof. Let f — T/±. Then, we have Df{.r) = wT± and kwDfi.r) is now a direct application of the coarea formula.

— L(l. The result O

2.7 A TRANSFORMATION RESULT FOR SURFACE AREA MEASURE

49

Note that for d = n, Tan[X, x] = Rn, and a i , . . . , an-q is an orthonormal basis of L^. Therefore, for d = n, r

G(Tm[X,x},Lq)=det{\

T

i

■

[ffll)... ,an-q]}1f2 = 1

T L^n—gJ

and the transformation formula of Proposition 2.10 reduces to the usual translative decomposition of Lebesgue measure. Note also that if g = 1, then the result of Proposition 2.10 reduces to / G(Tan[X, x],Lq)dxd

= f A ^ n + « ( X n (Lq +

x

y))dyn~^

Li-

which is the generalization of translative decomposition of Lebesgue measure.

2.7 A transformation result for surface area measure In this section, we present a useful transformation result for A™-1. Proposition 2.11. Let e be a unit vector in Rn and let g be any non-negative measurable function defined on R. Then, l

J

5«e,u;)

2

1

1

1

) ^ " - = 2( 7 rp("- >r(i(n - l ) ) " Jg(y)y-*{l

5— i

-

yf^^dy.

0

Proof. We will show that the result is a consequence of the coarea formula. Let / : Rn -> R x —> (e,x) . Note that / has a simple geometric interpretation, viz. f(x)

= IKspan{eH|2-

The partial derivatives of the mapping / is Df(x) =

2(e,x)eT.

For (e,x) / 0, we have Df(x)(x)

= O & x e spanfe}- 1 .

50

2. THE COAREA FORMULA Let us now use the coarea formula with D = Rn, X = 5 n _ 1 and Y = R. Since Ar1({^e5-1:|(cJa;>|G{0,l}})=0

it suffices to find Jf(u; 5 n _ 1 ) for those v, satisfying |(e,u;)| £ {0,1}. Let u E S n _ 1 be chosen such that |(e,o;)| ^ {0,1}- Since (e,uj) ^ 0, keiDf(cu) = spanje} . Furthermore, according to Exercise 2.5, Tan[5 n_1 ,o;] = span{o;} . Therefore, T a n [ S n - \ u ; ] n (kerD/(a;) n T a n f S ^ 1 , ^ ] ) 1 = span{u;} D [span{e} D span{u;} J-1. Since |(e,u;)| / 1, this subspace has dimension 1, cf. Exercise 2.11. Let e be a unit vector spanning the subspace, cf. Figure 2.9. Then, the Jacobian of / becomes Jf(w;Sn-1)

= \Df(w)(e)\ = 2-\(e,u,)\.\{e,e)\ = 2- \(e,u>)\- ||7rspan{e-}e|| = 2 • |<e,w)| • ||7rspan{w}xe - % a n { e } x n s p a n { w } x e | | = 2-|(e)W>|-|KpanMJ.c|| = 2.|(e,W>|-V/l-(elW>2.

From (2.4) we now get oo

1

J hifiwVJffaS"- )*!»"-^

J

S71-1

h(y)X^2(Sn-1nr1(y))dy

-oo 1

jh(y)Xr2{Sn-1nf-l(y))dy,

= 0

i.e.

h{{e^)2)-2-\{e^)\^l-{e^)2dujn-1

J 5»-i 1

= Jh(y)\r2(Sn~1nr1(y))dy. 0

2.7 A TRANSFORMATION RESULT FOR SURFACE AREA MEASURE

51

Figure 2.9. Illustration I for Proposition 2.11.

The quantity AJJ 2 (Sn 1 O / 1(y)) can now be e¥aluated as follows. First notice that for any y £ [0,1] we have S"- 1 n rl(ij)

= {- £ Sn~l : (e.w) = ± f 1 / 2 } *

The set Sn^1 n f~l{y) can be regarded as the union of two (n — 2)—dimensional spheres in span{e} , each of radius v /l — //, cf. Figure 2.10. Therefore»

A;;-2 (S"- 1 n rlUA) = 2 • A;;:2 (yr—^s»- 2 ) =

2

.(l_y)(-^An-2(sn-2)_

The result of the proposition now follows easily by setting

9(y) = %) * -\fy * V 1 -#•

52

2. THE COARBA FORMULA

Figure 2.10. Illustration II for Proposition 2.11.

2.8 Simplices Recall» that the convex hull of p -r 1 points in Rn is defined as conv{a:o,.ri... .,a>} = {XQXQ + Ai.ri + • • • + XpXplXQ > 0, • • •, Xp > 0, A0 + • • ■ 4- XP = 1 }■. We will denote this set a /^-simplex, if p < n. The points ar0, ari,.... xp are called the ¥eitices of the simplex. Note that a 0-, 1-, 2 - and 3-simplex is a point, a line-segment» a triangle and a tetrahedron, respectively» cf. Figure 2.11. In the coming chapters, we are interested in the p-dimensional Hausdorff measure (volume) of such simplices. More precisely, for x{ xp e /?", we consider V p (xi, ...,xp)= Note that Vi(.r) = ||J;||.

p!A£(conv{0. xx

xp}).

2.8 SIMPLICES

53

Figure 2.11. /7-simplices for p=0,l,2,3. We can derive a very simple formula for V p ( x i , . . . ,xp), using Proposition 2.7. ,xp G i? n ,

Proposition 2.12. For x\,...

yJdet{ATA},

Vp(xi,...,xp) = where A is the n x p matrix A = [x\,...,

Proof. The p—simplex conv{<9, x\,...,

Xpj.

xp} is a subset of the linear subspace

Lp = span{xi,..., x p }. Let us first assume that x\,..., xv are linearly independent. Let a\,..., ap be an orthonormal basis of Lp and let i? be an invertible p x p matrix such that r T ~\

r T ~\

\xl

a

•

= B

•

T

\JEp J

i

.

^

Lap J

Note that c o n v { 0 , x i , . . . ,xp} V

=

V

{^AÄ:A,>0,^A,<1} 2= 1

2=1

- { ^ ( £ T A ) Ä : A, > 0, J2 Xi < !}• 2=1

2=1

(2.6)

54

2. THE COAREA FORMULA

If we let p

X = {(Ai,..., Xp) e Rp : A2 > 0, J2 A* < !} i=l

we then get, using Proposition 2.7, A£(conv{0,a;i,...,:rp}) = AP({£TA:A2>0,^A,<1}) ( }

= |det{£}|A p (X),

where we at (*) have used Corollary 2.6. Because of (2.6),

yJdet{ATA} = |det{S}|. It remains to show that XP(X) = 1/pl Note that X is a subset of the unit cube in Rn. We find 1

Xp(X)=

l-Ai

1-Ai

Ap_i

f f '"

I

0

0

0 1

d\p--.d\2d\1

Ap_i

Ai

= / / • • • / dXp- -•d\2d\\ 0

0 1

0 1

1

= / / • • • / l{Ai > A2 > • • • > Xp}d\v ■ • • d\2d\1 oo o l

l

= — / 0

l

'" 0

d\p • • • dX2d\i 0

_ 1 The case where x\,...,

xp are linearly dependent is left to the reader.

□

Note that it follows from Proposition 2.12 that V p ( x i , . . . ,xp) = 0 in case xi,..., xp are linearly dependent. Also note that in case x\,..., xp are orthonormal vectors V p ( x i , . . . , xp) = 1.

2.8 SIMPLICES

55

Let us reconsider Definition 2.9. Under the assumption that the subspace (2.5) has dimension n — q > 0, the G—factors have a nice geometric interpretation. Thus, if A — [7r L ±ai,..., 7rL±an-q], as in Definition 2.9, we find G(Ld, Lq) = V n _ 9 (7r L J.ai,..., 7rL±an-q). The proposition below gives a useful reduction result for V p . Proposition 2.13. Let x\,...,xp e Rn and let xo E Rn satisfy xo e span{x 2 ,... ,xp}. Then, V p (x 0 + x i , x 2 , . ..,xp) = Vp(xi,X2,. ■ .,xp). Proof. Let A, B and C be the n x p matrices defined by A = [xo -\-Xi,X2,.

• . ,X p ]

5 = [x 0 ,x 2 ,...,x p ] C = [xi,x2,...,xp]. Then, for all i, j = 1 , . . . ,p T

+ (ATC)tl

(ATAV._j(A

B)a (A A)„ - | {ATBh

= {ATch

if j = 1 .f. = 2

^

Therefore, Vp(x 0 + x i , x 2 , . . . , x p ) = det{A T A} = det{A T £} + det{A T C}. For similar reasons det{ATC}

= det{BTC}

+ det{C T C}.

The columns and the rows of AT B and BTC, respectively, are linearly dependent and the corresponding determinants are therefore 0. We find Vl(xo + =

xi,X2,...,xp)

det{CTC}

= Vp(xi,x2,...,xp). .

D

In the proposition below, a decomposition result for two orthogonal sets of vectors is given.

56

2. THE COAREA FORMULA

Proposition 2.14. Let { x i , . . . , xr} and {x r +i, ■ • •, xr+q} be two orthogonal sets of vectors in Rn, i.e. (xi,Xj) = 0,2 = l , . . . , r , j = r + l , . . . , r + g. Suppose that r + q < n. Then, V r + g ( : r i , . . . , xr, x r + i , . . . , a; r+9 ) = V r ( x i , . . . , xr)Vq(xr+i,...,

x r + 9 ).

Proof. Let Ar = [ x i , . . . , xr\ Aq =

[Xr-}_1, . . . ,

Xr-\-q\.

Then, V r _ | _ ^ ( X l , . . . , Xr, X r - f l , . . . , X r -t-gJ

= V^(xi,..., xr)V^(xr+i,..., xr+9).

□ Finally, we have the duality result below, the proof of which is left as an exercise. Proposition 2.15. Let a i , . . . ,a n be an orthonormal basis of Rn and let Lq be a q—dimensional linear subspace of Rn. Then, V q , ( 7 T £ q a i , . . . , 7TLqaq)

= Vn_g(7TLJ.aq+l, . . . ,

7TL±an).

a 2.9 Exercises Exercise 2.1. Show that A° is counting measure, i.e. A°(A) = |A| for any i C i ? n . Exercise 2.2. Let a,b e Rn and let [a, 6] be the line-segment between a and 6, i.e. [a, 6] = {cm + (1 - a)b : 0 < a < 1}. Show that A*([a,&]). = ||a - 6||.

2.9 EXERCISES Hint. Start by showing that for any covering {Aj}

57 of [a, b], we have

Ik _ &I < y^diam(Aj). i Use this result to show that A* ([a, 6], e) = ||a - &|| for all e > 0. Exercise 2.3. Show that for any a G Ä \ { 0 } , a G i ? n and A C Rn, we have

A^(aA + a ) H Ä ( A ) . Exercise 2.4. Show that £ 2 is a 2-dimensional manifold, i.e. show that for any UJQ G S2 there exist a neighbourhood U of u;o and a homeomorphism (p : U ^ R2 such that <£?([/) is an open subset of R2. Hint. Recall that a neighbourhood is a set of the form U — B^(UJQ, r)C\S2, where £3(^0 , 0 is the open ball in R3 with centre a;o and radius r. Note that we can always choose a coordinate system in R3 such that coo G 5 2 \ { ^ G Æ 3 : ui > 0,u2 = 0 } . Show that we can then use ip = / _ 1 , where / : (0, TT) x (0, 2TT) -► 5 2 \ { O ; G Æ3 : CJI > 0, u2

=

0}

(0, (/)) —> (sin 0 cos 0, sin 0 sin 0, cos 0). Exercise 2.5. In this exercise, we show that T a n [ 5 n - 1 , c i / ] = span{o;} . Let us consider the transformation / from 6 = {(6>i,..., en-i)T

G Ä n _ 1 : 0 < Si < TT, i = 1 , . . . , n - 2, 0 < 0 n _i < 2TT}

into Æ n given by f(0) = (/i(ö), . . . , / n ( 0 ) ) r , 0 = (öi, . . . , 0n-lf

G 6,

where /i(0)

-cos^i

/2(0)

= sin0icos02

/3 (0)

— sin ^i sin #2 cos Ø3

/ n _ i (0) = sin 0i sin 02 • • • sin 0 n _2 cos 0 n - i /n(0)

= sin 0i sin 02 • • • sin 0n-2 sin 0 n _i

58

2. THE COAREA FORMULA

1. Show that / is 1-1 and that f{0) eSn~\

forallØeG.

2. Since / is differentiable the tangent space can be calculated as Tan[5 n _ 1 ,cj] = span{ai,..., a n _ i } , where Oi = ( ^ ( / - 1 H ) , . . . , ^ ( / - 1 H ) ) r , < = l , - , r » - l Find ai, i = 1 , . . . , n — 1, and show that a\ _L UJ for all i. Exercise 2.6. Show, using the coarea formula, that / h(cosO,sm0)d0 = / h{uj)dujl. sl

o Hint. Use (2.4). Exercise 2.7.

1. Let h be any non-negative measurable function defined on R+ U {0}. Show that oo

n

I h(\\x\\)dx

1

1

= XT (A"" ) j

Rn

h(r)rn-ldr.

0

Hint. Use the coarea formula with X = D = Rn\{0}, n

/ : R \{0}

-

Y = R+ and

R+

x -* \\x\\. It suffices to use Corollary 2.4. 2. Show that A™- 1 ^ 71-1 ) = 2(7r)^ n r(in)~ 1 . Hint. Let

h(r) =

-^—e-^l\

(27r) n / 2

and use that with this choice of h [ h{\\x\\)dxn = l. 3. Show that Xn(Bn) = 7T2nr(l + \n)~l.

Hint. Use the results in questions 1 and 2.

2.9 EXERCISES

59

Exercise 2.8. Let / be the mapping defined in Proposition 2.8. Show that

Df(x)(y) =

\\x\\-\y-^x).

\m\ Hint. Show that

*kix) = { M*1 - é) ij

l=J

= l,...,n.

Exercise 2.9. Let Ld and Lq be d— and q—dimensional linear subspaces of Rn, respectively. Show that dim(L d H (Lq n Ld)L)


Hint. The Grassmann dimension formula gives dim (Ld + Lq) = dim (Ld) + dim (Lq) - dim (Ld n Lg) and furthermore dim (Ld + Lq) < n. Exercise 2.10. Suppose that n — 2 and d = q = 1. Let Ld and L 9 be two lines in R2 through O, satisfying Ld n Lq = {O}. Show under this condition that G(Ld,Lq)

= sina,

where a is the angle between Ld and Lq. Exercise 2.11. This exercise elaborates on one of the details of Proposition 2.11. The notation below is as in Proposition 2.11. Show that if |(e,u;)| ^ 1 then the dimension of the subspace spanjcj}

n [span{e} Pi span{cj} J-1

is 1. Hint. We know from Exercise 2.9 that the dimension is either 0 or 1. Furthermore, since both e and UJ are unit vectors, we have |(e, o;)| = 1 if and only if u = ±e. Exercise 2.12. Let r and q be positive integers such that r + q < n. Let x i , . . . , xr, x r + i , . . . , xr+q G Rn and Lr = s p a n j ^ i , . . . , xr}.

60

2. THE COAREA FORMULA Show that = V r ( x i , . . . , x r )V 9 (7r L j.x r +i, • • •, 7rL±a:r+g). Hint. Use Propositions 2.13 and 2.14.

Exercise 2.13. Let the situation be as in Proposition 2.15. Let &i,...,& n be an orthonormal basis of Rn, such that Lq = span{6i,..., 6 9 }, h\ = span{& 9 +i,..., 6 n }. Let i?i and .82 be the n x q and n x (n — q) matrices defined by, respectively, B\ = [61,..., 6g], £ 2 = [bq+i, • • •, &n]Note that 7r£g = -Bi^Bf and 7rL± = B^B^. Correspondingly, we let M = [ai,- • -,a 9 ], M = K + i , . • -,fln]1. Show, using Proposition 2.12, that V g (7T Lq ai,...,7r Lg a 9 ) = |det{5^Ai}| V n - g ( 7 T L j . a 9 + i , . . . ,7TLJ.an) = | d e t { Æ f # 2 } | .

2. Let [A1IÆ2] be the matrix n x n [a\,..., ag, frg+i>..., &n]. Show that |det{[Ai|B 2 ]}| = |det{|^ 1 T l}det{[>l 1 |B2]}| = |det{B?Ai}| L^2 J and

uTi |det{[>li|B2]}| = |det{

1 }det{[^l 1 |^ 2 ]}| = | d e t { ^ S 2 } | . LA2 J Combining questions 1 and 2, we have proved Proposition 2.15. Exercise 2.14. This exercise concerns the geometric interpretation for n=3 of the G—factor, given in Definition 2.9, see also Exercise 2.10 and Figure 2.8. There are three non-trivial cases, corresponding to (d, q) — (1,2), (2,1) and (2, 2). Show under the assumption (2.5) that for any of these cases G(Ld,Lq)

= since,

where a is the angle between Lj and Lq. Hint. For (d,q) = (2,1), Proposition 2.15 is useful.

2.10 BIBLIOGRAPHICAL NOTES

61

2.10 Bibliographical notes The Hausdorff measures have been introduced with the purpose of measuring d—dimensional volume in Rn, cf. Hausdorff (1919). In Section 2.1, some of their important properties have briefly been mentioned. A more comprehensive treatment can be found in Rogers (1970) and Hoffmann-Jørgensen (1995b, Chapter 8). The latter reference also contains a rich collection of statistical applications of the coarea formula. The version of the coarea formula presented in Section 2.2 is less technical than some of the earlier versions which can be found in e.g. Federer (1969), Zähle (1982), Baddeley (1983), Zähle (1990), Jensen & Kiéu (1992b) and Kieu (1992). For simplicity, we have assumed that the sets involved in the transformations are differentiable manifolds. A version of the formula which is valid for Hausdorff rectifiable sets can be found in Federer (1969, Theorem 3.2.22) and other of the above mentioned references. Intensive use of multilinear algebra has also been avoided in the formulation of the coarea formula in Section 2.2. Instead, the emphasis is on the presentation of a procedure for calculating the Jacobian in non-standard situations which is as simple as possible. Those readers interested in multilinear algebra may consult Greub (1967). An account of geometric measure theory accessible to nonspecialists can be found in Simon (1983). The special case d — n presented in Section 2.3 has also been treated in Hoffmann-Jørgensen (1995b, Chapter 8). In this case, it suffices to calculate the matrix of partial derivatives of the mapping / . As emphasized by Baddeley (1983), the coarea formula plays an important role in classical stereology. As we will see in the coming chapters, the coarea formula plays an even more important role in the development of local stereological methods. From Section 2.4 and onwards, a collection of applications of the coarea formula is given. They have been chosen with the needs of the coming chapters in mind. The translative decompositions of Hausdorff measures given in Section 2.6 is not a new result, but can for instance also be found in Zähle (1982, 1.3.1. Corollary). In the latter paper, the geometric interpretation of the G-factors presented in Section 2.8 can also be found. The result of Proposition 2.11 can be interpreted as a distributional result for the projection onto a coordinate axis of a uniform random point, chosen on the unit sphere in Rn. As such, it is well-known, cf. e.g. Feller (1966) for the cases n = 2 and 3.

Chapter 3

Rotation invariant measures on £™ In this chapter, we will study more closely measures on the set of p—dimensional linear subspaces in Rn. In Chapter 1, we considered examples of sampling designs involving lines or planes with random orientations. In this chapter we introduce rotation invariant measures on the set of p—dimensional linear subspaces in Rn. For p = 1 or 2, these measures lead after proper normalization to the line and plane distributions considered in Chapter 1. A p—dimensional linear subspace will for brevity be called a p—subspace and will be denoted by Lp. Sometimes, we also write L™ for a p-subspace in Rn, if it is important to emphasize the dimension of the containing space. The set of p—subspaces of Rn will be denoted by ££.

3.1 Construction of rotation invariant measures on C™ The group of orthogonal transformations in Rn consists o f n x n real matrices 0(n) = {B : BBT = BTB = J n } . We consider the subgroup consisting of special orthogonal transformations SO(n) = {B E 0(n) : det(B) = 1}, which is called the group of rotations in Rn. A measure \i on £™ is said to be rotation invariant if / g(BLp)n(dLp)

= /

rn *~p

g(Lp)fi(dLp)

rn J~>P

for any B E SO(n) and any non-negative measurable function g on £™. Using the theory of invariant measures, it can be shown that there exists a rotation invariant 63

64

3. ROTATION INVARIANT MEASURES ON £ £

measure on £JJ which is unique up to multiplication by a positive constant, cf. Appendix. The following lemma is useful in the construction of these invariant measures. L e m m a 3.1. Let LP be a p—subspace of Rn and g a non-negative measurable function on Rn, Then, for any B e SO(n),

I

g{Buj)åJp'i =

g(uj)dujp^K

f

(3.1)

Proof. First note that for any B e SO(n), Sn^'Y C-. BLp can be identified with the unit sphere in HP and is therefore a [p - 1)-dimensional differentiable manifold in Rn. See also Figure 3.1.

Figure 3.1. Illustration for Lemma 3.1.

We will prove the result in the case where //(/) = 1{- € .4}..4 C fi". Then, (3.1) takes the form \I;rliBTA

n S"-1 n Lp) = ,yrl{A

n 5' 1 " 1 n BLp).

where BT A = {B1 ~ : „• e A}. Since BTA n Sn~l n Lp = BT(A r. BS"-1 r

= B (AnS"-

1

n BLP) nBI,,).

it suffices to show that for any B £ SO{n) and .4 C IV'. we have \$-l(BTA)

=

\r-l{A)

3.1 CONSTRUCTION OF ROTATION INVARIANT MEASURES

65

or equivalently, \rn-1(BA) = \Pn-\A). The important thing is now that the diameter of a set is not changed under rotations, viz. diam(jBA) = diam(A). Therefore, cf. (2.1), >C1(BA)=ma>C1(BAte) ejO

= UmA£- 1 (A,e)

D For p = n, we get in particular from Lemma 3.1 / g(Buj)dujn-1 = I g{uj)dujn-1. £n-l

5

(3.2)

n - l

We will now construct the rotation invariant measure on £™. We start by consid­ ering the cases p = 1 and 2 separately. In order to construct the rotation invariant measure on the set C[ of lines through O, consider the 2-1 mapping / : S n _ 1 -+ C[ UJ —>• spanju;}. Because of (3.2), a natural candidate to a rotation invariant measure on CJ[ is A™-1 lifted by the mapping / . Since / is 2-1, one usually chooses to lift ^A™_1 by / . The resulting measure /i™ on C[ is the unique measure satisfying the relation ^spanM)^"1

Jg(L1)fi(dL1) = ± j

for any non-negative measurable function g on C[. The measure pJ{ is indeed invariant under rotations. For B e SO(n), we get Jg(BL1)tf(dL1) = ± j

«KSspaiiM)^"-1

= l- j

g&imiBu)})^-1

{

Sn-1

=] \ J

^span^})^-1

£n-l

= J giLJfiidLi).

3. ROTATION MVARiANT MEASURES ON £^1

66

Let us next turn to the construction of the rotation invariant measure on £ | . An orthonormal basis u>i,u;2 for a 2™sttbspace can be generated by first choosing jj\ e 5 " " 1 and next UJ% € Sn^~l n span{wi} ± , cf. Figure 3.2.

Figure 3.2. The light grey plane is spanned by the orthonormal basis (jj'1^2' They are chosen on the unit sphere in the order mentioned.

Accordingly» we let jtif be the measure on £ | defined by

I g(L2)tä(dL2) ^2

= ~

f

I

^spanfa/i,^})^-2^!1"1-

•Sn-1S»-irlspaii{.j1}-L

The choice of the constant 1/4T is explained in Proposition 3.2 below» The measure juj is rotation invariant. To see this, let for Li € £"

f(Li) =

/

g(Lt 0 span{w 2 l)dw^ 2 ,

where 0 as usual indicates orthogonal sum. Then,forB e SO(m), we get» using that

3.1 CONSTRUCTION OF ROTATION INVARIANT MEASURES Bspan{o;i}

67

= span{i?u;i} ,

[ g{BL2)i4(dL2)

= — f

= T~

^(spaiiiB^i,^^})^- 2 ^- 1

f

f

^(span^cji,^})^- 2 ^- 1

I

5«-i

= ^

/ 5

/(spani^})^- 1

n-l

= J g(L2)ti$(dL2), where we at (*) have used Lemma 3.1. It should now be clear how to construct a rotation invariant measure on £™. The result is given in the proposition below. Proposition 3.2. Let an = A ^ ^ ^ " 1 ) = 2 ( 7 r ) H \ ± n ) - 1 and let c\n,p) —

. GvGv-\

• • • G\

Let /I™ be the measure on C™ defined by

J g(Lp)$(dLp) = —

x / G

11 %

1

/

S"- S ^ n s p a n l u ; ! } -

••• 1

n

S~

/ 1

^(span{cJi,...,o;p}) ±

nspa.n{ujll...,Ljp-i}

dou^-p'--duj^-2duj^-\

(3.3)

where g is non-negative measurable function on C™. Then, ji™ is a rotation invariant measure on C™ and the total measure is fJ>p(£p) = c(n,p). Proof. The proof of the rotation invariance is by induction in p. For p = 1, we have already proved the rotation invariance. Now, suppose that /z^_1 is rotation invariant.

3. ROTATION INVARIANT MEASURES ON Cnp

68

Let B e SO(n). According to Lemma 3.1, we have / S71-1 Dspan{a;i,...

=

Bu)p})diop~v

ø(span{£u;i,..., BUJP-\, ,wp-i}

/

ø(span{Æa;i,...,

Bujp-i,ujp})dujp~p.

5n-1nBspan{u;i,...,u;p_i}

Since JBspan{o;i,..., ^p-i}" 1 = span{So;i,..., BtJp-i}1,

we therefore have

J g(BLp)^(dLp)

= — f fiBLp-JuZ^idLp-!), where f(Lp-i)

=

/

g{Lv-\ 0 s p a n { o ; p } ) ^ - p .

The result now follows from the induction assumption. The total measure //£(££) is obtained by using g = 1 in the defining relation (3.3) and noticing that for i = 1 , . . . ,p — 1 5 n _ 1 n spanjc^i,..., ui} is a unit sphere in span{o;i,..., uJi} which can be identified with the unit sphere in

Rn-\

□

The following duality result is very useful. Proposition 3.3. For any non-negative measurable function g on £™_p, we have

Jg(L$)$(dLp) = j g(Ln-p)tfl_p(dLn-p). rn ^P

(3.4)

rn *~n-p

Proof. Let /2™_p be the measure on C^-p defined by the left-hand side of (3.4), i.e. for any non-negative measurable function g on £^_ p , we have

J g(Ln-p)tt-p(dLn^p) = Jg(L^np{dLp). rn

rn

(3.5)

3.1 CONSTRUCTION OF ROTATION INVARIANT MEASURES

69

The measure ft^-p *s rotation invariant, since for B G SO(n), we have /

g(BLn-p)R_p(dLn-p)

n—p

= j g{BL$)nnp{dLp) =j

gdBLpf^idLp)

& J g(Lt)$(dLp) =

/

ff(^n-p)An-p(^n-p),

where we at (*) have used the rotation invariance of pJl. A rotation invariant measure on C^_p is unique up to multiplication by a positive constant, cf. Appendix. Therefore, the measures pQ_p and ^™_p are proportional. Furthermore, since c(n,p) = c(n,n — p), ~Hnn-PK-P)

= *$(*%) = c(n,p) = c(n,n-p)

= nnn_p{Cnn_p).

At (*), we have used (3.5) with g = 1. The two measures are therefore identical and we have proved (3.4). □ It is also of interest to consider the set of p—subspaces containing a fixed r-subspace L r , say, where 0 < r < p. This subset of Cp will be called £p(ry The typical example ££(1) *s illustrated in Figure 3.3. A measure which is invariant under rotations keeping L\ fixed is fi^m defined by / g(L2)fJ%w(dL2) = -

^(span{a;i,a;2})^~ 2 ,

/ 5n-1nspan{c(;i}

^2(1)

where UJ\ is a unit vector spanning L\. Thus, if B e SO(n) satisfies BL\ = L\, then

j g(BL2)^(1)(dL2) = J g(L2)tf{1)(dL2). rn ^2(1)

rn ^2(1)

70

3. ■ ROTATION MYAR1AMT MEASURES ON ££

Figure 3.3. A plane L> containing the ixed line L\ is spanned by ^'1,^2 where ^2 CE 5 n _ 1 DLf. More generally» we have the following proposition which is a generalisation of Proposition 3.2. Proposition 3,4. Let 0 < / • < / ; < //. Let Lr he a fixed r-suhspace with orthoeomial basis (JJI, jr and let SO(n.Lr)

= {B G SO(n) : DLr = Lr)

be the subgroup of SO(n), consisting of rotations keeping Lr (ixed. Let jtiV, be the measure on C^ defined for /• = 0 by /AQv = ^ and for /■ > 0 by

^:

I

-

/

g(spanfø,..., u)p})dw^p • • • d<+,r"~~\ where g is a non-negative measurable function on C\. and
= <"-r.l>-r).

O

3.1 CONSTRUCTION OF ROTATION INVARIANT MEASURES Note that span{cji,... ,u; r }

71

can be identified with Rn~r and

Sn~x Pi span{a;i,... ,u)r}1' can be identified with the unit sphere in Rn~r. Furthermore, for i > r Sn~l D span{o;i,.. . ^ } = [5

n_1

- 1

Dspan{o;i,... ,LUT} ] n span{o; r +i,... ,Ui} .

These are the intuitive reasons for the fact that the measure /xn, x can be identified with ^Zrr. More formally, we have the proposition below, the proof of which is Exercise 3.5. Proposition 3.5. Let 0 < r < p < n and let g be a non-negative measurable function on Cp. Let Lr e C™ and let as earlier £p(r) = {Lp € £p'■Lr C Lp}. Let T be the 1-1 linear mapping T : Rn~r -+ L^r n—r \%li

• - - y%n—r)

>

/

v

Si&ii

i=l

where a\,...,

a n _ r is an orthonormal basis of L^r. Then, / rn P(r)

9(Lp)tf{r)(dLp)

= j

g(Lr®T(Lp_r))ii;irr(dLp-r).

rn-r S-r

D Accordingly, we can identify fin^ with /ipZ^ • In w n a t follows, we allow ourselves to omit the mapping T in the above integral relation. Furthermore, from now on we write dLp instead of /ip(dLp) and dL™,, instead of n™,JdLp). When proving results concerning the rotation invariant measures on p—subspaces, it is convenient to apply results relating measures for different dimensions p. This is in particular important for proofs based on induction arguments. One result of this type which is a direct consequence of the definition of /J,p and /iV N is given in the proposition below. More results of this type are given in Section 3.4.

72

3. ROTATION INVARIANT MEASURES ON ££

Proposition 3.6. Let 0 < r < p < n and let g be any non-negative measurable function on £p. Then,

c(p,r)Jg(Lp)dL; = J j g{Lp)dL^r)dLnr.

□ For r = p — 1, Proposition 3.6 gives us the following useful relation

apfg(Lp)dL;= J

J

g{Lp.x ® s p a n ^ } ) ^ - ^ ! ^ .

(3.6)

3.2 Crofton's formula Crofton's formula relates the Hausdorff measure of a d—dimensional manifold to corresponding properties on intersecting flats. As an example, the area of a surface in R3 is related to the number of intersection points along a line or the length of a planar section curve. Likewise, the length of a curve in R3 is related to the number of intersection points on an intersecting plane, cf. Figure 3.4. The formula is presented in Proposition 3.7. Proposition 3.7. Let X C Rn be a differentiable manifold of dimension d. Let q be a non-negative integer satisfying n — d < q < n. Then, Xdn(X) = a(d,q,n) j j A ^ " + " ( X n (L, + y))dyn-*dl%,

where a{d,q,n) =

. Vq+d-n+l

&n+l ' ' ' &n-q+l

(3.7)

3.2 CROFTON'S FORMULA

73

Figure 3.4. Three special cases of Crofton's formula. Proof. Note first that for q = n, (3.7) reduces to the trivial identity Xft{X) = A^(A'). Below, we shall therefore assume that q < n. The proof will be based on the translative decompositions of Hausdorff measures» presented in Proposition 2.10. The first step of the proof is to see that / G(Tan[X, x], Lq)dl% = c(d} q, n),

(3.8)

i::; where c{d, q. u) is a constant depending only on the dimensions involved. Thus» the integral does not depend on X or the particular choice of .r G A". The proof of (3.8) is Exercise 3.7. Using (3.8), we get / / G[ Tan[X, x], Lq)dl%dxd = c(d. q, n)A*(X).

(3.9)

74

3. ROTATION INVARIANT MEASURES ON ££

On the other hand, using Proposition 2.10 with ^ E l w e get /%G(Tan[Xix],Lq)dL^dxd

/ X £%

= I f

G{T<m[X,x],Lq)dxddL^

£? x = j l

\dn-n+q{X H (Lq + y))dyn-*dl%.

(3.10)

Comparing (3.9) and (3.10), we get c(d, q,n)\l(X)

= JJ

\dn-n+q(X

H (Lq + y))dyn-*dl%.

(3.11)

The constant c(d,q,n) can now be determined by inserting into (3.11) a particular simple X for which the integrals on the right-hand side of (3.11) can be calculated directly. The details are given in Santalö (1976, p. 244-245). For simplicity, we will concentrate on the case d = n — 1, where we choose X — Sn~l. For y e L\

AT^S"- 1 n (Lq +y)) = { ^ ( [ l - IMI2]172^-1) if 112/11 < 1 10

otherwise.

Therefore,

J J \tl{Sn~l n (Lq + y))dyn-"dLnq = J J l{\\y\\ < 1}(1 - IMI2)'«-1)/V3/"-«dL£. The inner integral does not depend on Lq and is equal to

aq J ^ I M I ^ i K i - I M I 2 ) ^ 1 ^ ^ Rn-q

1

o

3.3 A RESULT ON PROJECTIONS

75

where we at (*) have used Exercise 2.7. We find

j j Ar1 (S*-1 n (Lq + y))dyn-*dl% l

n-q

= VqVn-q-Bi——,

g + l

w

v

—^—) C (n, tf)

_ ö"g &n+l ' ' ' Vn-q+l — ®n

= Ar1(5n-1)o(n-l,g>n)-1. Ö

Note that for d — n, the inner integral of the right-hand side of (3.7) does not depend on Lq. Indeed, we have for d — n,

J \"n{X n {Lq + y))dyn-" = Xn(X), Lisee also Section 2.6. Accordingly, a(n, q, n) I dUl = a(n, q, n)c(n, q) = 1.

Example 3.8. For n = 3, (3.7) contains the following interesting special cases

\&X) = IJ J $(X n (L2 + y))dyldL\ r3 rj_ t-2 ^2

*§(* n {Lx + y))dy2dL\

\\{X) = lfl

>\(X n (L2 + y))dyldL\.

Xl(X) = ^jj ^2

^2

D

3.3 A result on projections In calculating sampling probabilities in local stereology, the proposition below is useful.

76

3. ROTATION INVARIANT MEASURES ON ££

Proposition 3.9. Let x G Rn\Lr and let g be any non-negative measurable function on R. Then, for 0 < r < p < n, we have

7

c(n-r,p-r)

IKL^II2

1

= B((n-P)/2,(p-r)/2)/g(y)ygiC"1(1 "

^ " ^

0

Proof. First notice that, since \in, x can be identified with /i^Z^, cf. Proposition 3.5, and

where 0 denotes orthogonal difference, it is enough to consider the case r — 0. Setting e = x/\\x\\, we thus want to show l

[ yKg(\frL L±e\\*)Æ:L = }

J

"

c%

? " c(nlV)

L

B((n-p)/2,p/2)J

__ /yKU)y g(y)yn-^-\l -yj yf^dy V U o

or equivalently, cf. Proposition 3.3, l

dLn /

^ A ££

2

) ^

^

c

= B(p/2,(n-p)/2)

J ^f"1(1 - ^ " ^ 0

The proof of (3.12) is by induction in p. For p = 1, we have c(n,p)

_

TT^-1)/2

B(p/2,(n-p)/2)~r((n-l)/2y Furthermore, according to the definition of /xj,

Jg(hLle\\2)dL? = ± J

g{hsv^}e\\2)d^-1

= \ j gdcuj)2)**"-1 Sn-1

The equation (3.12) follows then for p — 1 directly from Proposition 2.11.

(3J2)

3.3 A RESULT ON PROJECTIONS

77

Now, let us assume that (3.12) is correct for p — 1. Then, we use that, cf. (3.6), J g(\\vLpe\\2)dL"p = ±- J Now, for e G Rn\Lp-i,

^ll^^espanKiell2)^-^^!.

j

(3.13)

let e = 7rL± e/||7rL± e||. Note that é is defined for p-i

p-i

/Xp_i-almost all L p _i G ££_!• Since cvp G £p_ l 5 we get II2

II 7

W*

Lp-i®spa,n{Lup}e\\

= lkL p _ie|| 2 + = IK£,p_1e||2 H= lkL p _ 1 e|| 2 + Therefore, using Proposition 2.11 on the

(e,o;p)2 <7rjL^_ie,c^p>2 lkLi|-_ie||22. inner integral of (3.13), we get

g(\\nLpe\\2)dL;

J

&n—p (Jp 1

Interchanging the order of integration and using the induction assumption, we finally get

J

g(\\irLpe\\2)dL;

_ Vn-p

c(n,p-

1)

ap B ( ( p - l ) / 2 , ( n - p + l)/2) l

l

x / J g(z + (l-z)y)z^-\l-z)^-1y-*{l-y)n^-1dzdy.

(3.14)

7/=0 2=0

By elementary means, cf. Exercise 3.8, the double integral on the right-hand side of (3.14) reduces to l

5(1/2, (p - l)/2) j

g(u)u^-\l

-

Evaluating the constants in (3.14), we finally get (3.12).

u ) ^

- 1

^ □

78

3. ROTATION INVARIANT MEASURES ON £%

3.4 Pairs of subspaces In this section, we will consider transformation results, involving pairs of subspaces. Using Proposition 3.5, we have the following result for a pair of subspaces (L p , Lr) for which Lr C Lp. Proposition 3.10. Let 0 < r < p < n and let g be a non-negative measurable function on Cp x C™. If Lp e C™, we use the notation £% for

Then,

j Jg(Lp, Lr)dL?dL; = J j g(Lp, Lr)dLnp(r)dLnr. ^V

*^r

*~r

(3.15)

^p{r)

u Proposition 3.10 is a result concerning the change of order of integration. The proposition can be proved, using uniqueness results in invariant measure theory, as we did in the proof of Proposition 3.3. Below, we give an alternative and more direct proof in the case n = 3,p = 2,r — 1. We avoid using polar coordinates which complicate matters unnecessarily. Proof, (n = 3,p = 2,r = 1) Since S2 D L
j

Jg{L2M)dL\dL\

L\

S2f)L2

We now use Proposition 3.3 and find / 2 / #( L 2, span{tJi})G^dLi] c\ s2nL2

c\

s2nL±

3.4 PAIRS OF SUBSPACES = -— /

/

^(spaii{^2}±<spaii{wi})dc4i|é4i| .

= - •- /

/

^(span{cii2}X,spaii{c4;1})dc4;|dc4ifJ

^

2

2

S nspaii{^i}

f9

(3.16)

x

where we at i * j have interchanged the order of integration. This step can be justiied using the coarea formula.

Figure 3.5. Illustration for Proposition 3.10.

Now, let B be a rotation with angle ir/2 around spanj^i}. Then, cf. Figure 3.5,

f(span{oi2}X,spaii{c4ii})dc4i2

/

=

/ 2

S nspan{aJi}

=

f(spaB{Boi2,wi}»span{c4ii})dki2 ±

/ 2

S nspan{u.-i}-

^(span{a^,wt"i}.span{a;i})da;2, L

80

3. ROTATION INVARIANT MEASURES ON C™

where we at the last equality sign have used the rotation invariance. Inserting this result into (3.16), we finally get j j

g(L2M)dL\dLl

= - I S

/ S' 2 nspan{a;i}

= 2/ S2

^(span{o;2,^i},span{a;i})(i(x;2^i

/ £

9(L2,spa,n{uJi})dLlwdujl

2(1)

= J J gi^L^dLl^dLl ^ 1 ^2(1)

D Proposition 3.10 contains two interesting special cases. g(Lp,Lr) = f(Lr). Then, cf. Proposition 3.4, J J f{Lr)dLldLnp

f(Lr)dLnr.

=c(n-rlP-r)J

On the other hand, suppose that g(Lp,Lr)

(3.17)

= h(Lp). Then,

c(p, r) J h(Lp)dL; = j rn

First, suppose that

rn

j

h{Lp)dLnp(r)dLnr.

(3.18)

rn

Note that the last result has earlier been proved directly in Proposition 3.6. The result of Proposition 3.10 can be generalized, as shown in the proposition below. Proposition 3.11. Let 0<s
Then,

J J g(Lp,Lr)dLpris)dL;{s)= rn rp P ( 0 r( S )

j

J

rn rn ^r(S) S ( r )

g(Lp,Lr)dL;{r)dL?(s).

3.4 PAIRS OF SUBSPACES

81

In particular, we have J

f{Lr)dIf(s)dLnv(s)

J

rn

= c(n-r,p-r)

}{Lr)dLnr{s)

J

pp

rn

and c(p-s,r-s)

h{Lp)dLnp(s) = J

J rn p(0

J

h{Lp)dLnp{r)dLnr{s).

rn rn r(s) S ( r )

Proof. Using Proposition 3.5, indicated by (*) below, and Proposition 3.10, indicated by (**), we get the following, where we for simplicity have omitted the mapping T from Proposition 3.5, j

j

g{Lp,Lr)dL>{s)dL%a)

L

J><»)

r(.)

( }

= j

s n 9{Lp,Ls®Lr-s)dV-_ sdL p{$)

J

~

( }

=

9\Ls © Lp-S, Ls 0

j

g(Ls 0 L P - a , Ls 0 Lr-s)dLnpZl{r_s)dLnrZl

J

Cn-s

Lr-s)dLr_sdLp_s

Cn-s

r-s

p-s(r-s)

Let us consider the inner integral of the last double integral. We find /

g(Ls 0 Lp-S, Ls 0

Lr-s)dLnvZss{r_s)

p-s(r-s)

(*)

=

= =

/

/

Q\L/S 0 J^i—s 0 J-'p—ri l^s ©

L/r—s)dLir

/ Q\Lir 0 Lp—r, Ls 0 Lr—s)dL/p_r c-r-i f

g(Lp,Ls®Lr^s)dL^{r).

(3.19)

82

3. ROTATION INVARIANT MEASURES ON ££

Inserting this result into (3.19) we find

I

g(Lp,Lr)dLPr(s)dL;{s)

I

P(-) ^ r ( . )

= j rn-a ^r-s

( }

= j

j g{Lp,Ls®Lr-s)dL;{r)dLnrZss £n P(r)

j g(Lp,Lr)dL;{r)dLf(s).

r(a) S ( r )

□ 3.5 Random subspaces In Chapter 1, we considered sampling designs involving lines and planes with random orientations. In this section, we will generalize these concepts. Definition 3.12. An isotropic p—subspace is a random p—subspace Lp with density with respect to fip of the form p(Lp) =

l/c(n1p),LpeC;.

□ A number of the results from the earlier sections of this chapter can now be given stochastic interpretations. One important example concerns Proposition 3.3 which has the following stochastic interpretation. Proposition 3.13. Let Lp be an isotropic p—subspace. Then, Lp is an isotropic (n — p)—subspace. Proof. Let Lp be an isotropic p-subspace. Then, according to Definition 3.12 and Proposition 3.3 we have for A C C^_p Ajn

/ J

c(n,p)

3.6 RANDOM GRIDS Since c(n,p) = c{n, n - p), the last integral is equal to P(Ln-p an isotropic (n — p)—subspace.

83 e A), where Ln-p is D

Using Proposition 3.13, it follows that an isotropic 2-subspace in R3 can be generated by generating an isotropic 1-subspace in R3 and use the direction of this line as the normal of the 2-subspace. Definition 3.12 can be extended as follows. Definition 3.14. An isotropic p—subspace containing a fixed r—subspace Lr is a random p-subspace Lp G £>™(r\ with density with respect to iin,, of the form p(Lp) = l/c(n - r,p-

r),Lp e ££ ( r ) .

□ Using Definition 3.14, (3.17) and (3.18) have the stochastic interpretations indi­ cated in the following proposition. Proposition 3.15. Let 0 < r < p < n. Then, (0

An isotropic r—subspace Lr can be generated by first generating an isotropic p—subspace Lp in Rn and next an isotropic r—subspace Lr in Lp.

(//)

An isotropic p—subspace Lp can be generated by first generating an isotropic r—subspace Lr in Rn and next an isotropic p—subspace Lp containing Lr. D

3.6 Random grids Random grids play an important role in global as well as local stereology. In Chapter 1, some examples from global stereology have been given. In Chapter 6, we will consider applications of random grids in local stereology. In order to introduce such grids, let Ls be an 5-subspace of Rn. Let D 0 be a bounded subset of Ls and let {UJ}JL0 be a sequence of points in Ls. Let Dj = Do + Uj,j = 0 , 1 , . . .

84

3. ROTATION INVARIANT MEASURES ON ££

The pair (Do, {^j}°l 0 ) *s t n e n s a ^ t 0 induce a lattice of fundamental regions in Ls if the sets Dj,j = 0 , 1 , . . . , are space-filling and non-overlapping, i.e. (a) \jf^D5 (b)

= L8 DjinDJ2=0,j1^j2.

Now suppose that (Do, {v>j}°°_0) induces a lattice of fundamental regions in Ls. A point grid in Ls, induced by (Do, {^j}°l 0 )' ls t n e n a s e c l u e n c e of points in Ls of the form {u + UJ}°°_Q, where u e Do, cf. Figure 3.6. Using the concept of a point grid in a subspace of Rn, we can define a q—grid in i? , which is a grid of parallel q—dimensional affine subspaces in Rn. n

Figure 3.6. A point grid in the j-subspace Ls.

Definition 3.16. Let Lq e C™ and suppose that (DO,{UJ}JL0) induces a lattice of n fundamental regions in L^. Then, a
□

Definition 3.16 is illustrated in Figure 3.7. Note that a 0-grid is simply a point grid in Rn.

3.6 RANDOM GRIDS

85

Figure 3.7. A g-grid in Rn. We will now consider random q—grids. This is done in a way which is analogous to the procedure used for subspaces. We thus define an invariant measure on a set of q—grids in Rn, which are identical up to rotations and translations. A normalized version of the measure defines the special type of random q—grids considered. Let us consider

D0 C Rn~q, Uj e Rn~qJ

= 0,1,...,

and let us suppose that {A),{^j}°L 0 } induces a lattice of fundamental regions in Rn~q. We furthermore assume that U — Uj^0{uj} is a discrete subgroup of Rn~q with respect to addition, i.e.

(i) uj.+uj» ezv, j'j"e {0,1,...} (a) -UJ euj e {0,1,...}. Now, {Do, {^}°^ 0 } defines a set of q—grids in Rn, which are identical up to rotations and translations. This set is denoted Gq or Gq({Do, {UJ}°°_0}) if it is important to emphasize the dependence of {Do, {UJ}^L0}, Gq = {Gq(u] ui,...,

LUn-q) - u G A ) , ^ l , • • •, ^n-q oithonormal vectors in Rn},

where n—q

Gq(u\u)i,...,u)n-q)

= {span{u;i,...,u; n _J

+ ^

(u + Uj)^

:j = 0 , 1 , . . . } ,

86

3. ROTATION INVARIANT MEASURES ON ££

cf. Figure 3.8. Note that if /{o,!,...,^-,}

: RU q

~

-* span{o;i,...

,un-q}

n—q \%li

• • • 5 %n—q)

* /

j

%i^i->

i=l

then Gq(u;uJi,... ,uon-q) is a q—grid in Rn, induced by span{a;i,... ,ujn-q}

and

(

{f{u;1,...^n^}Do^{f{uJl,...,un^}Uj} ^=0)-

Figure 3.8. One of the line grids in Rs which is associated with the indicated planar point grid. We will now define a measure 77™ on Q™ which is invariant under rotations and translations, i.e.

Jg(BGq+x)rf(dGq) = I g(Gq)^(dGq), for all non-negative measurable functions g on Qq, all B G SO(n) and all x G Rn. Considering the definition of the rotation invariant measure on subspaces in Rn, cf. Section 3.1, the following construction is natural

j

g(GqX{dGq)

= ^Zq— 11 i=l

a

i

X

J n x

S~

J n 1

••' J

S - nspa.n{tJi} -

J n 1

I

S - r\spa.n{ui,...,un-q-1}

±

9(Gq(u)UJ1,...,Un-q))

D

o

(3.20)

3.6 RANDOM GRIDS

87

For simplicity, we will in what follows write dG™ for jf^i^dGq). The total measure becomes, cf. Proposition 3.2, Vq(Gq) = \n-q{D0)c(n, n - q) = An-qc(n, q), say. In Exercise 3.12 and 3.13, it is shown that the measure 77™ is invariant under rotations and translations. By normalising 77™, we can define a special type of random q—grids. Definition 3.17. Suppose that {Do, {UJ}°?0} induces a lattice of fundamental regions in Rn~q. A uniform and isotropic q—grid, induced by (A), {^j}°^0), *s a r a n ^ o m q—grid with density with respect to 77™ of the form p(Gq) =

l/[An-qc(n,q)],Gqeg^ D

Note that the uniform and isotropic plane grid in R3, defined in Chapter 1, cf. Definition 1.10, is the special case n = 3 and q = 2 of the above definition. Using Crofton's formula, it is fairly easy to calculate sampling probabilities associated with random grids. The proposition below generalizes Buffon's needle problem discussed in Chapter 1. The manifold X in the proposition will in later applications be an infinitesimal d—dimensional volume element. Proposition 3.18. Let X C Rn be a d—dimensional differentiate manifold with A^(X) < 00. Let Gq be a uniform and isotropic q—grid. Suppose that d — n + q = 0 and that X°n(X n Gq) G {0,1} for all Gq G Qnq. (3.21) Then, P(X f l G g / l ) = -7^—Kd, q, n)\dn(X), ^■n—q

where b(d,q,n) = —

.

Proof. Using the definition of a uniform and isotropic q—grid, we get dGn /

88

3. ROTATION INVARIANT MEASURES ON £ £

where we at (*) have used (3.21). In order to calculate the latter integral note that if Ln-q = s p a n { o ; i , . . . ,ujn-q},

^n-q))dun^

I X°n(X n Gq(u;uu ■ • • Do oo

*

n-q

/ X°n(X n (Ltq + £ (u + uj)i«>i))dun-9

=£

Rn-q

»=1

AS(in(L„i_, + «))å»-«

= |

Inserting this result into (3.22), we get P(X DGq^$)

= A^

( ]

=

*

J

j

X°n(X n (Ltq

+

u))dun-HLnn_q

J—b(d,q,n)\dn(X), ■rt-n—q

where we at (*) have used Proposition 3.3 and at (**) Crofton's formula.

□

3.7 Exercises Exercise 3.1. Let C C R2 be the square with centre (1,0) and side length 1, cf. the illustration below. Let A be the set of lines hitting C, i.e. A={LX

eC\

:LiHC^0}.

According to the definition of ii\ we have

3.7 EXERCISES lÅ(A)

=

89

9 / l{span{u;} e A}dujl. S1

Let for 0 < 9 <

2TT

Li(0) = span{(cos#,sin#) }. Note that for 0 < 0 < IT, we have L\{0) = L\(0 + 7r). Show that ^i(A) = \ j l{Li(ö) n C / 0}do = TT/2. o Hint. Use Exercise 2.6. Exercise 3.2. Show that for B e SO(n) and Lp G ££ we have BL^ =

(BLp)^.

Exercise 3.3. Let c(n,p) be defined as in Proposition 3.2. Show that c(n,p) = c(n,n — p). Exercise 3.4. Show that the measure AC(r)> p > r + 1, defined in Proposition 3.4 is invariant under rotations keeping Lr fixed, i.e.

J g(BLp)n;{r)(dLp)

= J

C

g(Lp)n;{r)(dLp),

C

p(r)

p(r)

for any B e SO(n,Lr). Hint. The proof can be made by induction in p. Start by showing the invariance directly for p = r + 1. Exercise 3.5. Let the situation be as in Proposition 3.5. Note that for any x,y e Rn~q, we have (Tx,Ty)

= (x, y).

1. Show that g(Lr®T(Lp-r))^Irr(dLp-r)

j Cn-r

~" P~r J! 2= 1

2.

J

a

i

J

5 — i H L i - S n - 1nLjrnsp<m{ujr+1}-L

J Sn~1nL^nsp<m{ujr+u...,ujp-1}-L

g(Lr 0 span{u; r +i,..., LUP}) duUp~p ■ ■ • d u ^ J ~ ^ ^ + i ~ Hint. Use among other things that T ( 5 n - r " 1 ) = S71'1 n L;k Show Proposition 3.5 by using question 1 and the definition of fi^ry given in Proposition 3.4.

90

3. ROTATION INVARIANT MEASURES ON ££

Exercise 3.6. Prove Proposition 3.6. Hint. Use the definition of /i™ and /zVx given in Propositions 3.2 and 3.4, respectively. Remember that c(p,r) = c(p,p — r). Exercise 3.7. In this exercise we will show that /G(Tan[X,x],L,)dL^

does not depend on X or the particular choice of x G X. This result is used in the proof of Crofton's formula, cf. Proposition 3.7. Consider, for x G X, Ld — Tan[X, x] which is a d—subspace of Rn. For L 9 G £™, the dimension of Ld n (Lq n Z^)-1 is atmost n — q, cf. Exercise 2.9. In this exercise we use the following without proof ^{{Lq

G £nq : dim(Lrf n (Lq n L d ) x ) < n - ?}) = 0.

Therefore, for /j,™—almost all Lq G £™, G^L^detjÆ^}1/2, where A = [7r^±ai, • •, 7r^±an_9] and a\,..., (Lg n Ld)-1, cf. Definition 2.9. 1.

Show that for B G SO(n), G(Ld,Lq)

2.

a n _ g is an orthonormal basis of Ld n

=

G(BLd,BLq).

Hint. Note that if a i , . . . , an-q is an orthonormal basis of Ld n (L9 Pi L^) , then Ba\,..., Ban-q is an orthonormal basis of £ 1 ^ H (BLq n BL^) . Show that for £ G SO(n),

I G(BLd, Lq)dLnq = J G(Ld, Lq)dLnq. Hint. Use the rotation in variance of /i™.

3.7 EXERCISES

91

Since any d—subspace can be reached from any other by a rotation, it follows from the results in this exercise that [G(Tzn[X,x],Lq)dI%

does not depend on L^ — Tan[X,x\. Exercise 3.8. This exercise concerns the last part of the proof of Proposition 3.9. 1. Let Y and Z be independent Beta-distributed random variables

Y ~ B { \ ^ ) , Z ~ B ^ ^ ^ ± ± ) . Show that

Z+(1-Z)Y~B^,^). Hint. Note that (Y, 1 - Z) has the same distribution as , Y1 Y1+Y21Y1

[

Yl + Y2 + Y2 + Y3)

where Yi,Y2,Y3 are independent and Y1 ~

2 X

(l), Y2 ~ x\n

- p), y 3 ~ X2(P ~ 1),

cf. Johnson & Kotz (1970, p. 38). 2. Show that the result in question 1 implies that l

/

l

/ g(z + (1 - z)y)zE^-1(l

- z)11^1-1y-l2(l

-

yf^^dzdy

y = 0 2=0

1

= 5(1/2, (p - l)/2) / giu)^-1^

-

uf^^du.

Exercise 3.9. What is the density of the polar coordinates of the normal to an isotropic 2-subspace in R31 (The normal is represented as a point on the unit hemisphere.) Hint. Use that the line containing the normal is an isotropic 1-subspace in R3. Furthermore, (1.3) is useful.

3. ROTATION INVARIANT MEASURES ON Lnv

92

Exercise 3.10. Let Lp be an isotropic p-subspace, containing a fixed r-subspace Lr. Show that for any x e Rn\Lr, we have IKL^H 2

p>(n-p

p-r]

Exercise 3.11. Let x\,X2 G R3 be linearly independent and let L2 be an isotropic 2-subspace in R3, cf. Definition 3.12. Find the distribution of V2(nL2x1,irL2x2)2 V 2 (xi,x 2 ) 2 Hint. Use among other things Proposition 2.15. Exercise 3.12. Let

and let us suppose that {Do, {UJ}^L0} induces a lattice of fundamental regions in Rn~q. In addition suppose that U — UJL0{UJ} is a discrete subgroup of Rn~q with respect to addition, i.e.

(i) uj'+uj» eU, / , / ' e { o , l , . . . } (M) - ^ - e M j ' G { o , i , . . . } . Let /i be a non-negative measurable function on the set of point grids induced by {DQ,{UJ}^L0}. For all y e Rn~q show that / h(uJL0{u + y + uj})du n -* = / h(U™=0{u + Do

uj})dun-q.

Do

Hint. Use, among other things, that

u?=0(D0-Uj) = Rn-q. Exercise 3.13. Show under the assumptions (i) and (u) of the previous exercise that the measure 77™ defined in (3.20) is invariant under rotations and translations. Hint. Let B e SO(n) and x G Rn. Start by showing that j g(BGq(u;

Wl,...,

ujn_q) + x)dun~q = / g(BGq(u) uu . . . , w n _ g ))du n -«.

Here, the previous exercise is useful. Exercise 3.14. Let Gg be a uniform and isotropic g—grid and let Lq be its associated q—subspace. Show that Lq is an isotropic q—subspace.

3.8 BffiLIOGRAPHICAL NOTES

93

3.8 Bibliographical notes In this chapter, we have chosen to construct the rotation invariant measures explicitly, using Hausdorff measures on unit spheres. As demonstrated in Section 3.1, it is possible to give a simple proof of the rotation invariance, by induction in the dimension of the subspace. This approach turns out to be very convenient in the chapters to come. In Santalö (1976), the measures are also constructed explicitly, using exterior calculus. In other expositions, cf. e.g. Schneider & Weil (1992), the measures have been introduced by means of invariant measure theory. Crofton's formula is a classical formula in integral geometry, which has many important application in global stereology. The proof of the formula given in Section 3.2 is based on Santalö (1976, p. 244-245). Note that other integral geometric formulae are also sometimes called Crofton's formula, e.g. a formula concerning integrals of powers of chord lengths, cf. Santalö (1976, p. 237-39). The transformation result concerning projections, given in Section 3.3, has been derived in Jensen & Kiéu (1994). There are important generalizations of this result, concerning the projections of simplices instead of line-segments, cf. Nielsen (1996). In this case, the distribution becomes that of a product of independent Beta-distributed random variables.

Chapter 4

The classical Blaschke-Petkantschin formula The Blaschke-Petkantschin formula is a geometric measure decomposition of the q—fold product of Lebesgue measure in Rn. One of the main reasons to consider the Blaschke-Petkantschin formula in this book is that it plays a central role in the development of local stereological methods. Let us start with an example of this type.

4.1 A local estimator of planar area Suppose that X is an open and bounded subset of R2. Let us assume that we want to estimate its area A(X), using local methods. More precisely, we want to use information along an isotropic line through the origin O. Let us recall the set-up leading to the Horvitz-Thompson estimator. A finite population V — { l , . . . , i V } is considered. To each object i G V, a characteristic y(i) G R is associated. The population total

iev is then estimated by yv =

^y(i)lp(i), ies

where p(i) — P(i G 5) is the probability that the z'th object is sampled. The problem of estimating A(X) is not of this discrete type but we can use the Horvitz-Thompson procedure on an infinitesimal level. Thus, let us for x G X consider an infinitesimal neighbourhood U(x) of area dx2. We want to find an expression for 95

96

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

the probability that this neighbourhood is hit by the line. For this purpose, we consider the transformation from polar coordinates to Cartesian coordinates / : R x [0, TT) -+ R2 (r,6) —> (r cos#,r sin.0). It is well-known that dx2 = \r\drdO.

(4.1)

Therefore, we can make the following infinitesimal calculations ^/■rr/

P(U(x

/

nL^Ø

rtX

dO

1

dX2

= — TT' 7T = 7-Tr |r|ar

where dO is the length of the angle interval for which L\ hits U(x), cf. Figure 4.1. The Horvitz-Thompson procedure therefore leads to the estimator

£

dxyl\r\dr = *

S

|r|dr

'

c/(x)nLi#0 ' ' ' j7(x)nLi#ø The corresponding continuous version will be used as estimator of A(X), Ä(X)=7T

I

\\x\\dxl.

xnLi Decompositions of the type (4.1) therefore become important. Note that (4.1) can equivalently be written as dx2 = 11 a; 11 ctø1 dL?. The decomposition (4.2) is a particular case of the Blaschke-Petkantschin formula.

Figure 4.1. Sampling of area elements by means of an isotropic line Lx through O.

(4.2)

4.2 DECOMPOSITIONS INVOLVING LINES

97

4.2 Decompositions involving limes in Rn The Blaschke-Petkantschin formula can 'be formulated as follows

IJifa? = Vq(xu • • .^"-«lldxldL», 1=1

(43)

i=i

where Yq(x\, vq) is q\ times the (/-dimensional Hausdorff measure of the g—simplex with vertices 0.x\ rq. cf. Section 2.8. In particular, for q = 1 and 2, we have the following transformation results» cf. Figure 4.2,

I !!(*i)dx'{= I I i / 9(xi,T-2)dxZd.r? = / / I

gUMWr'dxldL'l g(xl.x2)V2(xl.j'2y}'~2dxidsidLl

Rn Rn a L-2 L Note that V2(0:1,0:2) is 2 times the area of the triangle spanned by O, x\ and x2.

Figure 4.2. Dlustration for two special cases of the Blaschke-Petkantschin formula. In'this section» we will concentrate on the case q = 1 of the Blaschke-Petkantschin formula» which can be derived from polar decomposition in Äm» cf. Proposition 2.8. Preposition 4.1. Let X be an open subset of Rn. measurable function g on Rn. [g(x)dx"=

I

I

Then» for any non-negative

g{x)\\x\\n-ldx1dL,l.

(4.4)

98

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

Proof. Let us consider the mapping / given by / : Rn\{0}

— S71-1

x->x/\\x\\. We get, cf. Proposition 2.8, fg(x)dxn

= ( }

-

f

g{x)\\x\\n-ldxldun-1

f

/

g(x)\\x\\n-ldxldun-1

/

Sn-lf-H-uj)

=\ J =\ j

I

g^Wxir'dx'd^-1

j g^Wxir'dx'd^-1.

5 n _ 1 span{u;}

At (*), we have used that A™-1 (A) = A^ _ 1 (-A) for any A C Rn. Using the definition of the rotation invariant measure on lines in Rn, we get I g{x)dxn= Rn

j j

g{x)\\x\\n~ldx1dL1l.

C^ Li

Substituting g{x) with l { x G X}g(x) we immediately get the result.

D

We can directly extend this result to q—subspaces, containing a fixed (q — 1)—subspace Lq-i, say. Such a subspace can be represented as Lq = L/q — l ©

L\,

where L\ C L^_v Using Proposition 4.1, a decomposition result for lines in Lql_1 can be derived and then translated to a result for L g , cf. Proposition 4.2 below. The case q = 1 corresponds to Proposition 4.1. In the proof of Proposition 4.2 as well as other propositions in this chapter, we use the following consequence of the translative decomposition of Lebesgue measure, cf. Proposition 2.10 for d — n. Let Lr and Ls be r— and s—subspaces, respectively, which are mutually orthogonal. Then, for any non-negative measurable function h : Lr 0 Ls -* Ä+ U {0}

4.2 DECOMPOSITIONS INVOLVING LINES

99

and any open subset X of Rn, we have

h(x)dxr+s

/ xn(LreLs)

h{y + z)dzsdyr.

=J

j

Lr

(X-y)nLs

(4.5)

Proposition 4.2. Let X be an open subset of Rn. Then, for any non-negative measurable function g on Rn,

Jg(x)dxn = J

J gix^L^xT-tdxUL^y

(4.6)

^Vi)xnL*

x

Proof. We expand on the left-hand side of (4.6), using (4.5) with Lr — Lq-\ and Ls = Lq_v

J g(x)dxn = J

j

g(y + z)dzn-^ldy^\

Now, let us use Proposition 4.1 on the inner integral. We identify L^_1 with and find

j

= J

g(y + z)dzn-i+1

j

g{y +

z)\\zt^^-x^dLr^\

(4.7)

Rn~q+1

100

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

since L\ C L^_v Inserting into (4.7) we get Jg(x)dx"=

j

=

j

£

(

*=)

n-

g +

l

z^zt-^-'dz^Lr^dy^-1

J

j

giy +

j

j

9{y^z)\\z\\n^dzldy^-1dLn1-q+1

X n

(

j

L g

_

i e L l )

J

g{x)\\vLUxT-HxHLlq_lY

where we at (*) have used (4.5) and at (**) Proposition 3.5. (See also the remark immediately below Proposition 3.5.) □ 4.3 Proof of the classical Blaschke-Petkantschin formula We are now ready to formulate and prove the classical Blaschke-Petkantschin formula. Proposition 4.3. Let X\,...,Xq

be open subsets of Rn and let

g : Rn x ■■• x i T -+ R+ U {0} be a non-negative measurable function where 1 < q < n. Then, / ••' / x1 xq = J

J

9(x1,...,xq)J[dx? i=1

•• J

g{xi,.-.,xq)Vq(Xl,...,xq)n-q\ldx^Lnq.

(4.8)

Proof. We prove the result by induction in q. For q = 1, (4.8) is just the decomposition (4.4). Now, let us assume that (4.8) is correct for q—l. Then, using the induction assumption on g(xi,...,

xq-i) = / g(xi,...,

xq)dXq,

4.3 PROOF OF THE FORMULA

101

Cmarginal' argument), we get

x,

i=1

xq

X\

Xq-l

= J

~

I

•"

/

Lnq_x X i D L g _ !

g(x1,...,xq-i)

Xq-!nLq-i q-1

n q+1

x V,_i(xi,..., xq^) '

11

dxf'dL^. 2=1

We now use (4.6) with Lq-i = span{xi,.. .,xq-{\

and find

g(xi,...,xq-i) = /

g(xu...,xq)dx^

£q( g _i)

XqnLq

This step could be called a 'conditional* argument. Since V 9 ( x i , . . . ,x g ) V g _ i ( x i , . . . ,x 9 _i) • ||7rL±_ xq\\, cf. Exercise 2.12, we finally get

/ '" / X,

= /

=

g{xly...,xq)Y[dx2 i=1

Xq

/

I

"

I

I ^l'---'^)

x V g (*i, • •. ,ar g ) n -«V g -i(xi,... , ^ - i ) d x | J ] ^

^ - 1 ) ^ - 1 -

i=l

Using Proposition 3.10 and the induction assumption once more, the result follows immediately. □ The proof of Proposition 4.3 can easily be illustrated in the case q = 2 and Xi = X2 = X. In this case, two decompositions are combined, one 'marginal'

102

4. CLASSICAL BLASCHKE^PETKANTSCHIM FORMULA

decomposition invoking the line L\ (Proposition 4.1) and one 'conditional* decompo­ sition involving the plane /,■_> containing the line L\ (Proposition 4.2), cf. Figure 4.3.

Figure 4.3. Illustration for Proposition 4.3.

The classical Blaschke-Petkantschin formula can be extended in various ways. The dimension of the subspace may be higher than the number of points» cf. Miles (1979), and the subspace may contain ixed lower-dimensional parts. We will now proYe a version of the Blaschke-Petkantschin formula involving subspaces containing ixed parts. Preposition 4.4. Let X\

XH be open subsets of Ft" and let (I : 7?" x • • • x R'\ — /?,_ . {()}
be a non-negative measurable function. Let < i <} be /■ orthonormal vectors in R" and let £^'. v be the set of ;;—subspaces containing Lr = span{ei ? ... ,e r } 5 where P = q — r. Then,

/■••///U. -V:

vq)f[dx? <=

.Y,

= /

/

C;„. X:'-l.„

•■■ / XqnLp

1

s(si.---.-'-./>S>(ri

cr.x1,...,xq)n-pfl
1

4.3 PROOF OF THE FORMULA

103

Proof. First, note that if we in (4.5) let h(x) = h(x,7rL±x) then h(x,7rL±x)dxr~*~s

/ xn(Lr©Ls) = J

h(y + z,z)dzsdyr.

j

Lr

(4.9)

(X-y)nLs

We now get, using Proposition 4.3, (4.5) and (4.9),

Xi

l=1

Xq

= /■•■/

/

•'•

= / • • • / /

/ r

Lr

Lr c^-

/

9(Vi + z1,...,yq

•••

(X!-yi)nLq

/

f[ dz\dL«-* f [ dy\ i=l

/

/

••'

r

C^~ XiflL r eL g

+ zq)

(Xq-yq)nLq

x V ^ , . . . , zq)
9(yi + zi,'.',yq

zq)ndzrrf[dyl

+

/

i=l

g{xi,...,xq)

XqnLr®Lq

x V ^ s i , . . . , *L±xq)n-r

f[

dxr+qdL^r

i=l

= J J c;ir)x1nLp

" J ø(åi>"-,*g)V g (7r L ±xi,^ xqnLp

* =1

Using that V p ( e i , . . . , e r , x i , . . . , xq) = V g (7r L ±xi,..., 7rLj.:r9), the result now follows.

□

The result in Proposition 4.4 can be extended to the case where p > q + r, as shown below. The proof is based on ideas from Miles (1979).

104

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA be open subsets of Rn and let

Proposition 4.5. Let Xi,...,Xq

g : Rnx

••• x / P -+ Ä + U {0}

be a non-negative measurable function. Let e\,..., er be r orthonormal vectors in Rn and let Cn/r\ be the set of p—subspaces containing Lr = s p a n { e i , . . . , e r } , where p > q + r. Then, setting c(n, 0) = 1,

c(n-q-r,p-q-r)

'"

tffci,

•. • , x g ) J J d a r ?

(4.10)

Proof. The result will be shown using Proposition 4.4 which is (4.10) in the case p = q + r. Let us therefore assume that p > q + r. Using Proposition 4.4 with n = p,

9 Hdx? i=l

q = V r + g ( e i , . . . , er,Xi, . . . , X 9 ) p - r " 9 I I d X i + 9 ^ r + g ( r ) 2=1

Next, let us use that, cf. Proposition 3.11 with (p, r, 5) = (p, r + g, r ) ,

dL

r+q(r)dLp(r)

=

dL

p(r+q)dLr+q(ry

Combining with (4.11) and using g • V™+£ as a short notation for

g(xi,...,

xq)Vr+g(ei,...,

e r , r r i , . . . , z g ) n ~~ p ,

C4"11)

4.4 LOCAL ESTIMATORS OF VOLUME

105

we rewrite the right-hand side of (4.10)

/

/ ••• /

C

p{r)

C

9-K^f[d^dL;{r)

X L

r+q(r)

^ r+q

%

XqDLr+q

~

= j I j ■■■ j ^^-?n^xv9)^w C

r+q(r)

= J C

r+q(r)

X

l^Lr+q

^(r+q)

j

-

XlHLr+q

%

XqHLr+q

j

~

9-^ri

J dL;{r+q)]f[dX^dL-+q{r) C

XqHLr+g

p(r+q)

= c(n-r-q,p-r-q)

j

J

£r+q(r)

= c(n-r-q,p-r-q)

...

Xl<~MT+q

^

J

g•V ^

f[ ?

XqClLr+q

dx^dL«+q{r)

~

/ . . . / g ]~[ dxf. Xr

l=1

Xq

D

4.4 Local estimators of volume Let X be an open and bounded subset of Rn. Let us suppose that we want to estimate its volume, using information in an isotropic p—subspace containing a fixed r—subspace Lr, say. Using Proposition 4.5 with q — 1, we get c(n - 1 - r,p - 1 - r)dxn = \\^ux\\n~p

dxp dLnv{r).

The Horvitz-Thompson procedure therefore leads to the following local estimator of volume

dL;{r)

y>7, •^—'

,

c(n — r,p — r)

= sTdxn- An-1-^p-1-r). 2-" 0~p—r

[

c(n-r,p-r)

dxn

]-i

\\irL±x\\n-PdxPi

106

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

where the sum is over all infinitesimal volume elements of X hit by the isotropic p—subspace Lp. We shall use the continuous version denoted by mJ)(XnL,;Ir) = ^

/

K x z i r W .

(4.12)

xnLp It is possible to express m^K in terms of m _ ~ / L as shown in the proposition below. (n)

Proposition 4.6. Let X be an open and bounded subset of Rn and let rrv ,\ be given by (4.12). Then, m§>r)(X n Lp] Lr) = j m^-^iiX

- y) n Lp e Lr;

0)dyr.

Proof. We have, using (4.9) at (*), ^zL.m<£(XnLp;Lr)

= J Kxxirw xnLp

\\nLi.x\\n-?dxP

J xn(Lr®(LpeLT)) =

/ Lr

\\z\\n-pdzp-rdyr

/ (X-y)n(LpeLr)

= ^zitl. vfn—r)—0

f m^((X

-y)nLpe Lr- o)df.

J Lr

D One of the most interesting cases from a practical point of view is the case where X is intersected by a line. Here, mg)(XnL1;0) =

f

^

y

J xnLi

||x|rW.

4.4 LOCAL ESTIMATORS OF VOLUME

10?

In case O € A" and X is star-shaped relative to O. i.e. X r L\ is a line-segment for all L\ £ Cl{, cf. Figure 4.4» we get m%(X n Li: O) = _^L-(||*_||» + | | . T + n .

(4.13)

where x_ and x + are the end-points of the line-segment A' n L1.

Figure 4.4. A' is star-shaped relative to O.

In case XC\L\ consists of a countable number of line-segments» we can generalize the above result. Let us divide L\ into a positive and a negative half-line» separated by {OK L\ = L\+ U {()} U I i _ . Since A'O L \ consists of a countable number of line-segments» dXnL\+ and are countable, sets. Let us consider dXnLl+

dXnL\-

= {.n..r 2 ....}.

where the points are numbered according to increasing distance from O. We now associate to each x either 0 or 1 in the following manner. Let ,

a(.vi)

,

(() if \0..r{ l : C A

= <

.

~

{ I otherwise. Furthermore, for / > 1, let (I

\fa(xi-{)

= Q

:

'■

1«

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

The sequence {a(xi), a f ø ) , . . . } is therefore an alternating sequence of O's and 1 *s. For x € dX O Li_, a(x) is defined in the same manner, see also Figure 4.5.

Figure 4.5. Numbering of the points on dX n L%. The line L\ is divided into two orientated half-lines, Li+ and Li_, separated by the origin 0, indicated by •.

Proposition 4.7. Let A" he an open and bounded subset of R'\ Let L\ e C" and suppose that A' n L\ consists of a countable number of line-segments. Then,

x f

;

xedxnLi

Proof. We have

^Ȁ(A-nl.iO)= /

||»irV

XDLi

= I n»irv+ I mr'dy1. XnLi+

A'nLi-

Let us consider the integral

/ iiwirv xni^ in more detail. First suppose that {O.xi) C A' such that -YnL1+ = (0,xi)U(T2jx3)U---.

4.4 LOCAL ESTIMATORS OF VOLUME

109

Then,

/ iiä/irv xm1+

= J \\y\\n-ldyl+

J \\y\\n-%i + ...

= f ibirv+(/ \\y\rldyi- j iMrV)+--(0,x a )

(0,x 3 )

(0,x 2 )

- £

(-DaW / iwrv.

Next, let us suppose that X Pi L 1 + = ( x i , X2) U (x3, X4) U • • •.

In this case, we also have

j \\y\rw = E c-1)^ / iwrw. Xf)Li+

xedXDL1+

(0æ)

For the negative part dX D Li_, the same formula holds. It remains to show that j

| M r W = ||s||7n.

(4.14)

(0,x)

Let UJ be a unit vector in i? n pointing in the same direction as x. Let / :Ä - Li r —> ra>. Then, it follows from the coarea formula, see also Section 2.4, that for any nonnegative measurable function h on L\ Jh(y)dy1 Li

=

Jh(f(r))dr. R

Setting h(y) = (4.14) follows immediately.

l{ye(0,x)}\\y\\n-\ □

110

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

The estimator (4.13) has an interpretation also in the case where A"'" L\ consists of more than one line-segment Thus» suppose that O e: A' and let starLY) he the part of A. which can be seen directly from O. i.e. star(A) = {.r G A : lO.s] C A } . cf. Figure 4.6. See also Serra (1982. 1988). Since stariA • " L\ consists of one line-segment for all L% € £f, (4.13) can be regarded as an estimator of start A ?.

Figure 4.6. The set *tar(A' > is shown hatched. Estimators based on subspaces of different dimensions are related by a so-called Rao-Blackwell procedure, i.e. one estimator can be obtained as a conditional meanvalue of the other. Using this result, it can be shown that the variance of the estimator decreases as the dimension of the subspace increases» cf. the proposition below. Preposition 4.8. Let Lp. be an isotropic pi—subspace containing Lr and let LP2 be an isotropic /w —subspace containing Lr, where 0 < r < pi < p-> < it. Then, Vax(m%\r)(X n I /}1 ; Lr)) > \ar(//^J r J (-Y n LP2: Lr)l Proof. From Proposition 3.11, we have / J

/(L;Ji,

j

J

£n , C!"2

t l l ^ c(n-r.pi-r)

dp-} - r,pi - r) r(n - r.p2 - r)

4.4 LOCAL ESTIMATORS OF VOLUME

ill

Therefore, we can generate an isotropic p\ —subspace LPl containing Lr by first gener­ ating an isotropic p2—subspace LP2 containing Lr and next an isotropic pi—subspace Lpi, contained in LP2 and containing Lr. Therefore, we have Var(m^r)(XnLpl;Lr)) = V a r ( £ ( m ^ r ) ( X n Lpi; Lr)\LP2)) + £(Var(m£j r ) (X n

Lpi;Lr)\Lp2)).

Now, E{rn(;\r){XfMPl;LT)\LP2) plWV

J

Pl

c(p2-r,Pl-r)

pit')

<j pi _ r

c(p2-r,pi-r)

<7Pl-r

J

C(p2-r,pi-r)

J

" L'

"

"M

y XnLP2

xnLP2 =

m

Sr)(InLP2;Lr),

where we at (*) have used Proposition 4.5 with (n,p,r,q) that Var(m^r)(XnLPl;Lr))

= {p2,Pi,r, 1). It follows

= V a r ( m ^ r ) ( X n LP2; Lr)) + £ ( V a r ( m ^ r ) ( X n Lpi;

Lr)\Lp2)).

>Var(mgr)(XnLP2;Lr)).

n Note that m$r)(X

n Lp;Lr) = V(X), if p = n,

which implies that V a r ( m ^ } ( X n Lp; Lr)) = 0, if p = n.

112

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

4.5 Local integral geometric formulae for powers of volume Let X be an open and bounded subset of Rn. In this section, we discuss local integral geometric formulae for V(X)q for general q, i.e. formulae where V(X)q is expressed as an integral with respect to p—subspaces in Rn. The term local is motivated by the applications we have in mind where X is a biological cell and O is the nucleus of X. We can regard X as a neighbourhood of O which is studied by a p—dimensional section through O. We concentrate on the case p = q and get from Proposition 4.3 (use g = 1, X{ — X, i = 1 , . . . , q) the following local formula / Pi

dLn m W ( I n y - ^ - ,

(4.15)

where m(n\XCiLq)

= c(n,q)

f '" J V g ( x i , . . . ,xq)n~q xnLg xnLq

f[dxq. i=1

Note that, in comparison with the notation used in Section 4.4, min\X

n L x ) = m JJ } (X n Li; O).

We can also express m^n^ (X D Li) as m ( n ) ( I n Lq) = c(n, g)A9(X n Lq)9E(V^

X n L 9 ),

where i?(Vg~ 9 ;X n L g ) is the mean-value of Vg(xi,...,x9)n~9, when x i , . . . , xq are independent and uniform random points in X n Lq. More explicit results are possible in case X is an n—dimensional ellipsoid centered at O. In this case the general local formula (4.15) reduces to the local formula given in the proposition below which is originally due to Furstenberg & Tzkoni (1971), see also Miles (1973) and Jensen & Møller (1986). A related formula can be found in Guggenheimer (1973). Proposition 4.9. Let X be an n-dimensional ellipsoid centered at O. Let be the open unit ball in Rn and cun = \n(Bn(0,1))

= TTHXI + i n ) " 1

Bn(0,1)

4.5 POWERS OF VOLUME

113

its volume. Then, V

^

qK

^J

q)

c(n,q)

Proof. Since X is an n—dimensional ellipsoid, there exists a n n x n positive definite matrix E such that X = lxeRn Furthermore, XnLq

: xTEx < l ] .

is a q—dimensional ellipsoid in Lq. Therefore, cf. Exercise 4.10, n

E{V q~q- XnLq)

= \{X

Lq)n-qE(Vnq-«-~TKTq),

n

where X Pi Lq C Lq denotes a set of unit q—dimensional Lebesgue measure which is identical to X n Lq up to a linear transformation in Lq. Since X n Lq is an ellipsoid, we can choose X n Lq as a ball, more precisely we choose

xnLq = Bn{0, ujql/q) n Lq, cf. Exercise 4.10. Therefore,

/

•••

Bn(0,w^1/9)nL,

/

V ^ ! , . . . , ^ - ^ ^ .

5n(0,u;-1/9)nLq

This multiple integral does not depend on Lq. Therefore, cf. Proposition 4.3, n q q q

E{v - -Trnr ) = c{n,qylj

j

...

^ Bn(ö:uj;1/q)nLq

^c^q)-1

J

••• l/q

Vq(x1,...,xq)n-qf[dxqdLq^

Bn(0,u;;1/q)nLq

J

f[dx? 1,q

Bn(o:uj; ) =

j

Bn{o,uj- )

l=1

c(n,q)-\UBn(0^1,9W

= c(n,q)~l{ujqn'qujn)q = c(n,q)~Xujqnujqn. Finally, we get m W ( I H Lq) = c{n, q) • Xq(X n L,) 9 • A,(X 0 L , ) 7 2 " « ^ g ) - 1 ^ ^ " "

=

^xq(XnLqT. D

114

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

4.6 Exercises Exercise 4.1. Let the situation be as described just above (4.5). Show that h(x)dxr+s=f

/

f

Xn(Lr0Ls)

Lr

h(y +

z)dzsdyr.

(X-y)nLs

Exercise 4.2. Go through all the details of the proof of Proposition 4.3 in the case g=2. Exercise 4.3. As demonstrated in Miles (1979), we can make iterated use of Proposition 4.5 and thereby express integrals such as

J Xpn(X n Lp)dL; in terms of integrals along lines. His result is [ \pn(XnLp)dL^

= c(n-l,p-l)

f

££

f

\\x\\p~1dx1dL7{.

£? XDLi

In this exercise, we show this result by combining 1. and 2. below. 1. First show that / Xpn(X n Lv)dLnv = c(n - l,p - 1) /" ££

\\x\\-{n-p)dxn.

X

Hint. Use Proposition 4.5 with q = 1, r = 0 and /

\

II

II-

g(x) — ||x||

v

(n—p)

yj

.

2. Next show that f\\x\\-{n-p)dxn= x

f

I

\\x\\p-ldxldLr{.

c\ xnLi

Exercise 4.4. Let

m;;(InL

p

;Lr)^

/

Up—r

J

xmp

H^sirW,

4.6 EXERCISES cf.

115

(4.12).

1. Show that if Lp is an isotropic p—subspace containing Lr, then

%J»(int p ;L r )) = y(i), where V = Xn = A^ is Lebesgue measure in Rn. Hint. Use Proposition 4.5. 2. Show that the variance of rrvVJX C\ Lp\ Lr) is 0, if X is a ball with centre O, i.e. X — Bn{0,R). Hint. Show, using the coarea formula, that for all Lp e £ V \ , / \\*Lirx\\n-pdx? xnLp

=

(a;?+1 + .-. + a;2)( n -P)/ 2 dx p -..da:i.

f BP(O,R)

Exercise 4.5. Let h G Rn and let X be an open and bounded subset of Rn. Let

xh = {xex

:x + heX}.

Then, V p O j is a deterministic analogue of the covariance of a random closed set, cf. Stoyan et al. (1995). Show that the Horvitz-Thompson estimator of V(Xh) based on an isotropic p—subspace Lp containing the fixed r—subspace Lr is

^ ^ &p—r

/

heX}\\irLi.x\\n-pdxp.

l{x +

J

xnLp Note that this estimator involves observation on Lp and Lp + h simultaneously. Exercise 4.6. Let us consider the local estimator of volume based on an isotropic plane L2 through O, cf. (4.12),

= 2 j \\x\\dx2. xm2

m^)(XnL2;0)

In this exercise, we will discuss subsampling of X n L2 by means of a uniform line grid in L2. 1. Let L\ be any line through O, contained in L2. Show that mg)(XnL2;0) = 2

/

/

L2GLi xnL2n(Li+u)

Hrrllda;^ 1 .

(•)

116 2.

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA Let Gi be a uniform line grid (1-grid) in L 2 , parallel to L\ and with distance A\ between neighbour lines. Show that m g ) j l ( X n L 2 ; 0 ) = 2Ai

\\x\\dxl

j xr\L2nG1

can be regarded as a Horvitz-Thompson estimator of

m^)(XnL2;0). (3)

Hint. Using (•), m ^ ( l n l 2 ; 0 ) can be regarded as a population total. The objects of the population are infinitesimal line-segments in L2 0 L\ and the characteristic associated with such a line-segment, centred at u G L2 0 L\, is

2[

/

Ikll^ 1 ] x du1,

xnL2n(L1-\-u) where c^ 1 is the length of the line-segment. The line grid is Gi = {Li + (£/ + jAi)o) :j = . . . , - 1 , 0,1,...}, where Co is a unit vector spanning L2QL1 and £/ is uniform in the interval (0, A±). Note that the probability that G\ hits a line-segment of length du1 is dv}/Ai. (3)

3.

The estimator wi^ro) i ( ^ n ^2; O), introduced in the previous question, is a sum of the contributions over the individual grid lines. Show under the assumption that Xf\{L\ + u) consists of a countable number of line-segments that the contribution from a specific grid line L\ + u, say, is

æea(xnL2)n(Li+u)

4.

0

where a; is a unit vector spanning L\. The function a is defined in the same manner as in Section 4.4, the only difference is that here X is replaced by X n L2 and O by u. Show that

V V2 +

/

hLiQL^dv

0

= i|(x,u,>| • iixll + ^ ^ 2

l o g (

2

K ^ ) l + INI). iFLaGLi^ll

4.6 EXERCISES

117

Exercise 4.7. The local estimator of volume based on a vertical plane L
n L2] Lm)

\\7rL±^x\\dx2.

= 7T J Xr\L2

Let G\ be a uniform line grid in L2, with lines perpendicular to L^y line through O, parallel to the grid lines. Show by the same type of techniques as in Exercise 4.6, that

m™)tl(XnLr,Lm)

= \AX

Let L\ be the

(-l)a(x)lkL^II2

Y. x£d(XC\L2)PiGi

is the Horvitz-Thompson estimator of m^U) ( ^

n

^2; ^i(o))> when sampling with G\.

Exercise 4.8. This exercise provides an alternative proof of Proposition 4.8. Let the situation be as in this proposition. Because E(m™r)(X

n Lpi-Lr))

= E(m^\r)(X

n LP2; Lr)) = V{X),

cf. Exercise 4.4, it is enough to show that n Ln- Lr)2) > E(m^\r)(X

E(m^\r)(X

n LP2; Lrf).

(.)

In order to show this relation, solve the problems below. 1. Show that m{n)(JXnLpi;Lr)2-

/

^

Pl(r)

>m£\r)(XnLp2;Lr)2. Hint. Use that J7-P2

m{n)(JXnLV2]Lr)= p 2 ( r )V

[ m{n] JXnLPl;Lr)-

P2»

T)

J

pi(r)V

pi,

^ rj

cf. the proof of Proposition 4.8. 2. Show (•) by using question 1 and r dLnpi{r) f{L

J

^c(n-r!Pl-r)

Piir)

r

= J

r

f(Lpl)-

J

C

P2(r)CPpl(r)

cf. the proof of Proposition 4.8.

dLp\,

EI(±_

-,

c^_r^^_ry

dLn,,

efc)

c(p2-r1p1-r)c{n-r,p2-r)

118

4. CLASSICAL BLASCHKE-PETKANTSCHIN FORMULA

Exercise 4.9. Try to find an expression for the variance of mJ(XnL2;0) = 2

f

\\x\\dx2,

XCiL2

when L
£(V£-*; XnLq)

XTTX^),

and that X f) Lq can be chosen as a g-dimensional ball. We can identify Lq with Rq and X n Lq with an ellipsoid Xq in Rq of the form Xq = {x e Rq : xTZqx

< 1},

where E 9 is a q x q positive definite matrix. Recall that

x

x

<=i

q{

q)

Below, we let Aq be the q x q matrix given by A.q

=■ LOq

Ltq

1. Show that AqXq = Bq{0, uj~1/q). Note that Bq(0, oo~1/q) has unit g-dimensional Lebesgue measure. 2. Show that Xq(Xq) = Hint. Use that \q{AqXq)

=

det{Aq}-\

det{Aq}Xq(Xq).

4.7 BIBLIOGRAPHICAL NOTES

119

3. Go through the following derivations E{Vnq-i-Xq)

= /

-

V9(Vm,..,V%)n_9det{Aqrn^M_

/

/\qS\.q

J\qJ\.q

J\qJ\.q

J\qJ\.q

= det{A,}-("-«> j J\qJ\q

=

•.. |

V,( y i ) ...,j,,r-«ndj,?.

J\qJ\.q

n

\q{Xq) -«E{V^AqXq).

4.7 Bibliographical notes The Blaschke-Petkantschin formula is due to Blaschke (1935b), see also Blaschke (1935a) and Petkantschin (1936). To be more precise, in Blaschke (1935b), an affine analogue of (4.3) is proven, but (4.3) is easily derived from the affine version and vice versa, as demonstrated in Miles (1971). In fact, (4.3) was evidently only fully stated for the first time in Miles (1971). An affine version of local stereology will be mentioned briefly in Section 9.2. The Blaschke-Petkantschin formula can be extended in various ways, cf. Miles (1979) and Propositions 4.4 and 4.5 of the present chapter. In Zähle (1990) and Jensen & Kiéu (1992b), a generalized version is presented where Lebesgue measures are replaced by Hausdorff measures. This generalized version is the subject of the next chapter. There are at least two types of proofs of the classical Blaschke-Petkantschin formula available, a proof based on manipulating differential elements, cf. e.g. Santalö (1976, p. 200-201), and a proof based on invariant measure theory due to Møller (1987). In this chapter, we have presented an alternative proof based on induction. The ideas behind this proof can be found in Miles (1971, p. 362).

Chapter 5

The generalized Blaschke-Petkantschin formula The generalized Blaschke-Petkantschin formula concerns a geometric measure decomposition as the classical one, but, instead of Lebesgue measures, Hausdorff measures are considered. The formula is relative new (from around 1990) and one of the motivations for deriving this formula has been a need for constructing local stereological estimators of length and surface area.

5.1 A local estimator of length in R2 Let X be a smooth planar curve of finite length L(X). Let us assume that we want to estimate L{X) by local methods, using an isotropic line through O. Let us consider an infinitesimal curve element U(x) around x G X. In accordance with the usual notation, the length of U(x) is denoted dx1. If a{x) is the angle between the curve element and the line L\, we have sma(x)dx1

dO = 27r||a;|| • — = ||x|| • dO,

(5.1)

where d9 is the length of the angle interval for which L\ hits U(x), cf. Figure 5.1. This infinitesimal relation can be derived by considering the 3 rays emanating from O and passing through x and the 2 end-points of the curve element U(x), respectively, as 'locally' parallel in the neighbourhood of x. The sampling probability of the infinitesimal curve element becomes .rr/ , rtx dO 1 sin a(x)dx1 P(U{x) H Li / 0) = — = rr-TT . 7T

121

7T

||X||

122

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

Figure 5.1. Sampling of curve elements by means of an isotropic line L\ through O.

In contrast to the case where area is the parameter to be estimated and more generally n—dimensional Lebesgue measure, cf. Chapter 4, the sampling probability may here well be 0. In the example shown in Figure 5.2, the sampling probability is always 0 since any of the line-segments emanates radially from O. We therefore need a regularity condition on the curve in order to ensure that the sampling probabilities are non-zero. We may simply assume that a{x) > 0 for almost all x E X. Under this assumption we can construct the Horvitz-Thompson estimator of length

The 'continuous' version is used as an estimator of length L(X)

= TT I

||ar||/sinof(a;)dar0.

XfUi

The transformation result (5.1) is the simplest example of the generalized BlaschkePetkantschin formula. Usually, the result is expressed as sma(x)dx1

= \\x\\dxQdL\.

5.2 PREREQUISITES CONCERNING G-FACTORS

123

Figure 5.2. Example where any curve element has zero sampling probability.

5.2 Prerequisites concerning G—factors As illustrated in the previous section, it becomes important how the manifold X is positioned relative to the intersecting subspaces. This feature is captured by the G—factors introduced in Section 2.6, see also Section 2.8. Let us recall the definition of the G—factors. Let Ld be a d—subspace and Lq a q—subspace of Rn such that 0 < q < n and d — n + q > 0. Suppose that Ld n (Lq H L d ) x

(5.2)

has dimension n — q. Then, G(Ld,Lq) where a i , . . . , an-q

= V n _ 9 (7r L j_ai,..

.,irL±an-q),

is an orthonormal basis of the subspace (5.2).

For n — 2 or 3, we have seen in Chapter 2 that G(L^, Lq) is sina where a is the angle between Ld and Lq, cf. Exercises 2.10 and 2.14. A similar interpretation holds for arbitrary n, cf. Baddeley (1984, p. 49). Below, we derive a relation between G—factors for different subspaces which is useful in the induction proof of the generalized Blaschke-Petkantschin formula. Proposition 5.1. Let Lj be a d—subspace of Rn and let Lq C Lp be a. q— and a p—subspace, respectively, such that 0 < q < p < n. Suppose that d—n+q > 0 and that

dim(Ld n(LqnLd)

) = n - q.

Then, dim(Ld

H (Lp n Ld)L)

=

n-p

124

5. GENERALIZED BLASCHKE-PETK ANTS CHIN FORMULA

and G(Ld, Lq) = G(Ldl Lp)G(Ld

n Lp, Lq).

(5.3)

Proof. The trivial cases q = 0 or p = n can be dealt with directly, using Definition 2.9. So let us assume that 0 < q < p < n. We have

Ld n (Lq n Ld)L = Ldn (Lp n Ld)L ®LpnLdn

(Lq n Z^)-1.

(5.4)

We know that the dimension of the left-hand side of (5.4) is n — q which is also the sum of the dimensions of the two subspaces on the right-hand side of (5.4). Furthermore, dim(Ld

n (Lp n Ld)L)


(5.5)

and dim(L p r\LdC\ (Lq n Ld)L)


(5.6)

cf. Exercise 5.1. It follows that the inequalities in (5.5) and (5.6) must in fact be equalities. Let a i , . . . , an-p be an orthonormal basis of Ldn(Lp fi Ld) and a n _ p + i , . . . , an-q an orthonormal basis of Lp n Ld n (L g flL^) . Then, G(Ld, Lp) = V n _ p ( 7 r L ± a i , . . . ,

7rL±an-p)

and G(Ld H Lp, L g ) = V p _ g ( 7 r L j . a n _ p + i , . . . ,

ixL±.an-q).

Furthermore, G(Ld, L g ) = V n _ 9 ( 7 r L ± a i , . . . ,

7rL±an-q).

Since ^

= ^

(5-7)

+ ^nZ^

and 7YL±ai = 0,i = n — p + 1,...

,n — q,

we get G(Ld,Lq)

= V n _ 9 (7T L j.ai,... , 7 r L j . a n _ p , 7 r L p n L j . a n - p + i , . . .

,nLpnL±an-q).

For 2 = 1 , . . . , n — p, {^LpnL^ii

^LpDL^n-p+lj • • ,

are linearly dependent, since dim(Lp irL±ai,i

^LpHL^n-q}

D L^) — p — q. Therefore, using (5.7) on =

l,...,n-p,

5.2 PREREQUISITES CONCERNING G-FACTORS

125

we get G{Ld,Lq) = V n _ 9 (7r L ±ai,... ,7T L ±a n _p,7r LpnL j„a n _p + i,.. • ,nLpnL±an-q) = V n _ p (7T L ±ai,..., 7TL±an_p) • Vp_ g (7r L p n L j.a n _p + i,..., nLpnLi.an-q) = G(Ld, Lp) • Vp_g(7r L j.a n _ p + i,..., -KL±.an-q) — G(Ld, Lp) - G(Ld n Lp, L g ).

D Note that G(Rn1 ■) = 1. The G—factor can also be simplified in case d = n — 1. Thus, we have Proposition 5.2. Let L n _i be an (n - 1)—subspace of Rn. Suppose that 0 < q < n and that L n _i n (L9 n Ln-i)- 1 has dimension n — q. Then, G(L n _i,L 9 ) = 117T£,q a^ 11, where a n is a unit normal vector of L n _i. Proof. Consider the following orthogonal decomposition of Rn

Rn = {Lq n Ln_i) e (Ln_i n (L9 n L^)- 1 ) © L^_x. Let ai,...,an-q be an orthonormal basis of L n _i n (L g f)L n _i) and let a n _ g + i , . . . , a n _i be an orthonormal basis of L 9 n L n _ i . Then, using Proposition 2.15 G(Ln-i,Lq)

= V n _ 9 (7T L j.ai,...

y7rL±an-q)

= V g ( 7 T L g a n - g + l , - • • ,7TL 9 «n-l,7rL g «n) = Vg(an_9+i, . . .,

dn-iiTTLqCln).

Since 7r£gan G (Lg n L n _ i ) , we get G(L n _i, L 9 ) = V 9 _ i ( a n _ 9 + i , . . . , a n _i) • Vi(7r Lq a n )

= IKL^nll-

n

126

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

5.3 Decomposition of a single Hausdorff measure The classical Blaschke-Petkantschin formula was proved in Chapter 4 by means of induction. The formula was first proved for lines. Then, two decompositions involving lines were combined to a decomposition for planes and so on. This idea cannot be used for a manifold X of arbitrary dimension d, since for such sets only subspaces with a dimension p satisfying p > n — d can be considered. Therefore, only in the case where d = n or n — 1 we can start the induction with lines. However, the induction technique can still be used. Thus, in this section, we present an induction proof for the generalized Blaschke-Petkantschin formula for a single manifold X. The idea is here to start with subspaces with the highest possible dimension, i.e. p — n — 1 and then move down in the dimension until the limit p = n — d has been reached. In the next section, we present an induction proof for the case where more than one manifold are involved in the decomposition. The following lemma will be useful. Lemma 5.3. Let Ld be a d—subspace and Lq a q—subspace of Rn such that 0 < q < n and d — n + q > 0. Then, åim(Ld H (Lq n Ld)1) = n-q,

(5.8)

G(Ld,Lq)>0.

(5.9)

if and only if

Proof. It follows directly from the definition of the G-factor that (5.9) implies (5.8), cf. Definition 2.9. Let us now suppose that (5.8) is satisfied. We then want to show (5.9). The case q = n follows directly from Definition 2.9 so let us assume that q < n. Then, (5.9) can be shown by induction in q starting with q = n - 1 and moving downwards in the dimension. Thus let us suppose that q = n-l. Then, (5.8) says that Ld n ( L n - i n L^1 has dimension 1. Let a be a unit vector spanning this subspace. The vector a does not belong to I / n - i , since a e L n _ i implies that

a e Ln_! n (Ld n (Ln_i n Ld)L) = {O}, which is a contradiction since a has unit length. Therefore, G(Ld,Ln-i)

=

IITT^^H

> 0.

5.3 DECOMPOSITION OF A SINGLE HAUSDORFF MEASURE

127

Next, let us assume that (5.8) implies (5.9) for all subspaces with dimensions at least q + 1. We want to show the result for Lq, satisfying (5.8). Let L 9 +i be any subspace containing Lq. Then, according to Proposition 5.1, G(Ld, Lq) = G(Ld, Lq+i)G(Ld n L g +i, Lq). The first factor on the right-hand side is positive because of the induction assumption. The second factor on the right-hand side is positive by similar arguments as the ones used in the case q — n — 1. The left-hand side is therefore positive. □ As mentioned earlier, the generalized Blaschke-Petkantschin formula can be proved by means of induction. The formula for one manifold is given in the propo­ sition below. Proposition 5.4. Let X be a d—dimensional manifold in Rn. Let g : Rn -> Ä+ U {0} be a non-negative measurable function. Let e i , . . . , er be r orthonormal vectors in Rn and let £ n / r \ be the set of p—subspaces containing Lr = span{ei,..., er}. Suppose that 0 < r < p < n and d — n +p > 0. Then, under the assumptions Af(InLr) = 0

(5.10)

and /

Xdn({x € X \ L p _ i : G(Tan[X, x], L p _i + span{x}) = 0})dLJ_ 1(r) = 0,

rn

(5.11) we have c(n — 1 — r,p — 1 — r) / g(x)dx x = 1 / ø(:r)||7rLx:r||n-^ c;{r) xmp

(5.12)

n Note that for n = 2, d = 1, p — 1 and r = 0 we get the decomposition result derived in Section 5.1 by elementary means. Also note that for d = n, Proposition 5.4 reduces to Proposition 4.5 with q = 1.

128

5. GENERALIZED BIASCHKE-PETKANTSCHIN FORMULA

The proof of Proposition 5.4 is by induction starting with the case p = // - i and r = n - 2 and then moving in the dimensions as shown, in Figure 5.3. We are not interested in pairs of dimensions (p, r) for which p < r, corresponding to the heavily hatched area in Figure 5.3. Furthermore, the case p = n is trivial and the case p
Figure 5.3. Illustration of the induction procedure used in Proposition. 5.4. Proof of Proposition 5.4. We start by giving a direct proof in the case (p, r) = (n - 1. n - 2). by means of the coarea formula. Without loss of generality we assume that X C /??'\Ln_L>. Consider the mapping / which takes x € Rn\Ln^2 into L„^-2 - spau{.r}. It is convenient to represent Ln--> -f span{x} by a unit vector wx € Sn^1 n .L;;„2 such that L r ,_ 2 -t- span{.r} = I „ _ 2 4- >pan{o;x}, cf. Figure 5.4. Then.,

The differential of the mapping / is x where Qx = [(x,e n _i}r„ - (^^"n)^n-i]/\\^L^ \\ *m a mit normal of Ln_2 + span{s}, and e w ^i and en have 'been chosen such that e i , . . . , e n is an orthonormal basis of Rn, cf. Exercise 5.2. Therefore,

keiDf(x)

= Ln-2 + span{.r}.

5 3 DECOMPOSITION OF A SINGLE HAUSDORFF MEASURE

129

Figure 5.4. Illustration for Proposition 5.4. It follows from the coarea formula and Lemma 5.3 that for x e X Jf(x: X) > 0 <=> G(Taii[X,x]. /,„_•> -f- span{.r}) > 0. Furthermore, if a,r is a unit vector spanning Tan|X,a:] O ((Ln^2 + span{.r}) n TauLY. .rl P thee

Jf(x;X) = ||D/(x)(a x )|| = ||7r L x_ 2 x|r 1 ||7r ( i n _ 2 + s p a i i { j . } r «,|! = l l T r ^ ^ l l - ^ C T a n f X , ^ . !„_•.> + span{.r}). Since / - 1 ( w ) U / _ 1 ( - w ) = (L„_ 2 ©span{^})\I„_ 2 for ^ € 5 " " 1 n L^_ 2 , cf. Exercise 5.2, a direct application of the coarea formula yields (g(x)Jf(x;X)dxd

=

f

=i

I

/"

girjcl./'-^lJ

I

g(x)dxd-1dui1

5»-i n £X_ 2 A'n(£„_ 2 espan{^})

=

J

J

g(x)dxd-idL^l{n_2y

130

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

Therefore, we also have, cf. (5.11) for p = n — 1 and r = n — 2,

J g(x)dxd x g(x)l{Jf(x]X)>0}dxd

= J x =

l

^n-l(n-2)

XnL

J^) 1 { J ^^ X ) > °} dX
l

n~l

The result (5.12) now follows because

J

J

= jl{Jf(x;X)

l{Jf(x;X) =

= 0}dxd-1dL"n_1(n_2)

0}Jf(x;X)dxd

X = 0.

The next step is to show that (5.10) and (5.11) imply (5.12) for p = n — 1 and arbitrary r, cf. Figure 5.3. Again, we assume without loss of generality that X C Rn\Lr. Using Proposition 3.9, we get, cf. Exercise 5.3, c(n — 1 — r, n — 2 — r) I g(x)dxd X

= c(n _ i _

r? n

_

2

- r)-1 f g(x)

= c(n-l-r,n-2-r)-1

f ^n-2(r)

^L^\dLnn_2{r)dxd

f

l{x £ Ln.2}

f

g(x) ^ * 1 \ d x d dLnn_2{r).

(5.13)

X

\Ln-2

Using (5.12) for (p, r) = (n - 1, n - 2) on X \ L n _ 2 (condition (5.10) is fulfilled and (5.11) is fulfilled almost surely), we find for almost all L n -2

/

gix)

ifo**ii dxä

X\Ln-2

= £

J

j

n-l<„-2)

(Jf\in-a)ni»_,

g{x)\\-K^x\\G{T^[X,x\,Ln^)-ldxd-ldLnn_1(n_2).

5.3 DECOMPOSITION OF A SINGLE HAUSDORFF MEASURE

131

Inserting this result into (5.13) and changing the order of integration, we get c(n — 1 — r, n — 2 — r) I g(x)dxd X 1

= c(n-l-r,n-2-r)-

J

J

x[ j

=

j

j

g(x)\\irLi.x\\G(Tm[X,xl

Ln-i)"1

l{x^Ln-2}dL^(r)]d2rd-1dL-.1(r)

^(x)||7rL^||G(Tan[X,x],Ln_1)-1^-1^_1(r).

ci_1{r) xnLn_i This is (5.12) for p = n — 1 and arbitrary r. Finally, we prove that (5.10) and (5.11) imply (5.12) for arbitrary (p, r). The proof is by induction in p, for fixed r, cf. Figure 5.3. Without loss of generality, we assume again that X C Rn\Lr. We have shown the result for (p,r) = (n — l , r ) . This is the induction start. Now, we assume that (5.10) and (5.11) imply (5.12) for (p + l , r ) . We then want to show that the same is true for (p, r). First notice that if (5.11) is fulfilled for (p,r), then (5.11) is fulfilled for (p + 1, r) (use Proposition 3.11, Proposition 5.1 and Lemma 5.3). Therefore, because of the induction assumption, c(n — 1 — r,p — r) j g{x)dxd X

Then, we want to use (5.12) with X replaced by X f i L p + i and (n,p, r) = (p + l,p, r). This corresponds to a case along the horizontal arrow in Figure 5.3. Notice that (5.10) is fulfilled because X C Rn\Lr, while (5.11)

J A^HxGlnVAV!: S-l(r)

G(Tan[X n Lp+i, s], L p _i + span{:r}) = 0})
132

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

is fulfilled for almost all L p + i , if (5.11) is fulfilled for ( n , p , r ) (use Proposition 3.11, Proposition 5.1 and Lemma 5.3). Therefore, for almost all L p + i ,

^(x)|KL^a:||n-p-1G(Tan[X,x],V1)-1^-n^+1

/ xnLp+1

= c(p-r1p-l-r)-1

J

I

^ll^xir-^Tan^o;],^!)-1

clt]xnLp x G(Tan[X n L p + i, x], Lp)~l

dxd~n+p

Combining these results, we get (5.12) for (n,p, r).

dLp^y D

For n = 2, d = p = 1 and r — 0 (the example of Section 5.1), the regularity condition (5.11) takes the form Xl({x e X\{0} : a(x) = 0}) = 0. 5.4 Decomposition of a product of Hausdorff measures The generalized Blaschke-Petkantschin formula is a geometric decomposition of a product of Hausdorff measures. Before proving this formula, we need the following lemma which is an extension of Proposition 5.4. The proof of the lemma is straightforward, but rather tedious and can be skipped at a first reading. The idea in the proof is to insert dummy sets of full dimension and use in various ways the classical Blaschke-Petkantschin formula together with Proposition 5.4. Lemma 5.5. Let the situation be as in Proposition 5.4, but instead of g we consider a non-negative measurable function h:RnxjC^->R+U

{0}.

Furthermore, below we let £™/ r+1 \ be the set of p-subspaces, containing L r + s p a n { x } , where x e X. Then, under the regularity conditions (5.10) and (5.11), we have

J J X

£

h(x,Lp)dL;(r+1)dxd

p(r+l)

= J J h^L^^xr-PGiT^X^lL^dx^+HL^y

(5.14)

5.4 DECOMPOSITION OF A PRODUCT OF HAUSDORFF MEASURES 133 Proof. Without loss of generality we assume that X C Rn\Lr. For p = r - f 1 , (5.14) is a direct consequence of Proposition 5.4 so let us assume that p > r + 1. Let h(xi,..

.,xp-r)

=/i(xi,span{ei,...,er,xi,.. .,xp-r}) x l { e i , . . . , e r , x i , . . . , x p _ r are linearly independent}.

Then, for any Lp containing Lr and any x\ G Lp\Lr, /

•••

Bn(0,i)nLp

/

we have

ft(rri,a:2,...,a;p_r)

JJdaf l=2

Bn{0,i)nLp

ujP~r~1h(xi,Lp)J

=

cf. Exercise 5.4. Therefore, we can rewrite the left-hand side of (5.14) as

I X

h(x,Lp)dL;(r+1)dxd

J C

p(.r+D

= -7+r+1J I

J ■■ J

X £ ; ( r + 1 ) Bn{0,l)C\Lp

Bn(0,l)nLp

p—r

Jl{xi,X2l ■ • .,Xp-r) Yl

dxP dL

i p(r+l)dxi-

(5'15)

i=2

Using Proposition 4.4 on the inner integrals, we get, since 6i, . . . , 6 r , —

—

IFL^III is an orthonormal basis of Lr + span{#i}, h(xi,X2,...,Xp-r)Y[dtfdLp(r+l)

/

/ ••• / c;(r+1)Bn{o,i)r\Lp Bn(o,i)nLp =

/

•'•

5„(0,1)

/

i=2

h(xi,X2,-..,Xp-r)

B„(0,1). p—r 1

x V p (ei,. • •, e r , / ^ * s 2 , • • -, * p - r ) " IFL^III

n+p

I T dx? f=2

Inserting into (5.15) and interchanging the order of integration, we find

134

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

I X

h(x,Lv)dLnp{r+1)dxd

I C

p(r + D

p—r

= ^ p+r+1 j

•-. j

B„(0,1)

g{x2,...,xp-r)l[dx?,

B„(0,1)

'

(5.16)

=2

where

ø f ø , • . . , Xp-r) = / h(xi,X2,

• • • , Xp_r)

x Vr>(ei,..., e r , - , X2,. • •, Xp—r) IFL^III

dx\.

We now use Proposition 4.4 once more and get P—T

^-

p+r+i

/

••• /

B n (0,l)

^2,...,* P _ r )n^? '= 2

5n(0,l)

C

;-Hr) 5n(0,i)nLp«i

Bn(0,i)nLp_!

x Vp-i(ei,..., er,x2j... ,xp_r)n-^+1 J ] ^ r 1 ^ _ 1 ( r ) . ( 5 . 1 7 ) 2= 2

Furthermore, using Proposition 5.4 (conditions fulfilled almost surely),

g(x2,...,Xp-r) =

h(xi,X2,.-.,Xp-r)

cnP,( P - I ) ^xmpp 7Tr±Xl

xVpei,..,er,-—-—f-,x2,...,xp_r)

_ n p

^

xllTr^^Xiir-PG^anlX^!],^)-1^-^^^.^.

(5.18)

5.4 DECOMPOSITION OF A PRODUCT OF HAUSDORFF MEASURES 135 Combining (5.16), (5.17) and (5.18), we get

I x

j

h{x,Lp)dL^r+l)dxd

L' i i -i \

= ^+'+'

/

/

-

c

p-nr) B»(o,i)nLP-i

h(xi,x2,.

/

/

/

B„(o,i)nLp_i c;(p_l} xmp n p

• •, x p _ r )||7r L i.xi|| " Vp_i(ei,..., e r , x 2 , . . . , xp-r) p—r

2=2

Interchanging the order of integration, using Proposition 3.11, and using Proposition 4.4 once more, we get (5.14). □ Note that if h(x, Lp) = g(x), then (5.14) reduces to the result in Proposition 5.4. We are now ready to formulate and prove the generalized Blaschke-Petkantschin formula for an arbitrary number of sets. The case where d = n is Proposition 4.5. Theorem 5.6. Let X{ be a a?*—dimensional differentiable manifold, i = 1 , . . . , q. Let g : Rn x ••• x i T -► Æ+ U {0} q

be a non-negative measurable function. Let e i , . . . , er be r orthonormal vectors in Rn and let L™, , be the set of p-subspaces containing Lr = span{ei,..., e r }, where p > q + r and d2■■ — n + p > 0, i = 1 , . . . , q. Suppose that for all i = 1 , . . . , q /.-• Xi

/ £

/ \i(Xin(Lr+spzn{x1,...,xi-1}))dxfs11'--dxd11

=0

(5.19)

Xi-i

Aj({xi G Xi\Lp_ g + i _i : G(Tan[X i} a:i],Lp_ g+ i_i + spanfø}) = 0})

p-g+i-l(r)

d^_,+i_1(r)=0.

Then,

(5.20)

136

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

c(n-q-r,p-q-r)

=

/

\ x1

" \ g{xi,... xq

,xq)J[dxf i=l

g(xii'--iXq)Vr+q(ei,...,er,X1,...,Xq)n~P

'"

C^XxViL,

XqHLp

f[ G(T^[Xi,Xi], 2=1

Lp)-1 f l dxf-n+pdLnp(ry

(5.21)

i=l

Proof. The proof is by induction in q, starting with q = 1. For q — 1, the result is just Proposition 5.4. Now, let us suppose that (5.19) and (5.20) imply (5.21) for q — 1 sets. Let us consider X±,..., Xq satisfying (5.19) and (5.20). Below, we use that I

A*({*i É X i W *

:

G(Tan[Xi,m],L p - g + span{xi}) = 0})dL£_ q(r) = 0

p-q(T-)

/ £

>&({xi e Xi\Lp_ g + i _i : G(Tan[Xi,xi],L p _ 9 + i _i +span{xi}) = 0})

p-g+i-l(r)

This can be seen by combining Proposition 3.11, Proposition 5.1 and Lemma 5.3. We first want to use the induction assumption to rewrite / "• / x2 xq

9(xi,X2,...,Xq)Y[dx*i i=2

in terms of an integral with respect to £ n , ^, the set of p—subspaces, containing Lr + span{:ri}, where x\ £ X\. For this purpose, we need to check the regularity conditions (5.19) and (5.20) for Y\ — Xi+\,i = l,...,q — l. Below, we let gi = di+i, i = 1 , . . . , q — 1. Since (5.19) is satisfied for X\,..., Xq, we have for i = 1 , . . . , q - 1 and A^1 -almost all x\ G X\, I Yi

I A ^ ( ^ n ( [ L r + span{xi}]+span{^ 1 ,...,^_ 1 })) n ^ f Yi-i

i

=1

=0.

5.4 DECOMPOSITION OF A PRODUCT OF HAUSDORFF MEASURES 137 Furthermore, for i = 1 , . . . , q—1, we have for A^1 —almost all x\ G X\, cf. Lemma 5.5, ^ndVi € Yi\LP~q+i : G(Tan[Yi, 2/i], Lp-q+i + spanfø}) = 0})

/

dL

p-q+i(r+l) = °-

Since p - q + i = p - (q - 1) + i - 1, (5.19) and (5.20) are fulfilled for Y i , . . . , Yq-i. Using the induction assumption, we therefore get for A^1 — almost all x\ G X\, c(n -(q-l)-(r

+ l),p-(q-l)-(r

X / "• /

9{xi,X2,---,Xq)J\dxf

X2

i=2

Xq

=

+ 1))

'"

9(xi,x2,--.,xq)

7TL±X\ _ X Vr-|_g(ei, . . . , 6r, j . rr, ^ 2 , • • • , Xq)

IKz^ill x H G(Tan[X,, *<], Lp)" 1 f j dxf ~ n + ^ Z £ ( r + 1 ) . 2= 2

(5.22)

2= 2

Let us abbreviate the right-hand side of (5.22) as J

A(a:i,L p )dL~ (r+1) .

Using Lemma 5.5 at (*) below, we now find c(n-q-r,p-q-r)

'"

9(xi,x2, •.. ,xq) J J cfcrf

p(r + l)

=> I

J

c;(T)x1nLp

h(x1,Lp)\\wux1\rPG(Tzn[Xux1},Lp)-1dxi1-n+pdL;{r)

138

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

=

/

/

"

£^{r) XxHLp X2nLp

9{xi,x2,.-.,xq) XqnLp

x V r + g ( e i , . . . , e r , irL±xi,x2, q

• ■ •,

xq)n~p

q

x Y[ G(Tan[X 2 , s j , Lp)~l 2=1

]J dxf~n+p

dL;{r)

(5.23)

2=1

Since V r + 9 ( e i , . . .,er,7TL±xi,X2,..

-,xq)

= V r + 9 ( e i , . . . , er,xi,

x*i,...

,xg),

(5.23) is (5.21) for q sets.

□

5.5 Local estimators of d—dimensional Hausdorff measure We can construct local estimators of d—dimensional Hausdorff measure according to the same principles as we used for volume in Chapter 4. Let us consider a d—dimensional differential manifold X C Rn with X^{X) < oo. Proposition 5.4 then gives the relevant decomposition (g = 1), c(n-

1 -r,p-

1 -r)dxd

= \\7rL±x\\n-pG(T<m[X,x},

Therefore, the probability that an isotropic p—subspace infinitesimal element of X becomes dL

p(r) _c(n-l-r,p-l-r) c(n — r,p — r) c(n — r,p — r)

Lpy1dxd-n^pdL^ry Lp, containing L r , hits an

G(Tan[X, x),Lp)dxd \\7rL±x\\n~Pdxd~n^~P

We will here assume that the regularity conditions of Proposition 5.4 are fulfilled. In particular, we assume that d — n + p > 0. Using these sampling probabilities, we can suggest a local estimator of X^(X), based on an isotropic p—subspace Lp containing a fixed r— subspace Lr, see also Section 5.1. Since c(n — 1 — r,p — 1 — r) c(n-r,p-r)

crp-r cr n _ r

we get the following estimator

m^\x,Lp;Lr)

= ^ ^ &p—r

f J

xnLp

\\*Lix\\n-''G(TtaL[X,x],Lp)-1dxd-n+P.

(5.24)

5.5 LOCAL ESTIMATORS OF HAUSDORFF MEASURE

139

In contrast to the local estimator of volume presented in Chapter 4, the estimator m^lMx, Lp; Lr) generally depends on the pair (X, Lp) and not only on their inter­ section X H Lp. For p = n, the estimator reduces to

m^(X,Ln;Lr)

= Xdn(X),

the trivial estimator. As mentioned above, we assume in this section that the regularity conditions of Proposition 5.4 are satisfied. If condition (5.10) is not satisfied, however, we may use the estimator (5.24) on X \ L r , since X^(X D Lr) can be observed directly in Lp. The modified estimator becomes

Xdn(X n Lr) + ^ ^

\\7rL±x\\n-pG(Tan[X, x],

/

Lp)'1dxd~n^'.

{X\Lr)nLp

For d — 0, the condition d — n + p > 0 implies that p = n. Thus, we have no non-trivial local estimator of counting measure. Alternative methods must be used, cf. Chapter 6. The case d = n — 1 is also of special interest. Here, p can take the values 1,.. .,n. In particular, we have a local estimator of X^~1(X) based on information along a line L\

1 m^- \X,L1-0)

= ^

-

£

llsir^CMA-.sl.L!)-1.

(5.25)

Note that if ax is a unit normal vector of Tan[X, x] and UJ is a unit vector spanning Li, then we find from Proposition 5.2 that

G(Tan[X,x],Li) = HTT^^H = \((J,ax)\ = sinof(x),

where a(x) is the angle between L\ and Tan[X, x], cf. Figure 5.5.

140

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

Figure 5.5. Illustration of the notation used for the local. estimator of ÅJJ""1(X)» based on an isotropic line L\ through O.

For d = n, m / ? = m ,K, the local estimator of volume presented in Chapter 4. Example 5.7 (the planar case). For n = 2 and p — 1. the condition d — n + p > 0 implies that d = 1 or 2. As special cases of the general estimator (5.24), we therefore ha¥e local estimators of area and length in the plane

mg ) (X.L 1 ;0) = 7T J \\x\\dJ m<jt$(X.Ll;0)

=n

]T xexnLi

\\,-\\/sma(x).

These estimators of planar area and length may be useful alternatives to the more tra­ ditional estimators» based on point and line grids» cf. Chapter 7 on the implementation of local methods. In Figure 5.6, we illustrate the measurements involved» in case two perpencular lines are used. Usually, such a systematic set of lines» passing through O, is used. Details will be described in Chapter 7. Example 5.S (the spatial case, fi=3). All possible cases of the estimator (5.24) are listed in Table 5.1. The angle a(x) in Table 5.1 is always the angle between Lp and Tan[X,x], For d = 3 and (p,r) = (1,0), the result given in Table 5.1 is Proposition 4.7.

5.5 LOCAL ESTIMATORS OF HAUSDORFF MEASURE

141

Figure 5.6. Estimating planar area and length» using a local systematic design. Note that the estimators of length and surface area involve the determination of spatial angles, which is difficult in practice» cf. Chapter 7. In the case of length» only estimators based on planar sections are available, in the case of surface area, one may choose between line and plane sections and spatial angles may be replaced by planar angles» cf. Section 5.6. D

d

(PS)

_

1

71

2

27T

YL xexnii

iWf/siaa(x)

3

!* £

(-ir"Vii s

zedxnLi

(2,0)

(2,1)

(LO)

J2 xexni2

\\wL±xiysmQ(x)

2

7T J ||irLxæ||/sina{x)cij1 xnL2

2

7T J

%

||7T£±XpJ2

Yl xexni2

J ||*||/ sin xni-2 2

Xf\L2

lkll/sina(x)

/ xm2

a(x)dx1

\\x\\dx*

Table 5.1. Local estimators of length, surface area and volume in R? (d=t,2 and 3), based on an isotropic p-subspace containing a fixed r-subspace.

As in the case of volume, it can be shown that if Lp is an isotropic p—subspace containing Lr, then E(m%$(X,Lp;Lr))

=

\dn(X).

142

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

Furthermore, if LPl and LP2 are isotropic p\containing L r , 0 < r < p\ < P2 < n, then

cf.

and £>2-subspaces, respectively,

Exercise 5.5.

5.6 An alternative estimator of surface area A™_1(X) Let us suppose that X is an (n—1)—dimensional differentiable manifold with finite surface area X^~1(X). The local estimator of A™-1(X) constructed in the previous section, based on an isotropic line L\ through O, takes the form m^-

1

)(Jf,L1;0) = -

Ikir'GCTan^x],^)-1,

^ J2 xexnL1

L{Jl/z,

where G(Tan[X, x], L\) = sin a(x) and a(x) is the angle between L\ and Tanpf, x], cf. Proposition 5.2. Note that a (a;) cannot be determined from information along L\ alone but requires knowledge of the tangent space Tan[X, x]. In practice (n = 3), spatial angles such as a{x) are difficult to determine, while angles in planar sections are easier to determine, cf. Chapter 7. Therefore, we want to replace a{x) with a planar angle, cf. Figure 5.7. More generally, with an angle measurement in a q—subspace. Let Lq be an isotropic #—subspace containing L\. Then, as we shall see below, a(x) can be replaced by a function of the angle (3(x) between L\ and Tan[X Pi Lq,x], cf. Figure 5.7. Note that G(Tan[X n Lq, x], L{) = sin p(x). We will determine a function h of G(Tan[X Pi Lq, x], L\)~ average get the right answer, i.e.

such that we on the

£(/i(G(Tan[X n Lq, x], Li) _ 1 )|Li) = G(Tan[X, x], L i ) - 1 The resulting estimator becomes n/2

rhq^\x,Lq;L1)

= Y^—

]T

\\x\r1

h(G(Tzn[X n Lq, x], LO" 1 ).

Note that condition (5.26) implies that E(m^(X

n Lq-Li))|Ii)

= mføj - 1 ) (X,

W,0).

(5.26)

5.6 AN ALTERNATIVE ESTIMATOR OF SURFACE AREA

143

As we will see below, there exists exactly one analytic function, satisfying (5.26), viz. a so-called hypergeometric function. Let us here give the definition and a few basic facts about hypergeometric functions. A more extensive list can be found in Gradshteyn & Ryzhik (1965).

Figure 5.7. For it=3» the alternative estimator of surface area uses the planar angle /3(x) instead of the spatial angle a(x). A hypergeometric series is a series of the form k-i

k-i

* n (n + o n (■*+') k 2=0

The coefficient of ;:A' is for k = 0 equal to 1. The parameters n. ,^7 can assume 'arbitrary real values except for ~ = 0 . - 1 . - 2 If a or .1 are zero or negative integers» the series terminates after a finite number of terms and is a polynomial in z, The series is absolute convergent if \z\ < 1. In case 0 < 3 < % we can also represent the hypergeometric series by an integral» cf. Exercise 5.10, 1

F(a, /?; r, z) =

B{^_0)

/ (1 - *3/)~ V - 1 ( l ~ 0

V?'^^-

144

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA Using Proposition 5.1 and Proposition 5.2, we can rewrite condition (5.26) as f J

h {

dL th^) «U =_ J _ , IkLia>x\\ c(n- l , g - 1) H^a^l'

(5.27)

9(1)

where a æ is a unit normal of Tan[X, x\. Furthermore, if (l-zf2h((l-z)-1<2)

h(z) = we can write (5.27) as /

WLqo>x \\h\—n q

J £

iT2—>>~7

XFL^xV

TT =

1

L

^5-28)

c(n-l,q-l)

g(i)

In the proposition below, we show that this equation has a unique analytic solution. Proposition 5.9. The equation (5.28) has the unique solution h{z) =

F{-\,-l-{n-q)-l-{q-l)-z)

among functions which have a series expansion oo

h(z) = ^2bkzk k=0

which is absolute convergent for \z\ < 1. Proof. Let us assume that oo

h{z) = YJhzk. k=0

Then, the left-hand side of (5.28) can be written as °°

/

II2Å-

,. „ || V - , WLqQL^x\VK , KX

„ £

II

rlTn dL

g(l)

" S ^ Ka.ll* c(„-l, g -l)

q(i)

^

*/ £

q(i)

IKax|| 2 W c(n- 1,9-1)-

5.6 AN ALTERNATIVE ESTIMATOR OF SURFACE AREA

145

Now, using that II

||2

i

.JK^II2

II

\\^Liax\r and Proposition 3.9, the left-hand side of (5.28) becomes

v^

}(\\vLtax\\2(i-y))k

i

^_^

.*=! ,

= E ^ m n z a g = i J i ^ « x i i 2 f c / (i - \\*Lt«*\?vrk+* k=0

{ 2 '

2 /

£

xjT?-1(l-3/)*+s*1-1dy i ^ ^ T f 2 ' ^ + ^ i - ) n/7

E°° L ii

2

MI^M

fc=0

^2

^ 2 '

! n-g

,

n-1

..

f(* - 2--Ö^;* + - ö - ; Ikif^ll2)-

2 )

We now insert the expansions of the hypergeometric functions i

_

_ i

.°°.

+ ~ ; ||7rL,ax||2) = £ ^lbr^H 2 '-,

F(k - \ ^ k

i=o where

nV-^+onW+o , (fc) _ i=o n(fc +

i=o 2-i + ? )

_£_

2=0

If we let B(^,k

= k

k

it follows that (5.28) can be written as oo

oo

fc=0 j=0

..,.

+ q

B(^, -^)

^) '

146

5. GENERALIZED BLAS CHKE-PETKANTSCHIN FORMULA

Therefore, we can determine the coefficients bk (or bk) from the equations

6o40) = i i

J2hc\% = 0,l = l,2,.... k=0

From these equations, we find that, cf. Exercise 5.11,

7

h=

2=0

l

2=0

^r—

w

2=0

which implies that

k-) =

F(-\,-\(n-q);±(q-l);-),

as postulated.

□

Example 5.10. Let n — 3 and q = 2. Then, the relevant hypergeometric function is F(

~ 2 ' " 2 ; 2; Z)

=

^

^

+

^

(

2"

arccot

(-/f=))>

cf. Exercise 5.12. The function entering the alternative estimator becomes /i(G(Tan[XnL 2 ,x],Li) _ 1 )

D

5.7 Exercises Exercise 5.1. Show the inequality (5.6) dim(L p H Ld n (L9 n Ld)-1) Hint. Write the subspace as

Mn(L ? n M)-1


5.7 EXERCISES

147

where M = Lp n Lj. Use then Grassmann dimension formula, cf. Exercise 2.9, and dim(M + Lq) < p. Exercise 5.2. In this exercise we derive some of the properties of the function / in Proposition 5.4. 1. Show that the differential of the mapping / is D

= |kL^_2^irl7rspanK}-

f(x)

Hint. Use that

2. Show for UJ G S n _ 1 n L^_ 2 that / _ 1 M U f^i-uj)

= (L n _ 2 0 span{o;})\L n _2.

Exercise 5.3. Check that

c(n-r,n-2-r)

B(\, ^ f ^ ) _ B(l,2^)~C[n

c(n-l-r,n-2-r)'

T U

'

T)

'

This result is used in the proof of Proposition 5.4 in the case p = n — 1 and r arbitrary. Exercise 5.4. Show for x\ G Lp\Lr / Bn(0,l)nLp

•■•

/

that

l | e i , . . . , 6r, ^ i , . . . , Xp_r

Bn(0,l)nLp p—r

are linearly dependent} TT dx\ — 0. i=2

Exercise 5.5. Let v ;

&p—r J

xnLp cf.

(5.24).

1. Show that if Lp is an isotropic p—subspace containing L r , then E(m^(X,Lp;Lr)) = Xdn(X). Hint. Use Proposition 5.4.

148 2.

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA Show the extension of Proposition 4.8, i.e. if Lpi is an isotropic p i - s u b s p a c e containing Lr and LP2 is an isotropic p2~subspace containing L r , where pi < p2, then V a r ( m J ; ( f ) ( X , L P l ; Lr)) > V a r ( m £ $ ( X , LP2;

Lr)).

Hint. Start by showing that

m^){X,LP2;Lr)=

j

m™(X,

L

p

M

^

^

_

f).

Pi I'-)

In this step, Proposition 5.1 is useful. Then, use the same technique as in Exercise 4.8. Exercise 5.6. Let Lp be an isotropic p—subspace containing Lr. Find under regularity conditions an expression for the variance of rrvn,\\X, Hint. Because m/n/Mx,Lp\Lr)

Lp\ Lr) in the case p > r + 2.

has mean-value X^(X),

cf. Exercise 5.5, it is

enough to find an expression for the second moment of mit\'(X,

Lp\ Lr).

Here,

Theorem 5.6 is useful. Exercise 5.7. Let L\ be an isotropic line through O in the plane. Let X be a planar ellipse

where a > b. 1.

Show that if L\ makes an angle 0 E [0,7r) with the x—axis, then m?$(X,

L i ; O) = TT/I-L + ( 1 - 1 ) sin 2 6»].

2. Show that the probability density of m [ J ( I , Li; O) is in the case a > b f(x) = ih10 3. 4.

^Wx-rf'

if^2<*<™2 otherwise.

Show directly, using this density, that the mean-value of rn^Mx, L\\0) is equal to nab, the area of X. Hint. Express the mean-value as a Beta-integral. Also discuss the case a = b.

5.7 EXERCISES

149

Exercise 5.8. This exercise concerns systematic replication, cf. Figure 5.6. We will here only consider the case where X is an open bounded subset of the plane, which is star-shaped with respect to O G X. For 0 G [0,27r), let 1(0) be the distance from O to the boundary of X in the direction given by 0. 2ir

1. Start by showing that A(X) = / f(0)d0, where f(0) = l(0)2/2. Hint. Use that o m

(2 2)

i(o) ( ^ ' ^ 1 ' ^ ) n a s mea n-value A(X), when L\ is an isotropic line through O. 2. We can now estimate A(X) by measuring along n half-lines. The natural estimate becomes n *—' n k=o where 0 is uniform in the interval [0, 2ir/n). Show that for n=2, A(X) coincides

with mf$(X,Li\0)

and that EA(X) = A(X).

The estimation of the variance of A(X) will be discussed in Chapter 7. Exercise 5.9. This exercise concerns the relations between the angles a(x) and f5(x) in R3. Thus, let the situation be as in Figure 5.7 and let us assume that n = 3 and q = 2. Show that cot/?(x) = |cosø(a;)| cota(x),

(•)

where (#) G [0,7r) is the rotation angle of L(x) = 0, L2 is a vertical plane (i.e. a^ G L2) containing Li. Note that (•) implies that a(x) < P(x) < f. Hint. Go through the following reasoning: without loss of generality, we can assume that ax = (0,0,1). Let UJ G R S be a unit vector spanning Li, with polar coordinates (^ — a(x),<5(x)), i.e. a; = (sin(

a(x))cos6(x),sm(— — a(x))sm6(x),cos(— — a(x))) z z z = (cos a(x) cos <5(x), cos a: (a;) sin 6(x), sin a(x)).

Let a;i = (—sina(x)cos6(x), — sina(x) sm6(x),

cosa(x))

uj2 = (sin 6(x), — cos 6(a:), 0). Then,

U,UJI and

^2 is an orthonormal basis in R3 and L2 is spanned by u and (COSØ(X))CJI -f- (sinø(x))o;2-

150

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

The unit normal vector of L 2 is 7T

7T

Co = cosOO) - -)u)\ + sinOO) - TT)^2The relation G(Tan[X, x], Li) = G(Tan[X, x], L 2 )G(Tan[X n L 2 , z], £1) can now be translated as sin a(x) — (l - sin2 (#) cos2 a(ir))

sin /3(x)

and therefore cot2 /?(x) = — 9 - 1 = cos2 ø(x) cot2 a(x). sin (5{x) Exercise 5.10. Show that for 0 < /? < 7 1

F(Q

' ^; 7; z) = B{/3,\-0) I(1 " ^ ) _ a ^ 1 ( 1 ~ ^) 7 _ / J _ 1 ^0

Hint. Use that 00 k—i

^ ^.

fc=0 i=0

Exercise 5.11. This exercise concerns the last part of the proof of Proposition 5.9. More precisely, the determination of the coefficients b^ (or bk), k = 0 , 1 , 2 , . . . , from the equations

&o40) = 1 /

Y^bkc\kJk = 0,l = l,2,...

(•)

k=0

These equations are satisfied for exactly one sequence of coefficients bk,k = 0 , 1 , 2 , . . . , viz. the sequence defined by the recursive relations (CQ = 1J = 0,1,2,...) l-l k=0

5.7 EXERCISES

151

In this exercise, we show that the sequence of coefficients suggested in Proposition 5.9 is in fact the unique sequence, satisfying equations (•). 1. Show that if

7 h =

2=0

-1

2=0

^T—

fei'

2=0

then i

k =

i=0

l

»=0

fe=i

F

2=0

2. By convention, the first equation of (•),

IOCQ

= 1, is fulfilled. Now show that

(k)

with bk as in question 1 and c; have for I = 1,2,...,

as defined in the proof of Proposition 5.9, we

k=0 i-i

H ( 2 ^ 1 + i ) fc=0 i=0

V

J

2=0

2=0

It therefore remains to show that /

/ 7 x

Ac—1

l-k-1

E I n<-^+.>ii(^>=»fc=0

V

y

2=0

2=0

This is shown in question 3 below. 3. Show that fi(x) = 0 for all x G Ä, and all Z = 1,2,..., where the polynomial

fc=0 ^

Hint. Use induction in /.

'

2=0

2=0

fr.R-^Ris

152

5. GENERALIZED BLASCHKE-PETKANTSCHIN FORMULA

Exercise 5.12. This exercise concerns Example 5.10. We will derive the explicit form of F(—7£, —\\ \\ z) by combining a number of known results. In Gradshteyn & Ryzhik (1965, 9.121.13 and 20), it is stated that

F(~,i;i;z) = v T ^ „,113 ^2'2'z)

arcsinv/^ ^r-

N

F

(a)

=

(b)

Furthermore, the following relation is valid for hypergeometric functions 7F(a,

/?; 7; z) - 7 F ( a , /? + 1; 75 *) + ^ F ( a + 1, 0 + 1; 7 + 1; *) = 0,

(c)

cf. Gradshteyn & Ryzhik (1965, 9.137.11). 1. Show that relation (c) is correct. 2. Derive the explicit form of F(—\, — \\ \\ z) given in Example 5.10 by combining (a), (b) and (c). Hint. Use that • r * arcsm J z — 2

4- V^ arccot—==. vT^I

Exercise 5.13. This exercise concerns a rotational version of Crofton's formula, compare with Proposition 3.7. 1. Show that Lemma 5.5 implies the following result for p = q and r = 0 J \\x\\-{n-q)hdq(x;

X)dxd

= J \i~n+«(X

n Lq)dLnq1

cnq

x where hdq(x;X)

= J

G(T^[X,xlLq)dLnq(1)

and £ \ v is the set of q—subspaces containing L± = span{x}. Note that h™{x]X) = c(nl,q-l). 2. Find an expression for /^ _ 1 (:r; X) in terms of a hypergeometric function. Hint. Use Propositions 5.2 and 3.9.

5.8 BIBLIOGRAPHICAL NOTES

153

5.8 Bibliographical notes Earlier proofs of the generalized Blaschke-Petkantschin formula, cf. Zähle (1990) and Jensen & Kiéu (1992b), have been based on a product version of the coarea formula and abstract constructions such as Hausdorff measures on the vector space of multivectors. The approach in the present chapter is thus much simpler and hopefully more insight is gained into the geometry of the problem. Some of the local estimators of volume and surface area presented in Section 5.5 have been introduced already in Jensen & Gundersen (1989). In that paper, local estimators of powers of volume and surface area and of curvature are also discussed. Related model-based results may be found in Ambartzumian (1974) and Stoyan (1986), see also Chapter 8. Using Lemma 5.5, it is possible to derive a rotational version of Crofton's formula, cf. Jensen (1995) and Exercise 5.13. There also exist a dual translative version of Crofton's formula and a Crofton's formula for projected thick sections, cf. e.g. Schneider (1981, 1993).

Chapter 6

Local slice formulae Some of the local estimators of d—dimensional Hausdorff measure presented in Chapters 4 and 5 depend on local n—dimensional information about X (the G-factors) which cannot be collected on X n Lp. The exceptions are d = n (Chapter 4) and d — n — 1 (Section 5.6). Furthermore, the case d = 0 cannot be treated by the methods developed so far. There is therefore a need for expanding the local techniques. Inspired by Gokhale (1990), the obvious idea is to include more information by considering a slice, i.e. a p—subspace with some thickness. The extra information can either be used to determine the G—factors or to construct alternative local estimators. We will adopt the latter approach in this chapter. Let us start with an example.

6.1 A local estimator of number in R3 Let X be a finite subset of R3, X =

{xi,...,xN}.

We want to estimate the number N. This problem is of the well-known type with V = { 1 , . . . , W } , y(i) = 1,2 e V, and N

w = Yl yd) = iev

as parameter of interest. Let us estimate N, using a random slice of thickness It through O, T2 = L2 + £ 3 ( 0 , t ) , cf. Figure 6.1. We suppose that L2 is an isotropic plane through O. The sample is S = {ieV:xie 155

T 2 }.

156

6. LOCAL SLICE FORMULAE In order to and the sampling probabilities, note that

IN! < t =* P(i) = P(xi € r 2 ) = l. Let us therefore assume that ||xt|| > t. Since

we find, using Proposition 3.9» that p(i) = Pin e T2) fj T

r

3

= /l{lk,^:|i<0^| -Co

- AH)/1«»<SF1'*"11-"'"'* 0

The Horvitz-Thompson procedure therefore suggests the following estimator

Xi€T2

Note that the sampling probabilities only depend on the distances between the X{Js and O. In the sections to follow, we extend these ideas to estimation of d—dimensional Hausdorff measure based on observation in a random slice in Rn,

Figure 6.1. A 1-slice T\ and a 2-slice fj in R3,

6.2 A LOCAL SLICE FORMULA

157

6.2 A local slice formula for d—dimensional Hausdorff measure A p—slice in Rn of thickness It is a set of the form Tp = Lp +

Bn(0,t),

where Lp is a p-subspace in Rn and Bn(0,t) and radius t. The mapping r :Lp-+Lp

is the open ball in Rn with centre O

+

Bn(0,t)

is 1-1, such that two different p-subspaces cannot generate the same p—slice. Note that, as in Section 6.1, x e r(Lp) <£> ||7TL±x|| < t. T£ry

The set of p—slices whose corresponding p—subspaces contain Lr is denoted We have

%

= MLP) : Lp e C;(r)}.

We can construct a rotation invariant measure vn,, on T 7 ^ by lifting the rotation invariant measure on ^l(r\ by the mapping r. The measure is defined by the relation

/ g{Tp)v;{T){dTp) = J g(r(Lp))dL;{r), P(r)

S(')

for g a non-negative measurable function on TJjr\- For simplicity, we will use dT™,, as short notation for i/™,JdTp). We can also define random p—slices by the same type of construction as was used for subspaces. Definition 6.1. An isotropic p—slice of thickness 2t, containing a fixed r—subspace Lr, is a random p-slice Tp with density with respect to z/Vx of the form p(r p ) = l / c ( n - r , P - r ) , T p e % .

□ Note that if Lp is an isotropic p-subspace, containing Lr, then r(Lp) is an isotropic p—slice, containing Lr. Sampling probabilities are given in the proposition below. As in Section 6.1, they can be derived, using Proposition 3.9.

158

6. LOCAL SLICE FORMULAE

Proposition 6.2. Let Tp be an isotropic p-slice of thickness 2t, containing Lr. Let x G Rn and let Fa^ be the distribution function of the Beta distribution with parameters (a/2,6/2). Then, dTn

f

P(xeTp)=

/ i{xeTp}-

J

(\ c(n- r,p — r)

^—r

T

p(r)

— hp^{x,Lr), say, where

{

1

if p = n

Fn-p,p-r(t2/\\7TL±x\\2) l{xETp}

if r
□ Note that the sampling probability is never 0 if r < p . In Table 6.1 below, the sampling probabilities are given in all cases of interest in R3. r

p

0

1

2

3

0

i{x e T0}

—

—

—

1

i_(i_t2/W2)1/2

i{x e 7i}

2

t/\\x\\

|arcsin(t/||7r L ±x||)

i{x e T2}

—

1

l

l

3

1

1fable 6.1. The sampling probability h^r)(x,Lr)forxeR\

satisfying t < \\7rL±x\\.

Using Proposition 6.2 it is easy to derive a local slice formula, which is analogous to Proposition 5.4. Proposition 6.3. Let X be a d—dimensional differentiable and bounded manifold in Rn. Let g be a non-negative measurable function on Rn. Then, for arbitrary p,r, satisfying 0 < r < p < n we have, c(n-r,p-r)Jg(x)dxd=

J

J

g(x)h(^r)(x,Lr)-1dxddT^r).

(6.1)

6.3 LOCAL SLICE ESTIMATORS

159

Proof. Let us rewrite the right-hand side of (6.1)

= I

I l{x e

T^gix^^Uy'dx^T^

= J J l{x e

Tp}g(x)h^r)(x,Lr)-ldT^r)dxd dTn

/■ /

g{x)h^r){x,Lr)-\ X

j

l { s e T,}

*>

jd^

T" p(r)

— c(n — r,p — r) /

g(x)dxd.

X

Note that in contrast to Proposition 5.4, Proposition 6.3 contains no condition concerning the relation between p and d.

6.3 Local slice estimators of d—dimensional Hausdorff measure Let us consider a d—dimensional differentiable manifold X with A^(X) < oo. We want to estimate \^{X), based on an isotropic p—slice of thickness 2t, containing Lr. The Horvitz-Thompson procedure suggests the following estimator

J2dxd/h^r)(x,Lr), where the sum is over all infinitesimal d—dimensional volume elements of X in Tp. We will use the continuous version m%$(XnTp;Lr)

= J

h{£r)(x, Lr)~ldxd.

(6.2)

XnTr

Note that XCBn{0,t) m(^{X

n Tp; Lr) = Xdn{X), for all Tp e Tpn(r).

So in this situation, we get the trivial estimator. For d > 0, the integral of rav?(X nTp;Lr) can be discretized, using grids of points, lines, planes, etc.. This is the subject of the next sections.

160

6. LOCAL SLICE FORMULAE

6.4 The special case d = n In this case» we consider an open and bounded subset X of Rn. estimator of its volume is» cf. Figure 6.2,

m^\XnTp:Lr) = J h^^L^dx".

The local

(6.3)

xnTp We will now describe how the set A" n Tp can be subsampled by means of grids. At this point, it may be useful to recall the basic facts about random grids gi¥en in Section 3.6» in particular Definition 3.16 and Figure 3.7.

Figure 6.2. The local estimator m^^(X

flTp:Lr)

is based on information in X D Tp.

Because A' n Tp has dimension n, it suffices to consider random grids with fixed orientation. Detrition 6.4. Let Lq e £^ and suppose that (DO,{UJ}JLQ) induces a lattice of fundamental regions in L^. Then, a uniform q—grid in Rn, induced by Lq and (A)>{tøj}jl 0 )' is a random q—grid distributed as Gq(U) = {LfI + U + Vj : j = 0 . 1 . . . . } . where U is uniform random in Do. Below, we write G<{ as short for Gq(U)y when convenient.

D

6.4 THE FULL-DIMENSIONAL CASE

161

Let us use the Horvitz-Thompson procedure to find an estimator of (6,3) based on subsampling of X n Tp with a uniform g-grid. So, let U(x) be an infinitesimal volume element of X n Tp, chosen so small that atmost one affine subspace of the grid is hitting Fix), Then, with An-q = A"~ 9 (A)). Xn-q(-f±U(-r)) -fi -.

k

P(L (.r) O Gq # 0) =

Therefore, the Horvitz-Thompson procedure suggests the following estimator of (6.3)

2-*

4- 1 rf.r»-9 --*»-«

2-/

Vr1^"1

f/i

•

We use the continuous version m

Sw'/ A " n r/»: ^ = -4»-*

/

*>?)(*< ^ r ) " 1 ^ .

(6.4)

xnfPnaq Conditional on Tp, the subspace Lq can be chosen parallel or perpendicular to Tp» cf. Figure 6.3.

Figure 6.3. The direction of the g-grid is indicated, by Lq which can be chosen parallel or perpendicular to Tp, For q = 0, the estimator (6.4) takes the form m%$0(XnTp.Lr)

= An

£ xexnTpr\Go

hft^Lr)-1.

which can be used if X is represented in terms of voxels in the computer.

162

6. LOCAL SLICE FORMULAE For q = 1, we can write the estimator as

m^X

H Tp; Lr) = An^

J

h^r){x,

L^dx1.

xnrpnGi Below, we show, under regularity conditions, that we can replace the integrals along the grid lines with a discrete sum over the boundary points of X D Tp. An analogous result was derived in Chapter 4, Proposition 4.7. To see this, we will assume that for any u G L^, X Pi (L\+u) consists of a countable number of line-segments. This implies that the same is true for X r\Tpn (Li + u). We now associate to each x G d(X D Tp) C\ {L\ + u) the number a{x) G {0,1}, in the same manner as we did in Chapter 4, cf. in particular Figure 4.5, the only difference is that here X is replaced by X Pi Tp and O by u. Proposition 6.5. Let X be an open and bounded subset of Rn. Let L\ G C[ and suppose that for all u G L^, X fl (Li + u) consists of a countable number of linesegments. Then, if L\ is spanned by the unit vector to, we have for 0 < r < p < n,

m^(xnr p ;L r ) max{0,(a;,ct;)}

xed(xnrp)nG1

mi n{o(x,c>}

Proof. It is enough to show that the result holds for integration along a particular line. Let u G L^ and decompose L\ + u into a positive and a negative half-line, separated by u, Li+u

= ( L 1 + + u) U {^} U (Li_ + 7/).

We get

h^Lryldyl

j xnT p n(Li+u)

h(£)(y,Lr)-1dy1+

j xnr p n(Li++n)

J

/$)(l/,£rrV.

xnr p n(Li_+u)

Let us concentrate on the integral along the positive half-line. Suppose that d(X n Tp) n ( L i + + u) = {xi, x 2 , •. .}.

6.4 THE FULL-DIMENSIONAL CASE

If (u,xi)

C X nTp,

163

we have

x n Tp n (Li+ + u) = 0, xi) u (x2,x3) u • • •. Correspondingly, we have (omitting the integrand hwAy,Lr)~l,

when convenient)

xr\Tpn(L1++u)

-hh(u,xi)

(x2,x3)

(u,Xi)

{u,X3)

(u,X2)

E

("^ /

xed(XnTp)n(L1++u)

h^Lr)-%\

^x)

In the other case XnTpf)

(Li+ + u) = (xi,x2) U (x 3 ,x 4 ) U • • •,

and the resulting expression

xnTpn(L1++u) 7 (u x)

*ed(xnr p )n(L 1+ + w )

also holds here. It remains to show that for u G Lj- and x G L\ + u max{0,(x,w)} 1

1

J h^l)(y,Lr)- dy = (u,x)

j

Fn-p^rif/WTr^ivw

+

Tr^W2)-1^.

min{0,(a;,u;)}

This transformation result is a consequence of the coarea formula. Thus, consider the mapping / :R-+

Li+u

v —► VUJ + u.

164

6. LOCAL SLICE FORMULAE

Then, the coarea formula implies that

j k{y)dyl = Jk{f{v))dv. Li+tt

R

Let us choose (u,x)}Fn-p,p-r(t2/\\7rL±y\\2)-\

Kv) = HV e cf. Proposition 6.2. Note that u =

TTL±X

and that

X — U = TTLi% — ( X >

^)^'

We therefore get

f(v) e (u,x)

t VLU G (0,X

— U)

t min {0, (x,cu)} < v < max{0, (X,LU)}

and k(f(v)) =l{min {0, {x, to)} < v < max {0, (x, UJ)}} x Fn-Pø-r(t2/\\irL±(vuj + ^xx)!!2)-1

D

6.5 The case 0 < d < n Let X be a ^-dimensional differentiate manifold with A^(X) < oo. The local estimator of A^(X), based on an isotropic p—slice Tp, containing L r , is mJJj.jflnTpjL,.), cf. (6.2). The case n = 3, p = 2 and d = 1 is illustrated in Figure 6.4. Below, we discuss subsampling of X D Tp by means of grids. For d < n, it is necessary to randomize the orientation of the grids.

6.5 THE LOWER-DIMENSIONAL CASE

165

Figure 6.4. Illustration of tie sampling reieYant for the estimator mxU/J(X f\Tp;Lr)j in the case ii=3» p=2 and d=l,

Accordingly, let Gl{ be a uniform and isotropic (/—grid. The Horvitz-ThompsoB procedure thee suggests that if we subsample X n Tp with Gq. then

xhTP should be estimated by» cf. Proposition 3.18» £ * $ ) ( * ' Lrrld.rd/[A-iqh(d, xexnTFnGq

q, a)dxd\

x£Xf\TpnGq (

= m "f (A'nrp;Ir), say, where L/j

\

b(d,q,n) =

a

q+d-n+l

&n + l

.

Note that if i is chosen so large that X C Tp, then m^f^iX

n 7],: I r ) reduces to

1«

6. LOCAL SLICE FORMULAE

The sampling procedure is illustrated in Figure 6.5. Above we have tacitly assumed that Tp and Gq are independent. We can also derive the final estimator by first sampling X by Gq* and then subsampling X 0 Gq by Tp. For some applications, it is convenient that Tp and Gq are dependent. If q < p, one procedure of this type is first to choose Gq as a uniform and isotropic q—grid. This (/-grid consists of parallel q—dimensional affine subspaces. If Lq is their parallel q—subspace, then choose Tp as an isotropic p—slice containing Lq. For a d—dimensional manifold X with d - n — q = 0, \lt[{ X) can then be estimated by h(
Y,

h

%{x-L
<6-8>

cf. Jensen & Kiéu (1994, 'Proposition 1).

Figure 6.5. Subsampling of Tp by means of an independent uniform and isotropic f-grid Gq. If n — q < p, an alternative procedure is to first choose Gq as a uniform and isotropic q—grid and then Tp as an isotropic p—slice containing L~% where again Lq is the q—subspace associated with Gq. The estimator of ,V,' (X) becomes {d—n+q = 0)

b{d,q,n)-lAn^ J2

CV^'P"'-

(6 9)

-

xexnrpnGq cf. Jensen & Kiéu (1994, Proposition 2). The two procedures are illustrated in Figure 6.6. In practice, Tp and Gq are generated in the opposite order.

6.6 SOME FURTHER DEVELOPMENTS

167

Figure 6.6. Subsampling of Tp by means of a dependent f-grid Gq.

6.6 Some further ie¥elopmeiits for n = 3 For /? = 3, it is possible to deYelop the slice techniques somewhat further. These developments are related to so-called total projections. We will mainly discuss the case d—\ and thereafter shortly mention the other interesting case d = 2. Until now, we have deYeloped two local slice estimators of length. The irst one is based on an isotropic slice T2, containing Lr, where r = 0 or 1, and an independent uniform and isotropic plane grid G2. If A is the distance between neighbour planes in G2, the estimator is

x€XnT2nG2

see also the right part of Figure 6.5. The second one is based on a dependent pair T2 and G2, cf. (6.9) and the right part of Figure 6.6. Let us in more detail describe the assumed' simultaneous distribution of (T2,G 2 ). According to the definition of a uniform and isotropic grid, G2 can be generated as G2 = {spanjQi}- -f ( r -*- jA)Jli

: j = . . . , - 1 , 0 , 1 . . . .}.

where Q\ is an isotropic direction on the unit sphere S2 and U is an. independent uniform, random ¥ariable in an interval of length A. (We call O an isotropic direction on S 2 , if the density of O with respect to the 2-dimensional Hausdorff measure on S2 is constant (and therefore equal, to lAtør).) Secondly, T> can be generated as T 2 =span{fii,n 2 }H-B3(O.0.

168

6. LOCAL SLICE FORMULAE

where 0 2 is an isotropic direction on S 2 n s p a i i j ^ } ^ . The estimator of length suggested in (6.9) then becomes

where L\ = span{fii}. We can re¥erse 'the order of generation of T> and G 2 . Notice that L \ = span{Oi} is an isotropic line through () in R:] and L-2 = s p a i i f O i , ! ^ } is an isotropic plane containing L\. According to Proposition 3.10. we obtain the same simultaneous distribution of (Li. !•_)) if we first generate 1% as an isotropic plane through O in Ä 3 and then L\ as an isotropic line through Ö» contained in L2. As a consequence, we can start by choosing T2 as an isotropic 2-slice and secondly G2 as a random grid of parallel planes, perpendicular to the isotropic line L\ in I2, cf. Figure 6.7. Notice that if the thickness of the slice is so large that the spatial cur¥e X always is contained in T2, then the estimator reduces to 2AA§(A"nG 2 ).

'

(6 JO)

In some cases, the total projection ~L2X can be observed. Instead of determining the number Aft A' n G'-j) of intersections between the curve X and the grid G2 of planes, one can then instead count the number of intersections between the projected cur¥e 7T£,2X and L2 O G2, cf. Figure 6.7.

Figure 6.7. Illustration of the sampling relevant for a local slice estimator of length in Jc1.

What happens if T-2 is an isotropic 2-slice, containing a fixed line L^? Is it still possible to subsample with a dependent plane grid? This question is unsolved at

6.6 SOME FURTHER DEVELOPMENTS

169

the moment, but there exists a partial solution for the case where the curve is always contained in the slice. Equivalently» we suppose that t = oo. The design is illustrated in Figure 6.8. The plane L2, the mid-plane of the infinitely thick slice, is rotated an angle <j> around the ¥ertical axis Lt^. The grid planes are now perpendicular to L-2 with a unit normal a;, making an angle 9 with the Yertical axis. The coordinates of u then become u) = (sin 9 cos > sin 9 sin $, cos 9). If the angles are chosen independently with densities p(0) = - s i n 0 , 0 € (0.7r).p(o) = z>9£ 2

(0.7T),

n

then span{o/} is an isotropic line through O in Ä 3 and G2 can be regarded as a uniform and isotropic plane grid and the estimator (6JO) can also be used in this situation. As before» if the projected curve TTI2X can be observed one can instead count the number of intersections between the projected curve wi2X and L% H G2, cf. Figure 6.8. In some cases, LL> " G'-j is replaced by a so-called cycloid test system, cf. Cinz-Orive & Howard (1991). See also Roberts et al. (1991) and Howard et al. (1992). Corresponding results also exist for surface area. Thus» in Cruz-Orive & Howard (1995), the so-called vertical spatial grid has been developed.

Figure 6.8. Illustration of the sampling relevant for estimation of length in R3, based on projections onto a vertical plane.

170

6. LOCAL SLICE FORMULAE

6.7 Exercises Exercise 6.1. Show that the mapping T : Lp-+ Lp +

Bn(0,t)

is 1-1. Exercise 6.2. Check the results presented in Table 6.1. Exercise 6.3. Let X{ be a ^—dimensional differentiable manifold in Rn, i = 1 , . . . , q. Let g : Rnx • • ■ x ^ -> Ä+ U {0} be a non-negative measurable function. For Tp an isotropic p—slice containing L r , let hpn{yj{xi,...,xq]Lr)

= P(xi eTp,...,xqe

Tp)

be the q—order sampling probability. Show that

ce-r,,-,,/.../* = /

/

•'• /

T;{T) X1HTP

„ ^ ,^,n^

9(xi,.--,xq)f[da*
xqnrp

i=1

The first-order sampling probability is always positive. This is, however, not true for higher-order sampling probabilities. Give an example where

J'-'Jl{h^\xlj..^xq;Lr) x1

= 0}f[dxt>0. i=1

xq

Exercise 6.4. Let the situation be as in Section 6.3. Show that if Tp is an isotropic p—slice containing L r , then

Em(;$(XnTp;LT) Hint. Use Proposition 6.3.

= \dn(X).

6.7 EXERCISES

171

Exercise 6.5. Let X be a d—dimensional differentiable and bounded manifold in Rn with Xn(X) < oo. Let c be a positive constant chosen such that ||7rL±a;||
n Tp; L r ) satifies the inequality

/ c2)~l x A^(X).

Exercise 6.6. Let d = n. Show that the variance of rrrn^\x is a ball with centre O.

n Tp; L r ) is 0, if X

Exercise 6.7. This exercise concerns the estimator m^V^X DTp;Lr) of the volume of X, cf. (6.4), based on information in a p—slice, containing L r , subsampled by a uniform grid of lines. We are interested in the special case where r = 0. 1. Show that for r = 0, the integral appearing in Proposition 6.5 can be reduced to |<* > W )|

J

Fn^p(t2/(v2

+ ||7rLx*||2)rW

(•)

0

2. Consider the case n = 3 and p = 2. Let us use the following short notation di = \(x,u)\ = ||7rLl:r||,d2 = \\7rL±x\\,d3 = \\x\\. Show that in this case, the integral (•) can be rewritten as di

j

F^/^+dDrUv

0

(di d3
3

+ ^log(^)]

Hint. Use that for 0 < z < 1, Fi>2(z) = y/z.

2

t
172

6. LOCAL SLICE FORMULAE

Exercise 6.8. This exercise concerns again the estimator m^/V^X nTp;Lr) of the volume of X, based on information in an isotropic p—slice, containing Lr, subsampled by a uniform grid of lines. As in the previous exercise, we are interested in evaluating the integral max{0,{x,w)}

/

Fn-p,p-r{t2/\\-KL±{vuj

+ irL±x)\\2)-ldv.

(•)

min{0,(^,^)}

In this exercise, we concentrate on the case n = 3, p = 2 and r = 1. We use the notation L 1 ( 0 ) for Lr, the fixed 1-subspace contained in the slice, in order to avoid confusion with L\, the direction of the grid lines. The 3 distances defined in the previous exercise, d\, d2 and cfø, also enter into the formulae given below. 1.

Suppose that L\ — Lx^.

Show that in this case, the integral (•) can be written as

J F^/diyHv 0

2.

_ ( di

d2
~ | |di/arcsin(^)

t < d2.

Suppose that L\ C L^,Qy Let Z — |[7T/r 11

_ x-L^II.

(L 1 ( 0 )+Li)

"

Show that in this case the integral (•) can be written as di

yVi i i(*v(" 2 +* 2 ))- 1 ^ 0

yjd\ + Z2 < t

f d1 =

I Vt2 - z2 + §

/

1/ arcsin(t/V^ 2 + z2)dv

z
+ z2)dv

t < z.

Jd2

+ z2

di

f / 1 / sacsm(t/Vv2 v o

Exercise 6.9. Let d - n + q = 0. Show that the estimator

m^q(XnTp;Lr)

= b(d,q,nr1An-q

J2 x€XnTpnGq

h{£r)(x,Lr)-\

«

6.8 BIBLIOGRAPHICAL NOTES

173

based on an isotropic p—slice Tp, containing L r , and an independent uniform and isotropic q—grid Gq, has mean-value X^(X). Show also that the estimators (6.8) and (6.9) have mean-value A^(X). Hint. Start by calculating the conditional mean-value of (•) given Gq and then use (3.20), Proposition 3.3 and Proposition 3.7 (Crofton's formula). Exercise 6.10. This exercise discusses the possibility to construct a local estimator of length based on an isotropic 2-slice T2 in R3 and subsampling with a grid of planes G2 in R3 parallel to the slice. For this purpose, let U(x) be a line-segment in R3, of length I and centred at x e R3. We suppose that I < A, where A is the distance between neighbour planes of G2. Atmost one plane from G2 can then hit U(x). Let us sample the line-segment if x £ T2 and U(x) n G2 / 0. Below, we show that the sampling probability depends not only on the length of U(x), but also on the orientation of U(x). A Horvitz-Thompson estimator of length, based on this design, therefore involves determination of orientation which is difficult in practice. 1. Suppose that x = (0,0, ||x||) and that U(x) is parallel to the z—axis. Show that the sampling probability becomes

p(xeT2Mx)nGi^) = {4mW

11 f{

Hint. Let L2 be T2's mid-plane such that T2 = L2 +

B3{0,t).

This plane is an isotropic plane through O. The planes of G2 are parallel to L2. Given L2, the position of the planes of G2 is determined by a uniform random variable in an interval of length A. 2. Suppose that x — (0,0, ||x||) and that U(x) is parallel to the x—axis. Show that the sampling probability then becomes

P(xeT2,U(x)nG2^H) I s k i 1 " I ai
Ml > t.

6.8 Bibliographical notes The results presented in this chapter is mainly based on Kiéu & Jensen (1993) and Jensen & Kiéu (1994). The subject has however been further developed here. In this chapter, it is emphasized that grids are used as a means of subsampling. If the

174

6. LOCAL SLICE FORMULAE

parameter to be estimated is volume then grids with fixed orientation can be used, cf. Tandrup et al. (1997). Otherwise, there are two types of subsampling, independent subsampling and dependent subsampling with grids with random orientation. The former possibility is attractive with the observation equipment described in the next chapter. The slice design described in this chapter is the only local design among the designs described until now which can be used for estimating number. The idea of using such a design is due to Stephen Evans, cf. Gundersen et al. (1988b) and Evans & Gundersen (1989). Estimation of surface area in using a spatial line grid, has been suggested by Sandau (1987). The application of a confocal scanning light microscope for such estimation has been discussed in Howard & Sandau (1992).

Chapter 7

Design and implementation of local stereological experiments 7.1 Optical sectioning The local methods described in the previous chapters have mainly been developed with the analysis of biological tissue in mind. The full power of the local methods can be exploited if the tissue is transparent and physical sections can be replaced by optical sections. The quality of such sections is highest if a confocal microscope is used, but for many tissue types, conventional light microscopy works satisfactory. The concept is illustrated in Figure 7.1, showing a light microscope with a tissue block on the stage. Below, to the left, the tissue block is shown magnified, containing just a single biological cell, for simplicity. On the computer screen to the right in Figure 7.1, the focal plane (thin optical section) is shown. By moving the focal plane up and down, the picture changes on the computer screen and in fact, a whole continuum of sections is generated. In essence, we are travelling in 3-dimensional space. With this type of observation technique it is usually no problem to identify the different connected parts in a section that belongs to a particular cell. Indeed many stereological estimation problems can be solved easily, using optical sectioning. For instance, in order to estimate the number of cells of a certain type, it is not necessary as in Section 1.2 to assume that the cells have a minimal height. Instead, a reference point can be associated to each cell and then, a sample can be collected, using optical sectioning, as those cells with reference points in a sampling box, see also Chapter 8.

7.2 Implementation of local designs The simplest local design is that of an isotropic line L\ through O, where O here is the cell nucleus or an identifiable subset of the nucleus, like a nucleolus. Such a line can be generated using random number generators. The intersection between L\ 175

176

7. DESIGN AND IMPLEMENTATION

and the moYing focal plane will appear on the computer screen as a mewing point» cf. Figure 7.1. The line itself is in fact not generated. Instead 'its shadow' in the form of the moving point on the computer screen is shown. The line intersects the boundary of the cell X, say» when the point on the screen hits the boundary of X. Such intersection points are observed when the point mo¥es from one side of the boundary of X to the other. When an intersection point is found, a mouse is used to indicate the position of the point and the coordinates of the intersection point is stored in the computer.

Figure 7.1. Optical sectioning of a cell. An isotropic line L\ through a reference point in the cell is also shown.

7.2 IMPLEMENTATION OF LOCAL DESIGNS

177

If the focal plane through O can be regarded as an isotropic plane (e.g. because the tissue block has an Isotropic orientation in the original tissue), thee L\ can be generated in the focal plane» as a line making a uniform angle with a ixed line in the focal plane through O. The next simplest local design is that of an isotropic plane L2 through O. Unless L2 is parallel to the focal plane, the intersection between L2 and the focal plane will, when the focal plane moYes up and down, appear as a moYing line» cf. Figure 7.2. In fact, the plane L2 is not generated in the tissue block, but only its 'shadow* in the form of the moYing line. This design can be expanded by using a uniform line grid on 1/2. Unless the lines are parallel to the focal plane» they will appear on the computer screen as a grid of points on the moving line, cf. Figure 7.2. Sometimes» the focal plane through O may be used as L2.

Figure 7.2, The local plane design.

178

7. DESIGN AND IMPLEMENTATION

The local slice design will appear on the computer screen as a moving band» cf. Figure 7.3, unless the slice is parallel to the focal plane. If the information in the slice is subsampled by a line grid» such a grid may appear as a set of moving points inside the band. Points may move to the boundary of the slice band and disappear and other points may be bom. In other cases, it may be useful to subsample information in the slice with a plane grid, which may appear as a set of moving parallel lines (segments) inside the band. If the focal plane through O can be regarded as an isotropic plane» then the slice can be chosen parallel to the focal plane.

Figure 7.3. The local sice design.

7.3 Local stereological estimators In use In this section» we give a survey of local stereological estimators in Ä 3 . They are all associated with local designs which can be implemented with the type of techniques described in Sections 7.1 and 7.2. Furthermore, the measurements required for determining the estimators given in this survey can be collected with the techniques described in the previous sections. Using computer generated lines, planes, slices etc., it is possible to make independent replications.

7.3 ESTIMATORS IN USE

179

In order to abbreviate the notation, we use the term vertical plane in Tables 7.1-7.4. We call Z/2 a vertical plane, if Z/2 is an isotropic plane, containing a fixed line L^y This line is called the vertical axis, although £1(0) does not need to be the z—axis. In fact, the suitable choice of £1(0) will depend on X. Furthermore, some of the designs given in the tables below involve a line grid G\. The line parallel to the grid lines, passing through O, is denoted L\ which should not be confused with the vertical axis I/i(o). The vector LJ in Table 7.1 is a unit vector spanning L\.

Design

I:

Estimator

Isotropic L\

h

E

(-i)a(x)IMI3

xedXDLx

U:

Isotropic Z/2 Uniform G1CL2

III:

Vertical L2 Uniform Gx C L 2 Li -L Z/i(o)

IV:

Isotropic T2 Uniform G\

V:

Vertical T2 Uniform G\

2AX

1 ix a? ) 1

E

(-i)«(*yj"jV2+\\7rL2eLiXfdv

xed(XnL2)nGl

\AX

o

E (-l) a ( x ) ||7r L l x|| 2 xed(xnL2)nGi

æGo(XnT 2 )nGi

0

IITT^II 2 ))- 1 ^

^2

£ (-i)a(x) xEö(xnr2)nGi

/ 0

^i,i(* 2 /(^ 2 +

l 11*7(£i(o)+Li) r r ^A?))~ dv " " 11

Table 7.1 . Local stereological estimators of volume. The local estimators of volume given in Table 7.1 have been derived in Proposition 4.7, Exercise 4.6, Exercise 4.7, Exercise 6.7 and Exercise 6.8, respectively. All of these

180

7. DESIGN AND IMPLEMENTATION

estimators have special names in the applied stereological literature, cf. Gundersen (1988), Jensen & Gundersen (1993) and Tandrup et al. (1997). See also Cruz-Orive & Roberts (1993). In the order indicated in Table 7.1, the names are the nucleator, the isotropic planar rotator, the vertical planar rotator, the isotropic optical rotator and the vertical optical rotator. Below, the corresponding table for surface area is shown. The first estimator is the only one with a special name in the applied stereological literature, viz. the surfactor, cf. Jensen & Gundersen (1987a). We have derived this estimator in Section 5.6. The measurements required for determining the surfactor are only easy to collect if the focal plane through O can be used as L2. The last three estimators in Table 7.2 can be found in Section 6.5 as m^^X n T2; O), rnf^^X n T2; L 1(0) ) and (6.8) with d = 2. o = l, n = 3 and v = 2. See also Proposition 6.2. Estimator

Design

I:

Isotropic L 2 Isotropic Li C L2

II:

Isotropic T2 Uniform and isotropic G\

III:

IV

Vertical T2 Uniform and isotropic G\

Isotropic T2 Gi || T2

2ir

£ (l + c o t & ( f - / y ) | | : r | | 2 xexnLi

2A2

2A2

£

xexnT2nGi

£ xeXr\T2nGi

2^

£

iMtVlMI2)"1

F u (t 2 /lk Li . dl 2 )- 1 1(0)

Fx^Vll^arll 2 )- 1

xexnT2nGi

Table 7.2. Local stereological estimators of surface area. In Table 7.3, the local estimators of length are shown. They can be found in Section 6.5 as m^2(X n T2; O), m ^ 2 ( X n r 2 ; L 1(0) ) and (6.9) with d = 1, g = 2, 7i = 3 and p = 2. The line L\ in the last row of Table 7.3 is the line through O perpendicular to G^.

7.4 PARTICLE AGGREGATES

181

The estimator of number given in the first row of Table 7.4 has been derived in Section 6.1. The derivation of the estimator in the second row of Table 7.4 is immediate.

Design

I:

II:

III:

Estimator

Isotropic T2 Uniform and isotropic G
Vertical T 2 Uniform and isotropic G
Isotropic T2

2Å!

xeXnT2nG2

(

x€XnT2nG2

2A,

G2±T2

Fi^VlNI 2 )" 1

E

'

Fu^/lkLxxH2)-1

E xexnT2nG2

Table 7.:3. Local stereological estimators of length.

Estimator

Design

I:

Isotropic T2

^(«VIMI2)"1

E xeXPiT2

II:

Vertical T2

£

xexnT2

2 1 Fu^/lkL^^II )" 1(0)

Table 7.4. Local stereological estimators of number.

7.4 Particle aggregates Any of the estimators given in Tables 7.1-7.4 can be applied locally to each particle in a sample of particles, thereby obtaining information on a particle size distribution. We use here the name particles instead of cells to emphasize that the local methods

182

7. DESIGN AND IMPLEMENTATION

are applicable not only to biological cell populations. A model-based treatment of the problems discussed in the present section can be found in Chapter 8. Let us consider a particle aggregate which is a union of A disjoint d—dimensional particles Xi,..., Xjy. We suppose that X^Xi) < oo for all i. For d = 3, the typical example is biological cells. For d — 2, the particles may be cell boundaries, for d = 1, linear or very thin tubular structures inside biological cells. For d = 0, the particles may simply be clusters of points sitting around reference points. Let us suppose that we want to estimate stereologically the distribution of the d—dimensional Hausdorff measure of the particles. This distribution has distribution function 1 N 2=1

In particular, we are interested in moments of this distribution

/^ = ^ X > ^ , g = i,2,... i=l

The first step is to collect a sample of particles. Let us suppose that the sample consists of those particles with index in the random subset 5 of { 1 , . . . , A } . In the applications we have in mind the particles are not selected independently, but it is known that P(i e S) does not depend on i e { 1 , . . . , A } , i.e. P(i <E S) = p. Designs of this type based on the disector have been discussed in Section 1.2. Since P(i G S) = p, the sample distribution of particle d—dimensional Hausdorff measure is identical to the population distribution. To see this, let the sample distribution be defined by the distribution function G, satisfying

E J2 1{AÄ) < y} = cG(y), y e R, ies where c is the mean number of sampled particles. Then,

cG{y) =

EYJ1{<{Xi)
=

Ej2HieS}l{Xi(Xi)
= £P(ieS)l{A^)<2/} i=l

=

NpF(y),yeR.

Since F(oo) = G(oo) = 1, F = G.

7.5 APPLICATIONS

183

Thus, if we can estimate A^(Xi), i G S, with high precision, we can directly study a sample from the distribution of interest. If the particles are of spherical shape, then we can measure the size directly on a central section. The general situation is however that the particles are of varying and unknown shape. In this case it is in principle possible to estimate precisely AJ*(Xi), i G 5, with the observation techniques described in Sections 7.1 and 7.2, using for each sampled particle the average of independent replicated estimates of the type given in Tables 7.1-7.4. (It is here generally believed that estimates based on information in a slice are more precise than estimates based on information in a single section.) There may be technical problems in doing this, however, using conventional light microscopy on a population of biological cells. The problems occur at the top and bottom of the cells. In these parts of the cell, measurements may be difficult because cell borders are cut tangential by the focal plane and therefore have a fuzzy appearance. This may cause over- or underprojection. In cases with such technical difficulties, we can still estimate some of the moments \idq by applying a local estimator on each sampled particle X^i G 5, and then take the sample average

ieS

Here k is the number of sampled particles. For q = 1, we can use one of the estimators in Tables 7.1-7.4 and we thereby obtain an estimate of mean size. For the estimation of second-order moments (q = 2), the results in Section 4.5 and Chapter 5 can be used.

7.5 Applications of local stereological methods Local stereological methods are now in world-wide use in many areas of biology, most importantly in neuro science and cancer grading. Significant applications in neuro science can be found in Pakkenberg et al. (1991), Badsberg Jensen & Pakkenberg (1993), Tandrup (1993), Larsen et al. (1994), Regeur et al. (1994), Korbo & Andersen (1995), Oster et al. (1995) and Tandrup (1995). The use of local methods in cancer grading has been studied in Sørensen (1989), Ladekarl (1995) and Ladekarl et al. (1995). In this section, we will discuss a comparative study of local methods published in Tandrup et al. (1997). The goal of this study was to estimate the size of neurons from the dorsal root ganglion in rat. The center of the nucleolus was chosen as reference point for a neuron. Isotropically tissue blocks were cut from the ganglion. The focal plane in such a tissue block can therefore in this case be considered as isotropically orientated. A total of three to four neurons were sampled from each tissue block. This sampling scheme resulted in 44 neurons, with equal representation of 12 isotropic directions. Neurons

184

1. DESIGN AND IMPLEMENTATION

collected from the same tissue block are cut in the same direction. The structure is illustrated in Figure 7.4. As size parameter, both volume and surface area of the neuron were used. For ¥olume, three different designs were examined, two of which were genuine local designs» viz. Design II and IV of Table 7.1. The third design was the global design ' described in Section 1.3 as the spatial point grid design. Surface area was estimated using two local designs, Design I and IV of Table 7.2, and the global design described in Section 1.4 as the spatial line grid design. The distances between planes etc. were chosen such that the sampling density was not dependent on the size of the neuron.

Figure 7.4. A section through the nucleolus of a so-called type A neuron in the rat dorsal ganglion (diameter 50 |im). From Tandrup et al. (1997), with permission.

In Figure 7.5, the optical rotator estimates (Design IV) are compared to the estimates based on the global designs (spatial point and line grid). The estimates based on the global designs give consistently lower results. This is in accordance with the experience of the 'observer', who found it difficult to observe at the peripheral parts of the neuron. It appears as if some observations are missing in this part. The estimated 'bias* was 22% and 17% for volume and surface area, respectively.

7.5 APPLICATIONS

Figure 7.5. Dataplot illustrating the observational 'bias'. Design IV refers to Table 7.1 for volume and Table 7.2 for surface area, respectively. A total of 44 neurons have been analyzed (from Tandrup et al. (1997), with permission).

185

186

7. DESIGN AND IMPLEMENTATION

In Figure 7.6 below, the distribution of the estimates of volume and surface area are shown. The a¥erage in each of these sample distributions (except for the two in the first row) can be regarded as an estimate of the mean population size, as explained earlier in Section 7.4.

Figure 7.6. Distributions of estimates (from Tandrup et al. (1997), with permission). H e designs can be found in Table 7.1 (¥olume) and Table 7.2 (surface area), respectively.

7.6 SYSTEMATIC SAMPLING

187

In the application we have described, the tissue blocks had an isotropic orientation and the focal plane could therefore be used as an isotropic plane. This kind of design is useful when mean size is the parameter of interest. If the whole size distribution is needed, then the sampling techniques illustrated in Section 7.1, involving computer generated random lines, planes, etc., should be used. If technically possible, we can then study the full size distribution and furthermore the observer has the advantage that it much easier to keep track of the structure, if the tissue blocks do not have random orientations.

7.6 Systematic sampling along an axis When designing a local stereological experiment, an important question is of course to decide the density of points, lines or planes, involved in the systematic sampling. In this section, we will consider the simplest possible case, viz. systematic sampling on the real line. Except for the fact that it is always a good idea to start with the simpler cases, the results of this section are also relevant for sampling in R? with a uniform plane grid and for subsampling in a plane by a uniform line grid. More precisely, in this section we discuss the problem of estimating an integral of the type oo

Q= J f(x)dx,

(7.1)

— CO

where / : R —> R is an integrable function. The estimator considered here is oo

QA = A J2 f(U + jA),

(7.2)

j=—oo

where U is a uniform random variable in the interval [0, A). This estimator can be regarded as a Horvitz-Thompson estimator. Local examples can be found in Table 7.1, Design II and III. The mean-value of Q A is equal to Q, which can be shown easily by standard techniques. The question is now how the variance depends on A. If / is squared integrable, then the variance of Q A is finite and can be expressed in terms of the covariogram g of f oo

9(y) = / f(x)f(x —oo

+ y)dx.

188

7. DESIGN AND IMPLEMENTATION

We thus have ~, Var(Q A ) = A

^

°° g(kA) - j

k=-oo

g(y)dy.

(7.3)

_00

This result can be shown by rewriting each of the two terms in

Vzi(QA) =

E(Qi)-E2(QA),

cf. Exercise 7.3. Note that according to (7.3), Var(QA) can be expressed as the error made by approximating the integral of g with a discrete sum. In Kiéu (1997), see also Souchet (1995), an approximation to the variance has been derived under regularity conditions on the function / . This approximation represents further developments of work by Matheron (1965, 1971). The regularity conditions on / are as follows. It is assumed that there exists a non-negative integer m and a finite set of points ai G R, i — 1 , . . . ,p, such that 1. 2. 3. 4.

/ is of class C™-1 on R. f is of class Cm~*~2 on R\{a\,..., ap} where [a\, ap] is the support of / . For i = 1 , . . . , p andfc= ra,m + l , r a + 2, the limits from the right / ^ ) ( a + j from the left f^(a~) exist and are finite. The derivatives of / upto order m + 2 included are integrable functions.

ancj

Note that m is the order of the first derivative of / which may have jumps. It is shown in Souchet (1995) that under the conditions 1-4, the covariogram is of class C 2 m on R and of class C 2 m + 3 on Ä\{a» - CLJ : i, j = 1 , . . . , p } . Since the covariogram is a symmetric function g(y) = g(-y),y€

R,

it follows that under the conditions 1-4

9uH-y) =

(-iY9U)(v),v€R,

j = 0 , 1 , . . . , 2m. In particular, ^ 2 f c +i)(0) = 0,fc = 0 , l , . . . , m - l . As shown below, the variance approximation depends on g(2m+l){b+)

_

ff(2m+l)(r))

f, £

{a.

_

a

.

: i?

j = 1, . . . , p } .

(7

.4)

7.6 SYSTEMATIC SAMPLING

189

Very often, it is difficult to calculate the covariogram analytically and it is therefore of great importance that (7.4) can be calculated in terms of the original function / . Again under the conditions 1 ^ , we have for b e {ai - a,j : i, j = 1 , . . . ,p}, cf. Kiéu (1997),

= (-l) m + 1

[f{m)(4) - / ( m ) K-)][/ ( m ) «) - / (m) (aj)].

£ 2

{{^j)e{K^p} \ax-aj=b}

In particular, since 0 £ {ai — CLJ : i,j = 1 , . . . ,p}, we get ø( 2m+1 >(0 + )

-i(^ ( 2 m + 1 ) (o + )-^ ( 2 m + 1 ) (o")) = (-l) m+1 ^ E[/ ( m ) (^ + ) - / (m) K~)] 2 -

(7.5)

Example 7.1. Let / W

~ \ 0

|z|>l.

The function / is thus the area of the intersection between the unit ball in R3 and a plane at distance |a;| from O. The parameter of interest CO

Q = f f{x)dx — CO

is then the volume of the unit ball. Note that / satisfies the conditions 1-4 with m = 1, p = 2 and a\ = —l,a2 = 1. By elementary, but somewhat tedious calculations, we can derive the covariogram in this case. We get in particular

g(y) = K2 x (~\yf

+^|

3

- \\y\2 + ^ ) for \y\ < l.

Using (7.5), we find that £ ( 3 ) (o + ) = ^ ( [ / ( 1 ) ( - i + ) - / ( 1 ) ( - i - ) ] 2 + [/ ( 1 ) (i + ) - / ( 1 ) ( i - ) ] 2 ) = 4TT2.

The same result is obtained when differentiating g directly.

□

190

7. DESIGN AND IMPLEMENTATION

The variance approximation can be derived, using a refined Euler-MacLaurin formula. Recall that the Bernoulli numbers JE^, k = 0 , 1 , . . . , are defined by B0 = l ^ ( * k

= 4 t = 2,3,....

(7.6)

It can be shown that the Bernoulli numbers with uneven index all are 0, except for number 1. The Bernoulli polynomials are also used in the refined Euler-Maclaurin formula. The version of these polynomials used here is for k = 0 , 1 , . . . ,

' i/=0 ^

'

In some standard text books, one can also find the version

The Bernoulli numbers are Bk = k\Pk(0) = Bk(0). For later use, notice that for k — 1,2,... (use that P£ +1 (x) = Pk{x) and Pk+i(l) = ft+i(0)) l

j Pk(e)de = 0. o Elementary properties of the Bernoulli numbers and the Bernoulli polynomials are derived in Exercise 7.4. Below, we present the refined Euler-Maclaurin formula. We use the following short notation where [•] indicates integer part. Proposition 7.2. Let h : R —> R be a function satisfying conditions 1-4 with m > 1. Then,

A £ ) h(kA) - J h(y)dy fc=-oo

_co

= (-l) m A m + 1 5>< m >(a+) - ^m\a-)]Pm+1A(ai) 2=1

+ 0(Am+2).

D

7.6 SYSTEMATIC SAMPLING

191

The proof of Proposition 7.2 is discussed in Exercise 7.5. We can now derive an approximation to Var(QA) by applying Proposition 7.2 with h equal to the covariogram g of / , cf. (7.3). As mentioned earlier, when / satisfies conditions 1-4, then g also satisfies conditions 1-4 with m replaced by 2ra+l. The resulting variance approximation is formulated in the proposition below. Proposition 7.3. Suppose that / : R —> R satisfies conditions 1-4. Then, if B = {ai - CLJ : ij = l , . . . , p } and g is the covariogram of / , we have Var(Q A ) _

~

o A 2m+2

B2m+2

j2m+l)(n+\

[ö (2m + 2)! 5 > _A2m+2 J P 2m+2jA (ft)

66B\{0} x[fl(2m+l)(6+)_5(2m+l)(6-)]

+ o(A

2m+2

) D

Recall that the support of / is [ai,a p ]. The mean sample size is then oo

n = E Y^

!{U + jA e [auap]}

j=—oo

= (ap - ai)/A. 2m+2

Therefore, A corresponds to a term of order n~(2m~*~2\ The first term in the variance approximation is the first non-zero term of the type suggested by the correspondence principle in Matheron (1965, 1971). This term is called the extension term and denoted by E(A). Using (7.5), we find E(A) = - 2 A 2 m + 2

(2^

2m+

' ^(2m+1)(0+)

The second term in the variance approximation is the so-called Zitterbewegung or fluctuation term Z(A) = -A2m+2

]T beB\{0}

P2m+2A(b)l9{2m+1)(b+)-9(2m+1\b-)}-

192

7. DESIGN AND IMPLEMENTATION

Recall that P2m+2,A(&) = P2m+2& - [ £ ] ) with 0 < I - [I] of this term is zero in the sense that

< 1. The 'average'

l

/ , p2m+2(e)de = 0. 0

Example 7.1. (continued) Using the derivatives derived earlier, the first term of the variance approximation becomes, using that n = 2/A and B4 = - ^ , tf(A) = - 2 A < ^ < s > ( 0 + ) = ^ » - < . The contribution to the squared coefficient of error becomes

io n '

(i*)2

which is a well-known result, cf. e.g. Jensen & Gundersen (1987b). In Figure 7.7, the approximation to the coefficient of variation provided by the extension term is compared with the exact coefficient of variation. □ In practice, one tries to approximate Var(QA) by the extension term E(A) which is the central tendency of Var((§A) since the Zitterbewegung is made of functions oscillating around 0, cf. Figure 7.7. Robust estimation of the extension term, including the smoothness parameter m, is the subject of current research, cf. Kiéu (1997).

7.7 The circular case Systematic sampling along an axis does not cover all examples of interest in local stereology. In some cases, the parameter of interest can instead be expressed as an integral on R2, the planar unit circle or the unit sphere. In this section, we will concentrate on the circular case. The parameter to be estimated is then 2TT

Q = Jf(e)de, 0

7.7 THE CIRCULAR CASE

193

Figure 7.7. The coefficient of error of the estimator of the volume of a sphere, based on systematic sampling, is shown as a function of the average number of section planes hitting the sphere. The dotted curve is the approximation based on the extension term (from Jensen & Gundersen (1987b), with permission).

where / : [0, 2ir) —> R is an integrable function. The estimator considered is

0» = - £ / ( © + • - ) . n ^—'

n

where 0 is uniform in the interval [0, 27r/n). Again, if / is squared integrable, the variance of the estimator can be expressed in terms of the covariogram

(7(a) = y " / ( ö ) / ( ö + a ) d ö , a G [ 0 , 2 7 r ) , o where we use a periodic continuation of / . Note that g(a) = g(27T-a),ae

[0,2TT).

Furthermore, n-l

27r

Var(Qn) = — J2 9U-) ~ \ ÅW-

0-1)

194

7. DESIGN AND IMPLEMENTATION

It is possible to modify the theory developed for systematic sampling along an axis to the circular case. Here, we will just go into detail with a single local example, viz. the estimation of the area of a p—sided planar convex polygon X from measurements along an isotropic ray through O. The intersection between the ray and X is a line-segment. For a ray making an angle 0 e [0, 2TT) with a fixed axis, we let l+{0) and l-(0) be the largest and the smallest distance, respectively, from the end-points of the line-segment to the origin O. Then, the area of the figure can be expressed as

Q = Jf(9)d0, o 2

where f(0) = \(l+(0)

-

2

l-(0) ).

Let us first study the analytical properties of the function / , when O is in the interior of the polygon. Note that in this case l-(0) = 0 and f(0) = \l+(0)2. Suppose that we have a corner at 0 G [0, 27r), cf. Figure 7.8. Then, f(0

+

dO) = i - W

1

+ dOcot p(6-))2

\/(0)(l-d0cot/?(0+))2

, dO < 0 ,d0>0

and therefore / ( 1 ) ( 0 + ) - Z ( 1 ) ( n = -2f(0)(cot

/?(#+) + cot f3(0-)).

Figure 7.8. Notation used for the polygon example.

If 0 i , . . . , 0p are the angles associated with the p corners of the polygon, then the variance approximation based on the extension term becomes P 4 Var(Q n ) « -7T 4 £ f(Oi)\cot0(0+)

i=l

+ cot/?(0r)) 2 * n~\

7.8 EXERCISES

195

Next, let us consider the case where O is on the boundary of the polygon. Then, the function / is not continuous. It jumps at #o and #o + n, where #o is defined in Figure 7.8. The variance approximation based on the extension term becomes 2

Var(Q„) « y (/(0 O ) 2 + /(to + *?) x n~2. Finally, if O lies outside the polygon, the asymptotic behaviour can be of the order of magnitude of n~2 or n - 4 . The first situation occurs when some (atmost 2!) of the polygon sides lie on a ray from O.

7.8 Exercises Exercise 7.1. Explain in detail how the local designs shown in Figures 7.1-7.3 can be generated, using pseudo random numbers. Exercise 7.2. In this exercise, we review local estimators in R2. 1. Let X be an open and bounded subset of R2. Show that the available local estimators of its area are as indicated below. In the second row of the table, L\ is the line, passing through O, parallel to the line grid G\ and a; is a unit vector spanning L\. Design

Estimator

I

Isotropic L\ Isotropic T\ Uniform G\

Al

E

(-ir(x)iMi2

£ xedXOLi

xed(XDTi)nGi 0

2. Let X be a smooth planar curve with finite length. Show that the available local estimators of its length are as indicated on the next page. The angle between L\ and the tangent to X at x is denoted j x . We assume that 7æ / 0 for almost all

x e x.

196

7. DESIGN AND IMPLEMENTATION Estimator

Design Isotropic L\

7T ]T

Ikll/sinia;

xeXnLi

Isotropic T\ Uniform and isotropic G\

Fi^VlMI2)-1

£

\AX

3. Let X be a finite set of N points in R2. Show that the local estimator of N is Design

Estimator

Isotropic T\

E

W7IMI 2 )- 1

xexnri

Exercise 7.3. Derive the expression (7.3) of the variance of Q A in terms of the covariogram. Exercise 7.4. In this exercise, we will derive some of the elementary properties of Bernoulli numbers and Bernoulli polynomials. 1. The Bernoulli numbers have been defined in (7.6). Note that

Show that 2 B± = 30

6

, B5 = 0,B6 = —. 42

2. Show for k = 0 , 1 , . . . that k\Pk(x) is the unique polynomium f(x), satisfying

f(x + l)-f(x)

= kxk-\f(0)

= Bk.

(•)

7.8 EXERCISES

197

Note that this implies that P*(l) = P * ( 0 ) f o r A ; ^ l . Hint. Suppose that f(x) is a solution to (•). Let m

f(x) = Y,c*xli=0

Find a polynomial expression for f(x + l) - f(x) and compare its coefficients with those of kxk~l. 3. Show for k = 1 , 2 , . . . that Pfk(x) = 4.

Pk_1(x).

Show for k = 0 , 1 , . . . that (-l)kPk(l-x)

= Pk(x).

(..)

Hint. Consider the difference equation = (k + l)xk.

f(x + l)-f(x) We know that f(x) = (k + l)\Pk+i(x)

is a solution to this equation. Show that

/Or) = (-l) f c + 1 (fc + l ) ! P f c + 1 ( l - a ; ) also is a solution. Using the uniqueness of the Bernoulli polynomials, it follows that there exists a constant CQ such that {-1)MPM(1

-x)

= C0 + Pk+1{x).

(•••)

Differentiating on both sides of (•••), using question 3, we get (••). 5. Use the result in question 4 to show that Bernoulli numbers with uneven index are zero except for B\, i.e. S 2 i + i = 0 , j = l,2,....

198

7. DESIGN AND IMPLEMENTATION

Exercise 7.5. This exercise concerns the proof of the refined Euler-Maclaurin formula. We suppose that h : R —► R satisfies conditions 1 ^ - with m > 1. 1.

Show that P/A is continuous for / ^ 1 and that P I 5 A jumps at multiples of A with Pi, A ((A;A) + ) - P1A((kA)~)

2.

= - 1 for all k.

Show that oo

oo

i J h(y)dy + J h'(y)P1A(y)dy = ^ —oo

—oo

h kA

( ^

«

k=—oo

Hint. Start by rewriting the left-hand side of (•) as oo

/

(h(y)PiA(y))'dy.

—oo

Next, decompose R in disjoint intervals Ä = (-cx),6i)U(Ug 1 [6 i ,6 i + i)) such that h • PI 5 A is continuously differentiable in each of the intervals (6;, fø+i). Next, use partial integration at (••) below to show (bo = — oo) oo

/

{Hy)PiA(y))'dy

—oo oo

^ E w v + i ^ i ^ r + i ) - Hbf)p1A(bt)) oo

= -^^)[PljA(6+)-P1)A(6-)] 2=1

oo

= - 5 2 MfcA)[PliA((fcA)+)-P1)A((fcA)-)] &=—oo oo

= E M*A). &=—oo

3.

Next, show that

7.9 BIBLIOGRAPHICAL NOTES

199

oo

Af

h\y)P1A{y)dy

— OO

= (-l)mAm+1(I>(m)(^) -

h^\ar)]Pm+1A(ai)

i=l OO

+ f ^m+1\y)Pm+1A(y)dy).

(...)

— OO

Hint. Start by showing by the same type of technique as used in question 2 that OO

OO

I h"(y)P2A(y)dy + ± j h'{y)P1A(y)dy —oo

—oo

+

= - £[Ä'(a )-Ä'(a-)]P 2 , A (a), aeDh,

where LV is the set of points where h! jumps. If m = 1, we are finished, otherwise we continue along the same lines. The proof can be completed by evaluating the integral on the right-hand side of (•••). Exercise 7.6. Prove Proposition 7.3, using Proposition 7.2 and properties of g. Exercise 7.7. Derive the expression (7.7) of the variance of Qn in terms of the covariogram for the circular case.

7.9 Bibliographical notes Optical sectioning as discussed in Sections 7.1 and 7.2 is at its best when using a confocal microscope, cf. Petran et al. (1968) and Howard et al. (1985). There exists however quite a number of examples from biology where conventional light microscopy works satisfactory, cf. Gundersen (1986) and Gundersen (1988a, b). The sampling procedures and their associated estimators, presented in Tables 7.1-7.4, have been implemented in the C.A.S.T.-GRID computer program (Olympus, Denmark) which also contains routines for collecting the data needed in the global stereological estimation procedures, described in Sections 1.2-1.4. C.A.S.T. is short for Computer Assisted Stereological Toolbox. The program is an interactive, user friendly, data collection program which has been developed by Morten Bech, Olympus, in collaboration with theoretical and applied Danish stereologists, cf. Bech (1996).

200

7. DESIGN AND IMPLEMENTATION

Sections 7.6 and 7.7 contain a short account of the very recent developments in systematic geometric sampling, cf. Souchet (1995), Kiéu (1997) and Kiéu et al. (1997). This intriguing subject has earlier been studied from a stereological view-point by Cruz-Orive (1985, 1989a, 1993), Gundersen & Jensen (1987), Kellerer (1989), Matérn (1989) and Mattfeldt (1989).

Chapter 8

The model-based approach In the previous chapters, we have assumed that the structure under study is deterministic and the statistical properties of a random geometric sample of the structure have been discussed. This is the design-based approach. In the present chapter, we will reverse the situation and consider a random structure, having certain invariance properties. The intersecting lines, planes etc. are now non-random. This approach is often called the model-based approach although this terminology is somewhat misleading since the models considered are non-parametric models of a very general type. More specifically, we will consider a particle process, which can be represented as a so-called marked point process ^ = {fø; £;]} in Rn. Here, X{ is a point in Rn with mark E^ being a d—dimensional differentiable manifold in Rn with A^(Hi) < oo. The dimension d can take the values 0 , 1 , . . . , n. The point X{ is the reference point of the i'th particle X{ — xi + Hi, cf. Figure 8.1. It turns out that the results obtained in the previous chapters also can be used in the statistical analysis of such a random structure. We start by giving an introduction to point processes and marked point processes in Rn.

8.1 Point processes in Rn Let Bn be the Borel cr-algebra in Rn and let BQ be the bounded Borel sets. The outcome of a point process in Rn is a locally finite set, i.e. a member of Nlf = {(pC Rn\(f>n B is finite for all B e B%}. Note that if (j> e Nlf, then ø is atmost countable and closed. Below, we will use, when convenient, the notation <j>(A) for the number of points from in A G Bn, xE(f>

201

202

8. THE MODEL-BASED APPROACH On N1*, we let Aflf be the smallest a-algebra, such that {(/) E Nlf : 0(A) = n} G Afli",

for all A e BQ and all n = 0,1, process in Rn.

With these concepts, we can define a point

Definition 8.1. A simple point process $ in Rn is a random variable taking values in

{Nlf,Aflf).

D

The point process is called simple because no points from $ coincide. The distribution of $ is denoted P. We thus have P{C) = P ( $ eC) = P({u : $(£4;) G C}), C E Åflf. A point process $ = {xi} in Rn is said to be stationary, if for all x E Rn, the distribution of $ + x = {xi -\- x} is the same as that of $. Likewise, a point process $ in Rn is said to be isotropic, if for all rotations B G 50(72)

5 $ = {Bxi} and $ have the same distribution. The first-order properties of <3> can be described by the intensity measure which is the following measure A on (Rn,Bn), A{A) = E$(A) = f (A)P(d),Ae Bn.

If <£ is stationary, then A(A + x) = A(A), for all AeBn,x£

Rn.

Since the Lebesgue measure is, up to multiplication by a positive constant, the only measure on (Rn,Bn), which is translation invariant, cf. Appendix, we must have under stationarity

A(A) =

\V(A),AeBn,

8.1 POINT PROCESSES

203

where ¥(= An) is Lebesgue measure in Rn. The constant A of proportionality, is called the intensity of the process. We will from now on assume that 0 < A < oo.

Figure 8.1. The i'th particle is obtained by translating the primary particle z., with xi. According to the definition of the intensity measure, wc have E

YL 1Vr £ A\=

I 1U € A}AUr).A £ Bn.

This equation can be generalized as follows: let / be a non-negative measurable function on R11. Then,

Ej2fi^=

I fUIA(dx) = A / f(i)dxf\

under stationarity.

(8.1)

it"

The result (8.1) is usually called the Campbell theorem. An important concept for a stationary point process in R!! is the Palm distribution Pn. which is a probability measure on {Nl^\N1^). Let A £ Bn be any Borel set with 0 < \~{A) < y:. Then, the Palm distribution is defined for C G A' / ; by Pn{C) = E Y^ =

/

Yl

1{® ~ * e

C}/E$(A)

l(o-reC)P{do)/XV(A).

(8.2)

204

8. THE MODEL-BASED APPROACH

The Palm distribution P0 can be interpreted as the distribution of $, when the origin O in Rn has been chosen as a typical point from $. Because of the stationarity of $, P0 defined in (8.2) does not depend on the initial choice of the set A. Furthermore, if <$ is isotropic, then P0 is invariant under rotations. The Campbell theorem can be refined as indicated in the proposition below. Proposition 8.2. Let $ be a stationary point process in Rn and let h:Rn

x Nlf -+ Æ+ U {0}

be a measurable function. Then, E Y^ h(x, $) = A / xe^ Rn

/ K*, 4> + x)P0(d4))dxn. Nlf

Proof. Let g(x, 4>) — h(x, (j> + x). Expressed in terms of g, we want to show that E J2 9(x, $-x) XG$

=\ J j Rn

g{x, )P0(d(l>)dxn.

(8.3)

Nlf

If we can show (8.3) for g{x, ) = l{(x, ) G F}9 where F G Bn 0 Aflf, then the standard proof implies that (8.3) holds for all measurable functions g. Since {AxC

:AeBn,C

eAflf}

is a so-called (n/)—stable paving, it furthermore suffices to check (8.3) for g{x, 4>) = l{x G A, 0 G C}, AeBn,C

G Åflf',

cf. e.g. Hoffmann-Jørgensen (1995a, Section 1.7). For such a choice of g, (8.3) becomes

E J2 H*-x€C}

= \V(A)P0(C),

x£$nA

which is simply the definition of the Palm distribution. The result of the proposition now follows. □ The second-order properties of the point process can be described by the second factorial moment measure defined by aW(A1xA2)

=E

J2 Xl,X2£$

l{x1eA1,x2eA2},AuA2eBn.

8.1 POINT PROCESSES

205

The notation ^ above the summation sign is used to indicate that the sum is taken over pairs of different points from $. Note that the second-order moments E$(Ai)$(A2) can be expressed in terms of ofö and the intensity measure A. Under the assumption that $ is stationary, a^ can be expressed in terms of the so-called second reduced moment measure /C, defined by XK(A) = E0$(A\{0})

= J

^(A\{0})P0(#),AG

Bn.

Nlf

To see this, we use the refined Campbell theorem at (*) below a{2)(AixA2)

=E

]T

= J E1^ Nif

l{x1eA1,x2eA2} G

M}{A2\{x})P{d<j>)

xE(j>

x){A2\{x})P0(d^)dxn

- * [

[ l{x e Al}{^ +

= XI

j l{x G A1}(f)((A2 -

x)\{0})P0(d(j))dxn

Rn Nlf

= X2 I

JC(A2-x)dxn.

If in addition $ is isotropic, K is rotation invariant and it suffices to look at the function K(r) = JC(Bn(0,r)), where Bn{0,r) as usual is the open ball in Rn with centre O and radius r. Note that XK(r) = E0$(Bn(0,r)\{0}), the mean number of points from $ in a ball with radius r, centred at a typical point from <1>, which is not counted itself. The if-function is often used to describe clustering or inhibition compared to the Poisson process and is an important tool in the construction of parametric models for point process data. For a Poisson process, we simply have K(r) = V(Bn(0,r))

= ujnrn.

So-called local intensities can be derived from the K—function, x

XK(r2) - XK(n)

n

For a Poisson process, the local intensities are equal to the global intensity A. Sometimes, the relative local intensities are considered, i.e. A(ri,r2)/A. The relative

206

8. THE MODEL-BASED APPROACH

local intensity can be regarded as a discrete version of the so-called pair-correlation function, cf. Stoyan et al. (1995, p. 129). In Section 8.3, we discuss the stereological estimation of the K—function.

Figure 8.2. Examples of planar point process data. From the left, random, clustered and regular point patterns, respectively.

8.2 Marked point processes in Rn The particles will be represented by a marked point process in Rn, which is a random sequence ^ = {[x^S^]} such that the points <£ = {xi} constitute a point process in Rn (not marked). Furthermore, £; is the mark corresponding to Xi, where £; G Md, the set of d—dimensional differentiate manifolds in Rn with finite Hausdorff measure. The Borel a—algebra of Md is denoted MdThe i'th particle of the particle process is Xi + E{. The point X{ will be called the reference point of the z'th particle and E{ will be called the primary particle. We assume that O G E{. Examples for d = 0 and 1 are shown in Figure 8.3. Stationarity and isotropy are defined with reference to the particles. For instance, if the i'th particle is translated by x G Rn, then the reference point of the new particle will be Xi + x, while the primary particle is unchanged. Thus, a marked point process \P is said to be stationary, if for all x G Rn, the distribution of ^ + x = {[xi + x-i'Zi}} is the same as that of \I>. A marked point process is said to be isotropic, if for all rotations B G SO{n) BV =

{[Bxi\BZi]}

and \I> have the same distribution. Note that stationarity (isotropy) of \I> implies stationarity (isotropy) of <£.

8.2 MARKED POINT PROCESSES

207

Figure 8.3. Illustration of marked point processes with marks of dimension 0 and 1, respectively. The intensity measure of ^ is the measure A m defined by A m (A x K) ^EJ^H^

É A,Zi

G K},Ae

Bn,Ke

Md.

i

Note that A m (A x Md) = A(A), the intensity measure of the reference point process $ = {xi}. In case \I> is stationary, the intensity measure can be decomposed as A m (A xK)

= \V(A)Pm(K),

AeBn,Ke

Md.

(8.4)

Here, Pm is a probability measure on (Md,Md), called the mark distribution. We will also use the name particle distribution for Pm. The result (8.4) can be shown by using that stationarity of \I> implies that Am(- x K) is a translation invariant measure on Bn and therefore proportional to the Lebesgue measure, cf. Appendix. Note that if the E^ 's are independent and identically distributed and independent of <£, then Pm is their common distribution. Furthermore, if \£ is stationary and isotropic, then P m is invariant under rotations, i.e. Pm{BK) = Pm{K), B G SO{n), K G Md, cf. Exercise 8.5. The second-order properties of the marked point process can be studied by means of the second factorial moment measure, cf. Stoyan (1984c), a${A1xK1xA2xK2) = E

]T

l{xi G Ai,x 2 G A 2 ,Hi G K i , S 2

[ æi; Hi],[x 2 ;E; 2 ]e^

Note that a^\A1

x A2) = affliÅ! x Md x A2 x Md).

eK2}.

208

8. THE MODEL-BASED APPROACH

8.3 A few results from invariant measure theory In the next sections, we need the following results from invariant measure theory. They are here presented in condensed form and discussed in further detail in the Appendix. As in Chapter 3, we let SO(n,Lr)

= {B e SO(n) : BLr = Lr}.

It can be shown that there exists exactly one (left and right) invariant probability measure a?ry say, on SO(n,Lr). For any measurable non-negative function / on SO(n,Lr) and any B$ G SO{n,Lr), we then have f

f(BQB)afr)(dB)

SO(n,Lr)

=

I

f(BBo)a$r)(dB)

SO{n,Lr)

=

f

f{B)an{r){dB).

SO(n,Lr)

Furthermore, let £p(o) £ £n(r) b e a ^ xec ^ P~subspace containing Lr and let g be a measurable non-negative function on Cn
f

j g(BLm)afo{dB)= so(n,Lr)

dLn

( \

j g{Lv)-^—^—y c;(r)

(8.5)

Note that ££ ( r ) - {BLp{0) : B G SO(n,Lr)}. Likewise, let Tp(0) G TJ, be a fixed p—slice containing Lr and let h be a measurable non-negative function on Tn,y Then,

/ 50(n,L r )

h{BTm)an(r){dB)

=

r / h(Tp)-

dTn ^ — .

(8.6)

T-r)

Note that (8.6) is a consequence of (8.5) and the construction of the rotation invariant measure on p—slices from the rotation invariant measure on p-subspaces, cf. Section 6.2.

8.4 Estimation of the if—function of the reference point process We will throughout this section assume that $ is stationary and isotropic. Let D G #o b e chosen such that V(D) > 0. The usual estimator of A is, cf. Stoyan et al. (1995, p. 134), A = (D)/V(D).

8.4 ESTIMATION OF THE ^-FUNCTION

209

Note that A is unbiased and can be used directly with the observation techniques • described in Sections 7.1 and 7.2. Let us tore to the estimation of the K—function. Recall that XK(r) = EMBn(0,r)\{0}). Let D e BQ with V(D) > 0. Then» XK(r) can be estimated by 'the empirical analogue' \K(r)=

Y,

*(Bn(s.r)\{x})/$(D).

(8.7)

xG#nD

In some cases, the only information aYaiiable is # n £ ) and edge corrections are needed for x € # n D such that Bn(x,r) £ D, cf. Stoyan et al. (1995, p. 135) and Figure 8.4. In the biological sampling situation described in Sections 7.1 and 7.2, usually this problem does not occur.

Figure 8.4. Estimation of the ^-function from information in a bounded sampling window D involves edge corrections.

The estimator XK{r) is ratio-unbiased, i.e.

E E $(B*U.r)\{x})

^TW

= iA"M

<8 8)

'

210

8. THE MODEL-BASED APPROACH

To see this, we apply the reined Campbell theorem and get for the numerator of AA'( r) E ]T = A/

$(B n (,r.r)\{.r}) / \{x e D}(o - x)(Bn{s.

r)\{x})P0(d)dxn

FT- A"'-'

= A/

/ 1{J" e

D}o(Bni(Xr)\{0})TUde)dxn

/>'* A'*-'"

- XV[D) x £* t ) $(5„(0.r)\{0}) = A$(D) x A A » . It follows that (8.8) is Milled. The estimator XK(r) depends on complete observation inside balls centred at reference points. It has earlier been, investigated whether the if—function can be estimated using information from sections through the particle aggregate. It was found that this is only possible under speciic assumptions about the particle shape. In particular» the spherical model has been studied in e.g. Hanisch & Stoyan (1981) and Hanisch (1983). Here, inversion of an integral equation was required, which was found to be an ill-posed problem in the sense that small deviations in the observations could lead to large discrepancies in the estimate. We will now show that if we replace the plane with a slice centred at each of a sample of reference points, then the if—function can easily be estimated in a stereological fashion, without shape assumptions about the particles» cf. Figure 8.5.

Figure 8.5. 2-d illustration of a sampling design of the type described in Sections 7.1 and 7.2 which can be used for estimating the Z-functioe.

8.4 ESTIMATION OF THE /^-FUNCTION

211

Let us once more consider the estimator XK(r). We need to determine the number of points in the finite set $n(Bn(x, r)\{x}) for x e <3>nD. The corresponding designbased problem has been solved in Chapter 6. Here, we showed that the HorvitzThompson estimator of the number of points in a non-random finite subset X of Rn, based on an isotropic p—slice Tp through O, was, cf. (6.2),

E *$)(*.or1xeXnTp

We will use this result in a model-based fashion; the orientation of the slice is now fixed and isotropy of the slice is replaced by the isotropy of the Palm distribution. Proposition 8.3. Let $ be a stationary and isotropic point process in Rn. Let Tp(0), 0 < p < n, be a fixed p—slice in Rn of thickness 2t through O and let D e BQ be a bounded Borel set with positive volume. Then,

\%)=

Y,

E

h^iy-x^r'/HD)

(8.9)

xe$nDye§n(Bn(x,r)\{x})n(TpW+x)

is a ratio-unbiased estimator of XK(r). Proof. Let us find the mean-value of the numerator of (8.9). Using the refined Campbell theorem, cf. Proposition 8.2, we get

EE =\ j j D Nlf

=

A

/ /

*$)(*-«.or1

E

Hy&Bn{x,r)\{x}}h^Q){y-x10)-lP0{d(p)dxn

Yl yetø>+z)n(Tp(0)+æ)

E

l{2/eB„(0,r)\{0}}ft^ ) (2/,0)- 1 P 0 (#)^

!/€*nrp(0) It remains to show that the last mean-value with respect to the Palm distribution is equal to £ 0 ( $ ( £ n ( 0 , r ) \ { 0 } ) ) . To see this, let

f(y) =

l{yeBn(0,r)\{0}}.

Note that

f(By) =

f(y),BeSO(n).

212

8. THE MODEL-BASED APPROACH

The function h^Jy, O) has the same property. Using this and the rotation invariance of P0, we get for any B G SO(n),

/(»)/#>, or 1

Eo £ ye$nTp{0) T

yeB

= E0

$nTp{0)

]T

f{y)h%{y,oy\

ye$nBTp{0)

Using that a?Q) is a probability measure, cf. Section 8.3, we therefore get 1

Eo £

mh^oy

yE$nT p ( 0 )

= E0 f

Yl

f{y)h%{y,oylan{Q){dB)

SO(n) ^ « ( W r f c ,

-*P

1

ytv

/

HytT^h^oy —?Tn

dTn v

,yj

= E0YJf{v)ye$

□ In this section, we have assumed that $ is isotropic, i.e. B& and $ have the same distribution for all B £ SO(n). Under this isotropy assumption, the secondorder properties of the point process can be described by the K—function. Under more restricted isotropy where $ is invariant under rotations keeping an r—subspace Lr fixed, an analogous theory can be developed.

8.5 Estimation of moments in the mark distribution The moments in the mark distribution can be estimated, using the local estimators of d—dimensional Hausdorff measure developed in Chapters 5 and 6. The dual designbased situation has been described in Section 7.4. In order to estimate the first moment of d—dimensional Hausdorff measure in the mark distribution, we will make use of the local estimators developed in Chapter 5.

8.5 ESTIMATION OF MOMENTS

213

(Analogous results can be obtained with the local estimators from Chapter 6.) Recall that for X G Mj we have developed, under regularity conditions, the following local estimator of X^(X) d m^r )\x,Lp;Lr)

=^ ^ f llTr^xir^GCTan^x],^)-1^-^, &p—r J xnLp cf. (5.24). Here, 0 < r < p < n and d — n H- p > 0. The idea is now to use this estimator on each of a sample of particles from the marked point process. Before we can specify the resulting estimator in more detail, we need to show the in variance result given in the lemma below. Lemma 8.4. Let X G Md and Lr G £™ • T n e n > we have

for anv B E

50(71, Lr) and Lp G ££/ r ) ,

m%$(BX, BLP; Lr) = m%$(X, Lp; Lr).

Proof. According to Exercise 3.7, G(BTan[X,x],BL p ) = G(Tan[X,x],L p ), for any B G SO(n). Furthermore, £Tan[X, x] — Ta,n[BX,Bx] SO(n,Lr) and x G Rn, we have BTTL_LX =

and, for B G

irL±Bx.

(Use in various ways that TTL±X is the unique point in L^r closest to x.) Therefore, for B G SO(n,Lr) and Lp G ££ ( r ) ,

=

\\BTrL±x\\n-pG(Ta3i[X,x],Lp)-1dxd-n+p

^ZL f &p—r J XDLp

= ^-L j \{Bx eBXD &p—r J

BLp}\\7rL±Bx\\n-pG(TMi[BX,

Bx],BLpyldxd-n+lp.

We now use that the Hausdorff measures are invariant under rotations (see the definition of the Hausdorff measures in Chapter 2) and get

/ l{y G BX n BLp}||7TL;L2/|r-PG(Taii[5X,2/], B L p ) " 1 ^ - ^

= ^ ^

®p—r Jn

R

= m§fi(BX,BLp;Lr).

D

21.4

8. THE MODEL-BASED APPROACH

We are now ready to present the estimator of the first moment in. the mark distribution. We let E§ be a random manifold with distribution Pm and the parameter to be estimated is thus

Em\dn(Z0) = j

\dn(=.)Pm(d~).

Below» we assume that Ho satisles the regularity conditions of Proposition 5.4 such that

,

f J

, * p{1}

dL

'\ > c(n — i\p — r)

p(r)

Proposition 8.5. Let \[> = f [x^: Hz-]} be a stationary marked point process in Rn and let us suppose that the mark distribution is invariant under rotations in SO(ji. Lr). Let £p(o) G £">rx be a fixed p—subspace in Rn containing I r , 0 < r < p < n. and let D € BQ be a bounded Borel set with positive volume. Then, if d — n — p > 0,

i

is a ratio-unbiased estimator of EmXcl(E®).

O

The procedure of Proposition 8.5 is illustrated for r = 1, p = 2 and n = 3 in Figure 8.6 below. In the hgure, it is thus assumed that Pm is invariant under rotations keeping the indicated line Lr fixed.

Figure 8.6. Illustration of the sampling design used for estimating moments of the mark distribution.

8.5 ESTIMATION OF MOMENTS

215

Proof of Proposition 8.5. We will use that (8.4) implies that for any measurable, non-negative function / on Rn x Md, we have E)Pm(dZ)dxn.

E J2 /(**, Hi) = A y y / ( * , 1

Rn Md

Accordingly, the mean-value of the numerator of (8.10) can be rewritten as

i

D}m^\E)Lp{0)]Lr)Prn(dE)dxn

= \JJl{xe Rn

Md

=

\V(D)Emm^\Zo,Lp(0y,Lr).

It remains to show that Ernmp(r)

( ^ 0 ' ^p(O)5 Lr)

=

EmXn(Eo).

Using that Pm is invariant under rotations in SO(n,Lr) B e SO(n,Lr) Ernm

and Lemma 8.4, we get for

(S0,£p(0);£r)

p(r)

[BTEo, L p(0) ; Lr J

= Emm^ = Emm^

fa,

BLp{0)]

Lr).

Since a?^ is a probability measure, cf. Section 8.3, we get E

™mp(r)

{-o,Lp(0y,Lr)

=

Emm%$

I

(So,

BLm;Lr)afr){dB)

SO(n,Lr)

= Em

j

m^^BL^L^a^dB)

SO(n,Lr) (*) „

f

(n,d),~

p(r)

=

EmXn(Eo).

r

r

^_J^Mr)__

216

8. THE MODEL-BASED APPROACH

At (*) we have used (8.5) and at (**) we faa¥e used that the local estimator ra^?(So,Lp;Lr) Is» for a fixed Ho and an isotropic p-subspace Lp containing L r , an unbiased estimator of Å£(EO). O Note that for d = 0, Proposition 8.5 can only be used with the trivial choice p = n. In this case, the local estimator described in Chapter 6 must be used instead» see Exercise 8.6. This estimator has been applied in neuro science to quantify the phenomenon called satellites^ where small glia cells are distributed around neurons in the brain. An example of a section through this type of structure is shown in Figure 8.7. The neurons are the large elongated light grey ceils, containing dark grey nuclei. In some of the neurons» the nucleolus can be seen inside the nucleus as a small dark spot. The numerous spherical nuclei belong to various types of glia cells. The neuron nucleoi can be chosen as reference points and the mark associated with a particular reference point is the set of glia cell centres sitting at a distance at most R from the reference point. An estimated spatial distribution of glia cells around neurons in human temporal cortex is shown in Figure 8.8, see also Gundersen et al. (1988b).

Figure 8.7. A focal plane through a 40-/*m-thick section of the CA3-layer of a human hippocampus (from Jensen & Gundersen (1993), with permission).

ft is also possible to estimate the second moment in the mark distribution, i.e. EmXi(B0)\ Here, we can use Theorem 5.6, which concerns the decomposition of a product of Hausdorff measures in FT. For X € M4, we get, under regularity

8.5 ESTIMATION OF MOMENTS

217

conditions, e(n-2-r.p-2-r)\i(X)2 =

f.r.j-i.x2)fl~"p

Vr+2(ei £ n { f ) xhlp

Xf)Lp 2

2

x f[ G(Tan[X, .r t ], Lp)~l f[

dx^dLnp(r).

1=1

i=\

Here. 1 < r — 1 < /> < /? and d — n + p > 0.

Figure S.S. The local numerical density of glia cells as a function of the distance from a neuron nucleolus (from Gundersen et al. (1988b), with permission).

The obvious estimator of Aj((X) » based on an isotrapic p—subspace Lp containing Lr, thee is (Tn—r^n — r—i

—
I

/

I

/

—

f

\n—p

v r +2(f , i,....e r ,.ri.:r2)

J J Xf)Lp XDLp 2

2

i=i

?=i

218

8. THE MODEL-BASED APPROACH

say. Note that m^n,f)(X,Lp]Lr) is an unbiased estimator of A^(X)2, under the randomness described. This estimator has the same type of invariance properties as m^' (X, Lv\ Lr), see the lemma below. The proof of the lemma is left to the reader. Lemma 8.6. Let X G Md and Lr £ C7}. Then, for any B £ SO(n, Lr) and Lp e ££ ( r ) , we have m%$(BX, BLp; LT) = m§*\x, Lp; Lr).

U

Let us now construct an estimator of EmX^Eo) . We assume below that EQ satisfies regularity conditions of the type mentioned in Theorem 5.6 such that

J

p

v)

c(n — r,p — r)

p(r)

Proposition 8.7. Let ^ = {fø; E{]} be a stationary marked point process in Rn and let us suppose that the mark distribution is invariant under rotations in SO(n, Lr). Let Lp(o) £ £Vx be a fixed p—subspace in Rn containing L r , 1 < r + 1 < p < n, and let D e BQ be a bounded Borel set with positive volume. Then, if d — n + p > 0, Em\i{E0f

= Y^l{Xi e D}m§$(Zi,Lm;Lr)MD)

(8.11)

i

is a ratio-unbiased estimator of £"mA^(Ho)2. Proof. The proof is of the same type as the one used for Proposition 8.5. We therefore give a condensed version. The mean-value of the numerator of E m A^(S 0 ) 2 becomes

EY,n^^D}fh^(EuLm]Lr) i

= A J J l{x 6 D}m§$ (~, Lp(0y,Lr)Pm(dE)dxn R"Md

=

XV(D)Emm(pld))(E0lLm;Lr).

Furthermore, we have for B e

SO{n,Lr)

8.6 EXERCISES Ern

219

(S0,£p(0);£r)

™'p(r)

— £jmmp^

(£?

^o,^p(0);^rj

= Em,™p(r) (So, BLp(0y,Lr)

= Em — TP — £jm

J f I

m^

(So, £Lp(0); L r )a^(dB)

™(n'^V" T . T \ m , \ {^o, Lpj Lr)— p r)

J

^

P( r ) -

c(n — r,p — r)

The main interest in the estimator presented in Proposition 8.7 is for d = n — 3, p = 2 and r — 0. In this case, the estimator reduces to £ m A|(Ho) 2 = ^ l { x

s

D}m{^(Ei,Lm-0)/^{D),

€

i

where TTILQN ( S , L 2 ( 0 ) ; O ) = 27T

/

/

2 x

aiea,(0,xi,X2)dx2dxi

EC\L2(o) SnL2(o)

and area(0, xi, 2:2) is the area of the triangle with vertices O, xi and rz^- For general S, Monte-Carlo methods are needed in order to determine rh2(0) ("> ^2(0) 5 O).

8.6 Exercises Exercise 8.1. Let $ be a stationary point process in Æn. Show that the intensity measure is invariant under translations, i.e. A(A + x) = A(A), A G Bn,x G Rn.

Exercise 8.2. In this exercise, we will show that the definition of the Palm distribution P0 does not depend on the set A G Bn. Let

v(A,C) = E Y, xe$r\A

H®-x£C}-

220

8. THE MODEL-BASED APPROACH

Show, using the stationarity of <£, that v{A + y, C) = v(A, C), AeBn,yeRn,C

e Nlf.

We therefore have that v(A,C) = \cV(A) and

r(c)-^c)-Xc

' n i

°

j

~

AV(A) ~~ A

does not depend on A. Exercise 8.3. Suppose that $ is stationary and isotropic. Show under these assump­ tions that P0 is invariant under rotations, i.e. P0(BC) = P0(C), B G SO(n), C G Mlf. Exercise 8.4. Suppose that $ is stationary and isotropic. Show under these assump­ tions that the second reduced moment measure K is invariant under rotations. Also indicate how a histogram of relative local intensities will appear in the three situations illustrated in Figure 8.2. Exercise 8.5. This exercise concerns the mark distribution P m or the particle distri­ bution, as it also is called. 1. Show of $, 2. Show 3. Show

that if the Si's are independent and identically distributed and independent then Pm is their common distribution. that if \P is stationary and isotropic, then Pm is invariant under rotations. that P m ({~ G Md : O G H}) = 1.

Hint. Use that for the marked point process \I> = {[£;;£;]} we have O G H* for all i. Exercise 8.6. Let \£ = {[#;;£;]} be a stationary marked point proces in Rn and let us suppose that the mark distribution is invariant under rotations in SO(n,Lr). Furthermore, suppose that ^ G Mo, the set of finite subsets of Rn. Let Tp(0) be a fixed p—slice in Rn of thickness 2t, containing L r , 0 < r < p < n, and let D G BQ be a bounded Borel set with positive volume. Show that £

l{*i € D}m$) & i

n

Tm;LT)mD)

8.7 BIBLIOGRAPHICAL NOTES

221

is a ratio-unbiased estimator of ÆmA°(!Eo), where

æesnTp(o) cf. (6.2).

8.7 Bibliographical notes In this chapter, we have discussed the stereological analysis of marked point processes. Section 8.1 and 8.2 are a brief account of the theory of marked point processes, for a more comprehensive treatment, see Stoyan et al. (1995). The stereological estimation of the K—function based on information in a slice, cf. Section 8.4, has been discussed in Jensen et al. (1990a) and developed further in Jensen (1991) and Jensen & Kiéu (1992a). The idea behind this procedure is due to Stephen Evans, cf. Evans & Gundersen (1989). Stereological analysis of general random spatial structures has been studied in a number of papers, cf. e.g. Mecke & Stoyan (1980), Ambartzumian (1981), Stoyan (1981), Stoyan & Ohser (1982, 1985), Stoyan (1984a, b), Hanisch (1985), Stoyan (1985a, b), Jensen (1987), Schwandtke (1988), Cruz-Orive (1989b), Jensen et al. (1990b) and Kiéu (1991).

Chapter 9

Perspectives and future trends This chapter contains some general comments, which tie the different chapters together, extend the treated subjects and point to some future trends in stereology.

9.1 Mathematical and statistical aspects Many of the proofs presented in this book are new. This is in particular true for the proofs in Chapters 4 and 5. Induction in the dimension of the space of interest is an important tool, cf. e.g. Proposition 3.9, Proposition 4.3, Lemma 5.3, Proposition 5.4 and Theorem 5.6. The idea of using this type of induction can be found in Miles (1971, p. 362). Throughout the book, the stereological estimators have been constructed as Horvitz-Thompson estimators. I believe that this is the first systematic exposition of local stereological estimators as Horvitz-Thompson estimators. The importance of this concept has, however, been realized for some time, cf. e.g. Baddeley (1993). Classical Horvitz-Thompson estimators are constructed for finite populations. Except for the case where number is the parameter of interest, our populations are infinite and the local estimators are constructed in this book, using the Horvitz-Thompson procedure on an infinitesimal level; introductory examples were given in Sections 4.1, 5.1 and 6.1. These somewhat ad hoc considerations can most probably be formalized, cf. e.g. Cordy (1993), but I would be surprised if this led to new local estimators. One of the strengths of using a formal estimation procedure as the HorvitzThompson procedure is that it becomes possible to see whether a given sampling design leads to an estimator involving geometric measurements which are possible to collect, cf. Chapter 7. One example where this is not possible has been discussed in Exercise 6.10. 223

224

9. PERSPECTIVES AND FUTURE TRENDS

Many local estimators of a given type are related by a Rao-Blackwell procedure. For instance, cf. Proposition 4.8, the local estimators of volume are related by

m $ r ) ( X n LP2;Lr) = £(m
(9.1)

for LPl an isotropic pi-subspace contained in LP2. The consequence is that local estimators based on lower dimensional subspaces have higher variances. The application of the Rao-Blackwell theorem in stereology has more generally been discussed in Baddeley & Cruz-Orive (1995). Note that in (9.1), X n Lpi can be regarded as a randomized subsample of X D LP2. For non-randomized subsamples, there are 'counterexamples' in which estimators based on lower-dimensional subsam­ ples are more efficient than higher-dimensional ones, cf. e.g. Jensen & Gundersen (1982, Section 6) and Ohser (1990).

9.2 Affine version of local stereology Local stereology in the form presented in this book was first formalized in Gundersen (1988) and Jensen & Gundersen (1989). The inspiration came from the paper by Cruz-Orive (1987a). Before this, an affine version of local stereology had been studied for some years, involving subspaces passing through uniform points, cf. Jensen & Gundersen (1983, 1985), Miles (1983, 1985), Gundersen & Jensen (1985) and Jensen (1985). This affine version of local stereology has mainly been concerned with the estimation of the volume of an open and bounded subset X of Rn. Let us here give a short summary of the main results. In affine local stereology, the sampling design consists of a pair (Lq,xo), where Lq is an isotropic q—subspace and XQ is a uniform random point in X. Note that the affine subspace Fq = Lq +XQ is in 1-1 correspondence with its parallel q—subspace Lq and the orthogonal projection x^~q of XQ onto L^. Note also that when Lq is an isotropic q—subspace and XQ is a uniform random point in X then, given Lq and XQ~q, xo is uniform random in X D Fq. The estimators of V(X) used in affine local stereology can be obtained by a RaoBlackwell procedure applied to the estimators from Chapter 4. As an example, we

9.2 AFFINE VERSION OF LOCAL STEREOLOGY

225

get from (4.12) the following affine estimator of V(X) mq(X n Fq) =

E{m^){(X-x<))nLq-O)\Lq,xl-0)

= /

m%{{x-X0)nLq,o)^^

xnFq = AJ(InFq)-^

/

\\x\\n-qdxqdxl

/

xnFq (x-x0)nLq 1

= \l(XnFqy ^

[

\\x-x0\\n~qdx^dxl.

f

xnFq xnFq Note that the origin O is replaced by the uniform random point x0 in X. For q = 1, the estimator can be expressed in terms of the (n + 1)—line [L n + 1 ], defined in Miles (1983). We thus have rh^X n Fx) = Xi(X n F!)-\

2 n

° [L" +1 ], n[n + IJcTi

See also Cabo & Baddeley (1995). Affine estimators of V(X) have also been constructed, mainly for the purpose of estimating the second moment of volume in a particle population. One of the estimators takes the form c(n,2)A2(InF2)2V(InF2), where V(X fi F2) is the mean-value of

when xo, ^1, ^2 are independent and uniform random points in X D F2, cf. Jensen & Gundersen (1985, p. 93). For n = 3, the estimator can be written as ,

3V0

ao where ao = XJ(X D F2) and Vo is the mean area of a triangle with uniform random vertices i n I n F 2 . For convex X D F2, it has long been known that 35 1 0.0739 = —-=■ < Vo/ao < — = 0.0833, z 4ö7T

1Z

(9.2)

cf. Santalo (1976, p. 63-65) where the related Sylvester problem also is discussed. The lower and upper limits are attained for, respectively, ellipses and triangles, cf.

226

9. PERSPECTIVES AND FUTURE TRENDS

Blaschke (1917, 1923). It has recently been proved by Pfiefer (1990) that the lower limit of (9.2) still holds for not-necessarily-convex XnF2. The ratio Vo/a 0 is however not bounded from above in this case. In Jensen & Sørensen (1991), Vo/ao has been determined for a variety of biological particle shapes. The variability of Vo/ao was found to be remarkably low. For a recent application of affine local stereology in metallography, see Karlsson & Cruz-Orive (1992).

9.3 Curvatures and other parameters In this book, we have mainly been concerned with the estimation of d—dimensional volume in Rn. Many other parameters may however be of interest, e.g. the thickness of biological membranes, the orientation of bone surfaces or the Euler Poincaré characteristic of the capillary network, to mention a few, cf. Jensen et al. (1979), Odgaard et al. (1990). Local solutions to the estimation of such parameters have not yet been developed. Some recent developments in this direction can be found in Roberts & Cruz-Orive (1993). There is however one class of parameters where local estimation methods may in principle be developed easily, i.e. integrals of curvatures. Let X be an (n — 1)—dimensional differentiate manifold in Rn. At each point x e X, we have n - 1 principal directions and n — 1 principal curvatures K{X) =

(Kl(x),...,Kn-i(x)).

The i'th elementary symmetric functional on Rn, i = 0 , 1 , . . . , n, is defined by HS{z) =

l,zeRn

# f 0 ) = Yl ^(i)---^(i)^Gfi n ,i = l,...,n, where If = {a G N* : 1 < a ( l ) < • • • < a(i) < n}. The z'th integral of curvature, i = 0 , 1 , . . . , n - 1, is then, cf. Santalö (1976), M?(X)= ^

1

)

j H?-\K{x))dx»-\ (9.3) x A parameter of the type (9.3) can be estimated by the local methods presented in Chapters 5 and 6. In particular, using an isotropic p—slice Tp containing Lr, cf. Section 6.3, Mf (X) can be estimated by

^; 1 ) / Hr^^x^h^ur'dx--1. (9.4) xr\Tp

9.4 FUTURE TRENDS

227

Subsampling of X n Tp with grids along the lines described in Section 6.5 is also possible. There may however be some practical difficulties in determining K(X) at selected points x G X with the techniques described in Chapter 7. Curvatures have been studied from a dual translative viewpoint in e.g. Schneider & Weil (1986) and Weil (1989). The stereological estimation of mean shape from projections has been discussed by Weil (1993).

9.4 Future trends The major breakthrough in modern stereology has been the invention of local 3-d sampling, cf. Sterio (1984). With such sampling, particle size distributions can now in principle be estimated directly, and the need for solving ill-posed problems is not there any more. There may however be technical problems, using conventional light microscopy. At the moment, much current research in local stereology is concerned with the experimental investigation of the new techniques. Another important aspect of local stereological methods is that, in contrast to earlier methods, they can be applied without specific shape assumptions. This is an important advance in flexibility and power of the methods. The efficiency of systematic sampling has gained renewed interest after the significant work by Souchet (1995), Kiéu (1997) and Kiéu et al. (1997). The consequences for the applied stereologist of this intriguing asymptotic theory still remain to be assessed.

Appendix

Invariant measure theory The results of this appendix have mainly been taken from Barndorff-Nielsen et al. (1989). The reader is referred to these lecture notes for further details. See also Schneider & Weil (1992). Let A* be a topological space and G a topological group. A mapping 7 from G into the symmetric group S{X) over X, i.e. the set of one-to-one transformations of X onto X with composition of transformations as composition rule, is called an action of G on X if (/) (//)

7 is a homomorphism the mapping Gx X -> X (g,x) -+-y(g)(x) is continuous.

Below, we write gx as short for j(g)(x). The subset Gx = {gx : g G G} is called the orbit of x. The group G is said to act transitively on X if there is one orbit only, i.e. for every x\, X2 G X there exists a g G G such that gx\ = x (øz,x) is continuous and the inverse image under / of every compact set is compact. A measure \i on X is said to be G—invariant if

X

X

for all g G G and all non-negative measurable functions h on X. 229

230

APPENDIX

Let us now suppose that (G, X) is a standard transformation group, as defined in Barndorff-Nielsen et al. (1989, p. 12). We then have the following result which is a corollary of Barndorff-Nielsen et al. (1989, Theorem 4.1). Theorem A.l. Suppose G acts transitively and properly on X. Then, there exists one and, up to multiplication by a positive constant, only one G—invariant measure on X. U Example A.2. The additive group G = (i? n ,+) acts transitively and properly on X = Rn by GxX

-> X

(y,x) -> x + y. Therefore, since the Lebesgue measure is translation invariant, it follows from Theorem A. 1 that the Lebesgue measure is, up to multiplication by a positive constant, the only measure with this property. □ Example A.3. Consider the action of SO{n, Lr) on £ n ,x given by

so(n,Lr)xc;ir)^c;(r) (B, Lp) —> BLp. This action is also transitive and proper and it follows from Theorem A. 1 that there exists, up to multiplication by a positive constant, a unique SO(n, Lr)—invariant measure on Cn( v □ L 1

-

p(r)

Generally, we can define two actions on a topological group G , viz. left action 6 and right action e of G on itself 6:GxG->G (90,9) -» 909

and e:GxG-*G (go,g) -»99Ö 1 By Theorem A.l, there exist measures as and ae on G which are invariant, respec­ tively, under left and right action. We thus have / h(9og)Mdg) G

= / h(g)a6(dg) G

(A.l)

INVARIANT MEASURE THEORY

231

and j h(gg^)ae(dg) G

= J h(g)ae(dg)

(A.2)

G

for all go £ G and all non-negative measurable functions h on G. Note that we also have (with suitable choice of the arbitrary constants for as and ae) j h(g-l)a6{dg) G

= J h(g)ae(dg).

(A3)

G

This important formula is a consequence of the uniqueness of ae, say, since both the left-hand side and the right-hand side of (A.3) define a measure which is invariant under right action. If G is compact, it can be shown that the two measures coincide, i.e. a$ = ae = a, say. Combining (A.l) and (A.2), we then have / h{g0g)a{dg) = / h{gg0)a(dg) = / h(g)a(dg) G

G

G

for all go £ G and all non-negative measurable functions h on G. The compactness of G also implies that a may be chosen as a probability measure. Example A.4. The group SO(n, Lr) acts from the left and the right on itself. Since SO(n, Lr) is compact, there exists a unique probability measure a j \ on SO(n, Lr) which is invariant under both left and right actions. Note also that the invariant probability measure a?^ lifted by the mapping SO(n,L r )-*£p(r) B —> BL p ( 0 ), where Lp(0) G Cn< N is fixed, is a SO(n, Lr)—invariant probability measure on £™, x and (8.5) therefore follows by the uniqueness of such a measure. D

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Subject index curvature 153, 226

a notation 107, 162, 179

cycloid test system 34, 169 affine local stereology 119, 224 affine subspace 36, 44, 49, 72, 84, 224 area 95, 97, 140, 148, 149, 194, 195

design-based approach 201 differentiable manifold 38 disector 6, 33

Bernoulli numbers 190, 196 polynomials 190, 196 Beta distribution 91, 92 Beta function 74, 76 binomial distribution 30 Blaschke-Petkantschin formula

sampling design 6 duality 25, 56 edge effect problem 9, 209 ellipsoid 112 Euler-Maclaurin formula 190, 198 extension term 191

classical 100, 102, 104, 119 generalized 127, 135 bounded point grid design 14 Buffon's needle problem 18, 31

fluctuation term 191 focal plane 175 fractionator 34

C.A.S.T.-GRID 199 Campbell theorem 203 refined 204 circular window design 22 coarea formula 38, 61, 109, 128, 163 confocal scanning light microscopy 174, 199 convex hull 52 convex set 225 covariance 115 covariogram 187 circular 193 Crofton's formula 72, 88, 93, 152, 153

G-factor 48, 55, 59, 60, 73, 123 grid in Rn 84 Hausdorff measure 35, 61 Horvitz-Thompson estimator 2, 5, 7, 9, 10, 13, 16, 21, 24, 25, 28, 95, 105, 115, 138, 156, 159, 161, 165, 173, 187, 223 hypergeometric function 143 induction 68, 76, 93, 100, 128, 136, 223 intensity measure 202 245

246

SUBJECT INDEX of stationary point process 203

invariant measure theory 63, 69, 93, 119,202,208,229 isotropic 202, 206 band 27 direction in R3 19 line 96, 121 /7-slice 157 p-subspace 82 /7-subspace containing a fixed r-subspace 83

over- or underprojection 183

Jacobian 39

pair of subspaces 78, 104, 110, 117 pair-correlation function 206 Palm distribution 203 plane grid design 19 point process 202 Poisson point process 205 polar coordinates 19 polar decomposition 45, 97 powers of volume or surface area 112, 153 primary particle 206 projection 75, 93

^-function 205, 209

random sample 2

lattice of fundamental regions 84 Lebesgue measure 36 length 4, 18, 31, 121, 140, 180, 195 lexicographic ordering 10 line 225 line-segment 7, 56, 107, 108 linear subspace 63 local intensities 205 local stereology 27 look-up plane 6 manifold 38, 57 marked point process 206 model-based approach 34, 201 moments 182, 212 nucleator 180 nucleus 27 number 3, 9, 27, 155, 174, 181, 196 one plane sampling design 3 optical sectioning 175

Rao-Blackwell procedure 110, 224 ratio estimation 14 reference plane 6 reference point 27, 206 regression 18 rotation 63 rotator 180 sampling design 1, 33 sampling probability 2, 5, 7, 8, 10, 13, 17, 22, 24, 87, 138, 156, 157 sampling theory 1 second factorial moment measure 204, 207 second reduced moment measure 205 serial sectioning 6 shape 6, 183, 210, 227 simplex 52 slice 157 spatial point grid design 11 spherical shape 5, 183, 210 star 110 star-shaped set 107 stationary 202, 206

SUBJECT INDEX surface area 26, 49, 141, 180, 184 surfactor 180 Sylvester problem 225 systematic sampling 7, 34, 187, 200, 227 with disectors 7 with lines 25 with planes 21

247

spatial point grid 12 uniform and isotropic circular window 23 plane grid 20 4-grid 87, 165 variance 22, 29, 30, 32, 110, 115, 142, 187

tangent space 38 translative decomposition 46, 98

vertical design 26, 34, 169

uniform bounded point grid 15 q-grid 160

Wicksell's problem 5, 33

volume 11, 105, 179, 184

zitterbewegung 191