Heike Hufnagel A Probabilistic Framework for Point-Based Shape Modeling in Medical Image Analysis
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Heike Hufnagel A Probabilistic Framework for Point-Based Shape Modeling in Medical Image Analysis
VIEWEG+TEUBNER RESEARCH Medizintechnik – Medizinische Bildgebung, Bildverarbeitung und bildgeführte Interventionen Herausgegeben von Prof. Dr. Thorsten M. Buzug, Institut für Medizintechnik, Universität zu Lübeck Editorial Board: Prof. Dr.-Ing. Til Aach, RWTH Aachen Prof. Dr. Olaf Dössel, Karlsruhe Institute for Technology Prof. Dr. Heinz Handels, Universität zu Lübeck Prof. Dr.-Ing. Joachim Hornegger, Universität Erlangen-Nürnberg Prof. Dr. Marc Kachelrieß, Universität Erlangen-Nürnberg Prof. Dr. Edmund Koch, TU Dresden Prof. Dr.-Ing. Tim C. Lüth, TU München Prof. Dr. Dietrich Paulus, Universität Koblenz-Landau Prof. Dr. Bernhard Preim, Universität Magdeburg Prof. Dr.-Ing. Georg Schmitz, Universität Bochum
Die medizinische Bildgebung erforscht, mit welchen Wechselwirkungen zwischen Energie und Gewebe räumlich aufgelöste Signale von Zellen oder Organen gewonnen werden können, die die Form oder Funktion eines Organs charakterisieren. Die Bildverarbeitung erlaubt es, die in physikalischen Messsignalen und gewonnenen Bildern enthaltene Information zu extrahieren, für den Betrachter aufzubereiten sowie automatisch zu interpretieren. Beide Gebiete stützen sich auf das Zusammenwirken der Fächer Mathematik, Physik, Informatik und Medizin und treiben als Querschnittsdisziplinen die Entwicklung der Gerätetechnologie voran. Die Reihe Medizintechnik bietet jungen Wissenschaftlerinnen und Wissenschaftlern ein Forum, ausgezeichnete Arbeiten der Fachöffentlichkeit vorzustellen, sie steht auch Tagungsbänden offen.
Heike Hufnagel
A Probabilistic Framework for Point-Based Shape Modeling in Medical Image Analysis With a foreword by Prof. Dr. rer. nat. Heinz Handels
VIEWEG+TEUBNER RESEARCH
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Dissertation Universität zu Lübeck, 2010
1st Edition 2011 All rights reserved © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011 Editorial Office: Ute Wrasmann | Britta Göhrisch-Radmacher Vieweg+Teubner Verlag is a brand of Springer Fachmedien. Springer Fachmedien is part of Springer Science+Business Media. www.viewegteubner.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: KünkelLopka Medienentwicklung, Heidelberg Printed on acid-free paper Printed in Germany ISBN 978-3-8348-1722-8
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ωp ∈ R , Ω = {ω1, ..., ωn } ) * v1 , ..., vn ) *
M , Θ - , ! ⎞ ⎛ n ωp2 1 ⎠ p(M |Θ) = p(Ω|Θ) = p(ωp |Θ) = exp ⎝− 2λ2p (2π)n/2 np=1 λp p=1 p=1 n
)#.*
λp /
Sk , ωkp Sk
0 Sk
Tk = {Ak ∈ R3×3 , tk ∈ R3 } Ak ∈ R3×3 tk !" # Ekij ∈ RNk ×Nm ij si mj $ Tk mj
Tk mj = Ak mj + tk . # Mk
mj
M = Tk M.
%!&'
(
Q = {Ωk , Tk } # ) * )
) !&
$ !!+ ,
Θ = {M¯ , v , λ , n}
+
p
p
Qk = {Ωk , Tk }
Ek
S
Θ
Q
(Q, Θ) → p(S|Q, Θ)
Θ
Q = {Qk }
!
p(Q, Θ) = constant
(Q, Θ)
"
" # "
(Q, Θ)
(Q, Θ) → p(Q, Θ|S)
Q
Θ
$%&
' (
$%&
=−
N k=1
log(p(Qk , Θ|Sk )) = −
N k=1
log
p(Sk |Qk , Θ)p(Qk |Θ)p(Θ) p(Sk )
.
) *+
Θ ¯ ∈ R3Nm M
Nm Nm vpj λp ∈ R n n n ≤ N vp ∈ R3Nm : n
Ω vp Sk ωkp : n
¯ + Mk = M p(Mk |Θ) =
N
p=1
ωkp vp
(2π)n/2
1 n
p=1
λp
exp − n p=1
2 ωkp
2λ2 p
Tk Tk = {Ak ∈ R3×3 , tk ∈ R3 }
Ak tk
Mk = Tk Mk
Ek Ek ∈ RNk ×Nm j Ekij = 1
ski = Tk mj + N (0, σ)
Ekij
p(Sk ) p(Θ) Θ
Θ
C(Q, Θ) = −
⎛
N
⎞
⎝log(p(Sk |Qk , Θ) + log(p(Qk |Θ))⎠ .
!"# p(Sk |Qk , Θ) = p(Sk |Tk , Ωk , Θ) k=1
p(Sk |Qk , Θ) =
Nk Nm
1 p(ski |mkj , Tk ) Nm i=1
mkj = m ¯j +
n
ωkp vpj .
p=1
j=1
ski ! #
mj p(ski |mj , Tk ) =
1 3
(2π) 2 σ
exp(−
1 (ski − Tk mj )T .(ski − Tk mj )). 2σ 2
!$%%#
C(Q, Θ) ! # & ωkp ' !$(# )
⎞ Nm 2 s − T m 1 1 ki k kj ⎠ exp − C(Q, Θ) = − log ⎝ 3 Nm 2σ 2 2 j=1 (2π) σ k=1 i=1 ⎛ ⎞ N n n 2 ω kp ⎝log((2π)n/2 ) + log( ⎠ λp ) + !$%*# + 2λ2p p=1 p=1 Nk N
⎛
k=1
= α(n) + β(Nm ) − ζ(σ) +
N
Ck (Qk , Θ).
k=1
+ , , Nm α(n) = k log((2π)n/2 ) β(Nm ) = k Nk log(Nm ) ζ(σ) = N Nk log (2π)− σ −1 - Cglobal (Q, Θ) = Nk=1 Ck (Qk , Θ) 3 2
Ck (Qk , Θ) =
n p=1
log(λp ) +
2 ωkp
2λ2p
−
Nk i=1
⎛
log ⎝
Nm j=1
⎞ ski − Tk mkj 2 ⎠ exp − . 2σ 2
!$%$#
! {Q, Θ}" # $ % & ξ ¯ , vp , λp ) = log ξkij (Tk , Ωk , M
Nm j=1
ski − Tk mkj 2 exp − 2σ 2
' ' x m ∂ξkij (ski − Tk mkj )T =− γkij ∂x σ2
N
j=1
γkij
∂(ski − Tk mkj ) ∂x
s −T m 2 exp − ki 2σk 2 kj . = ski −Tk mkl 2 Nm l=1 exp − 2σ2
(
) * γkij Ekij
ski mj
+ " {Tk , Ωk } Θ = {M¯ , vp , λp , n}
% , Tk " Ωk Θ
Ekij
Tk
s −T m 2 exp − ki 2σk 2 kj , Ekij = γkij = ! ski −Tk mkl 2 Nm l=1 exp − 2σ2 " #!
$ Ek Tk % $ & tk & ' Ak
$ & tk (
" #! )
∂Ck (Qk , Θ) ∂tk
= +
Nk Nm
γkij
i=1 j=1
(ski − Tk mkj )T σ2
∂(ski − Tk mkj ) ∂tk
&
∂(ski − Tk mkj ) ∂ = (ski − tk − Ak (m ¯j + ωkpvpj )) = −I3×3 . ∂tk ∂tk n
p=1
*
∂Ck (Qk ,Θ) ∂tk
= 0 &
Nk Nm n 1 γ (s − t − A ( m ¯ + ωkp vpj )) = 0 j kij ki k k σ2 p=1
i=1 j=1
& ⎛
˜¯ j + tk = s˜k − Ak ⎝m
n
⎞ ωkp v˜p )⎠ .
+!
p=1
&
s˜k =
Nk Nk Nm 1 1 ˜¯ j = ski , m m ¯j γkij Nk Nk i=1
j=1
i=1
v˜p =
Nk 1 γkij vpj . Nk i=1
Ak tk !""
ski = ski − s˜k
˜¯ j + mkj = m ¯j −m
n
ωkp(vpj − v˜p ).
p=1
# "
Ck (Qk , Θ) = −
Nk N k=1 i=1
⎛ ⎞ Nm ski − Ak mkj 2 ⎠. log ⎝ exp − 2σ 2 j=1
Ak $ %&" ' Ck (Qk , Θ)
∂Ck (Qk , Θ) ∂Ak
= −
Nk Nm
γkij
i=1 j=1
2
∂ ski − Ak mkj =0 ∂Ak 2σ 2
# Ak
Nk Nm
γkij mkj mkjT =
Nk Nm
T γkij ski mkj
i=1 j=1
i=1 j=1
⇔ Ak Υk = Ψk , Υk , Ψk ∈ R3×3 .
% " Υk Ψk r s υ[r][s] =
Nk Nm
γkij mkj [r] mkj [s]
i=1 j=1
ψ[r][s] =
Nk Nm
γkij ski [r] mkj [s].
i=1 j=1
mkj [s] mkj s $ ( 3 × 3
Ω = {Ωk } Ωk
Tk Θ ! " ## ## $ %
"
! & '
%
Nk Nm ωkp ∂Ck (Qk , Θ) 1 = 2 − 2 γkij (ski − T mkj )T Ak vpj = 0. ∂ωkp λp σ i=1 j=1
% #
Nk Nm
dkp =
dkp
gkqp
gkqp = gkpq (
γkij (ski − tk − Ak m ¯ j )T Ak vpj
i=1 j=1
gkqp =
Nk Nm
T T γkij vqj Ak Ak vpj .
i=1 j=1 % #
ωkp
p (
n
σ2 ωkp − dkp + ωkq gkqp = 0. λ2p q=1 ωkp % ! Ωk ∈ Rn dkp # dk ∈ Rn % ! Gk ∈ Rn×n ( ⎛ 1 ⎞ 0 λ21 ⎜ ⎟ ⎟ Ωk − dk + Gk Ωk . 0 = σ2 ⎜ ⎝ ⎠ 1 0 λ2
' #
gkpq
ωkp
#
n
⇔ (Gk + Rnn ) Ωk = dk ! !
Gk
Rnn =
−2 σ 2 diag(λ−2 1 , ..., λn )
#
dk
)
# #
ωkp
k
% # *+
(Gk + Rnn )
" #
Qk = {Ωk , Tk } ! , ¯ , vp , λp } Θ Θ = {M
M¯ λ v Q m ¯ ! m " # ! p
p
j
k
j
k ∂Cglobal (Q, Θ) (ski − Tk mkj )T =+ γkij m ¯j σ2
N
N
k=1 i=1
! m !
∂(ski − Tk mkj ) = 0. ∂m ¯j
j
−1
$ % ! γ m ¯j =
Nk N
γkij ATk Ak
k=1 i=1
Nk N
γkij ATk (ski − tk − Ak
n
ωkp vpj )
p=1
k=1 i=1
kij
M¯ , v Q !&
λp
p
k
∂Cglobal (Q, Θ) ∂λp
=
N
k=1
2 ωkp 1 − 3 λp λp
=0
' () ⇔ λ2p =
N 1 2 ωkp . N k=1
¯ λ , M Q % * v ∈ R V ∈ R & 1 p = q v v =δ = 0 p = q vp
p
k
3Nm ×n
T p q
p
pq
3Nm
V T V = In×n .
Z ∈ Rn×n
Λ ∂Λ = 0 ⇔ V T V = In×n ∂Z
1 Λ = Cglobal + tr Z(V T V − In×n ) . 2
!"
# $ % vjp ∈ R3
vp mj # Cglobal & N Nk ∂Cglobal 1 =− 2 γkij (ski − Tk mkj )T ωkp Ak ∂vjp σ k=1 i=1
∂Cglobal = Bpqj vjq − qjp ∂vjp q=1 n
qjp =
N Nk 1 γkij (ski − tk − Ak m ¯ j )T ωkp Ak , qjp ∈ R3 σ2 k=1 i=1
Bpqj =
N Nk 1 γkij ωkq ωkp ATk Ak , Bpqj ∈ R3×3 ∀j. σ2 k=1 i=1
'$ vjp ∂ 1 tr Z(V T V − In×n ) = ∂vjp 2
∂ 1 tr ZV T V ∂vjp 2 n 1 = (zqp + zpq )vjq 2 q=1
zqp = zpq .
# n n ∂Λ = zqpvjq + Bpqj vjq − qjp. ∂vjp q=1 q=1
("
Ak AT k Ak =
I3×3
Bpqj Nk Bpqj = bpqj I3×3 bpqj = σ12 N i=1 γkij ωkq ωkp k=1 ∂Λ ∂
vjp = 0
n
(zqp I3×3 + bpqj I3×3 ) vjq = qjp
n
⇔
q=1
vjq (zqp + bpqj ) = qjp
q=1
vjp B [V ]{j} ∈ R3×n
! "
bpqj
j
# # V ∈ R3Nm ×n
[V ]{j} = [vj1 , ..., vjq , ..., vjn ]. Nm [V ]{j} $ !
[V ]{j} (Bj + Z) = [Q]{j} .
Bj ∈ Rn×n
qjp
½ºµ
¾ºµ
Z
&
[V ]{j}
bpqj [Q]{j} ∈ R3×n # [V ]{j} %
#
V [V ]{j} = [Q]{j} (Bj + Z)−1 .
&
'
Z [V ]{j} Z = [Q]{j} − [V ]{j} Bj ∀j.
˜ {j} [Q]{j} − [V ]{j} Bj = [Q]
( &#
j
) # !
˜ V Z = Q.
V ∈ R3Nm ×n Z ∈ Rn×n
˜ ∈ R3Nm ×n Q
'
'
V
#
*
) # V = U SRT U ∈ R3Nm ×n S ∈ Rn×n R ∈ Rn×n V T V ← U R . ˜ $ Z # Z = V T Q
Z=
1 T ˜ ˜ T . V Q + (V T Q) 2
' % #
vjp
Q˜ [Q]˜ = [Q] − [V ] B Z˜ = V Q˜ Z = (Z˜ + Z˜ ) V [V ] = [Q] (B + Z) V = U SR V ← U R
V − V ≤ . A A = I = 0 B ! ! " j p # ˜ v = q . B (z I + B ) v = q ⇔ $% {j}
{j}
1 2
T
{j}
j
T
{j}
{j}
−1
j
T
T
t 2
t+1
T k
k
∂Λ ∂
vjp
3×3
pqj
n
n
qp 3×3
pqj
jq
jp
pqj jq
q=1
jp
q=1
" ˆ ! # v # [V ] ∈ R {j}
p
3n
⎛
[Vˆ ]{j}
⎜ ⎜ ⎜ =⎜ ⎜ vjq ⎜ ⎝ vjn
& Bˆ [B ] ⎛
[Bj ]pq
vj1
⎞
⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
∈ R3n×3n
pqj
j pq
ˆ11j B
ˆ1qj ... B
ˆ1nj ... B
⎜ ⎜ ⎜ ˆ ˆ ˆ =⎜ ⎜ Bp1j . . . Bpqj . . . Bpnj ⎜ ⎝ ˆnqj . . . B ˆnnj ˆn1j . . . B B
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
#
ˆ {j} [Bj ]pq [Vˆ ]{j} = [Q]
' ( # # Z V )
# # A A ! # # T k
k
M¯ Sk !" M¯ Sk # # $ s˘kj mj Sk % ¯ , vp , λp } Θ = {M
s˘kj =
Ns i
E(Hkij ) (Tk−1 sik ). i E(Hkij )
&'()*
+ s˘kj
Sk $ s˘kj Sk + (T mj , s˘kj ) " !
vp λp , '-( + Q = {T, Ω} Ak = I3×3 tk = (0, 0, 0) k ωkp = 0 k p
.. M1 M2 $ M1 ..
M2 / ω1p % M1 = M2 + p ω1pvp ω1p << λ2p 01 (2234 5 λp # 6 vp
+ 7 5 '82
N Sk 5 Nm B ∈ R3N ×N
Cov(B) = BB T ∈ R3N ×3N % m
m
m
BB T = ESE T
S ∈ R3N ×3N BB T E ∈ R3N ×3N m
m
m
m
Value of global criterion
−1400
n=10
−1450
−1500
−1550
n=8
−1600
−1650
n=7
−1700
−1750
0
10
20
30
40
50
60
70
80
90
100
Number of iterations
n
10 7
Nm = 3000
Cov(B) ∈ R9000×9000
B !"#
B = U ΣV T
U $ U ∈ R3N ×3N V T $ V ∈ RN ×N Σ $ Σ ∈ Rmxn
σi % BB T m
m
BB T = U ΣV T V ΣT U T = U ΣΣT U T = ESE T .
!"&
' U U = E ΣΣT $ (
3Nm × 3Nm $ )
$ * +, -..&/ 0
1-
! " # $ % & "
' '" & && & & & mj &
! !" # "! " ! ! " " $ " " ! " % & ! "$ "! !!$" " # ' ! "! "$ " " ! !
""!" # & " $$" " " " $ $ " $ " $ "! $ " ( " " ' & ! "$ $ "!" $ !" ! !" & " " " " $ " ! " " ! "$ "! " " ! " " " $) " $ !" * $ " " + $ " "" " si ηsi "$ " + $ " "" " mj ηmj ( !$ " $) "! " " " " " " , " !$ " " ηsi − ηmj - ! " " & " " T $ " ! $" " " $ "$ " $ " " # " ! " " " " $ $ T ηmj = (A−1 )T ηmj . #& " + " " $" & " !
T ηmj =
(A−1 )T ηmj |(A−1 )T ηmj |
/ $) "! "
& # "$
si
!
p(si |T, mj ) =
! !
" #
#
η γkij
exp(−
γkij
ski −Tk mkj 2 2σ2
ηski −Tk ηmkj ) 2ση2
−
2 kl exp − ski −T2σk m − 2
= Nm
l=1
$
ηsi − T ηmj si − T mj 2 1 ) exp(− ). exp(− const 2σ 2 2ση2
"
mj
ηski −Tk ηmkj 2ση2
.
#
%
" ! &!
' (
!
# "'
$ ' ) *& ++,- # *
,..' ,../- 0 '
) !
k
"
3
1 2
456 #
*6
,../-
*3 +++-
3 !% ! *7 ,..-
8
" ! * ++9- "
si
d
!
;
ns s %! p(ns |s, si ) = p(|φ|, d)
ssi
!
φ
: ) , "
)
!
a2
< :
• •
si <
6
T =
j
exp(−4a2 |s
sj
%6
si sj si sj T i sj |) |si sj | ( |si sj | ) !
1mm
ns
d si
φ s
s si
•
T
•
nsi
! " #$
$
%& $ '
( $ ) % # * ++% , , , ' # - " , $
σ 2 ! " # $ %% % " & %%
' ( " ) !
& * # *
"
$
'
( $ "
' $ +
!
"#
$
% ! "&
"'
() ""
*
+ ",-
.
/( 01 (
&22& 3( &22& &22' 1 &22,4 % #
&
*
5
5
6 $ 7 7 5 $
*
H. Hufnagel, A Probabilistic Framework for Point-Based Shape Modeling in Medical Image Analysis, DOI 10.1007/978-3-8348-8600-2_4, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011
! ! ! ! " # $ % % & ' ( )' *++,- . ' / ! . & 0 12033 4 2 * 0 12033 $ 2
0 12033 . * *
0 120 33 5 * 1203 6 6 ! 7 & ! . & & . &
! "
! "
S M
d2CP (S, M ) =
NS 1 si − mi 2 NS i=1
NS S M !!" " " d Sk Nk ski Mdef Nm mj # $ % &'( d2 (Sk , Mdef ) =
Nk 1 ski − mki 2 Nk i=1
mki = arg minm
j ski − mj
)
d2 (Mdef , Sk ) =
Nm 1 skj − mj 2 Nm j=1
skj = arg mins ski −mj
* dmax (Sk , Mdef ) * Sk Mdef ski − mki mki = arg minm ski − mj dmax (Mdef , Sk ) ) % ki
j
H(Sk , Mdef ) = max (dmax (Sk , Mdef ), dmax (Mdef , Sk )) .
)
% + # # , * % " % # !!"
* ) !! '-" ./ 0 1 A B CT =
|A ∩ B| . |A ∪ B|
! "" # $ % &'(')
ωp
*
k=1
S1
%
% ! $ #
"" +
""
+
# ! ,- .
( ! # , - & ,-) & //() ! ,- # ! 0 # #
/ ! 1 / ' # #
' + # * 23
0
!
,-
0 "" 4(/ *
ωp # # % &//) % %
M
! 5 # 4/( 4' 4//
1 2
ICP
1 2
EM−ICP
!" ## $ % ¯ − 3λ1v1 M ¯ + 3λ1v1 M & ' !" ( ) ' * !" ( )
!
cm
0.207 ± 0.048 0.214 ± 0.058 0.431 ± 0.036 0.567 ± 0.186
0.139 ± 0.032 0.125 ± 0.030 0.415 ± 0.042 0.380 ± 0.044
" # $%% ! # & ! cm
0.102 ± 0.003
0.160 ± 0.022
! " ! !# $ " ! % ! & " ! ' ! ! ' ! " ! ( " ! ( )* ! % !& + ,- 70mm ! # 276 − 337 ! ! #& ! 0.24mm " ! % ! & " ! " σstart = 0.5mm& = 0.7& . ,/ - " 0 % & 1 ( # ! ! !+# " 11 # !
! ! + " + ,&- #& ! ! + " 2 #& 3 ! # ! ( # ! ! + 35%
0.139cm 0.125cm 0.207cm 0.214cm ! " # $ % " % & ' " % ( & ! " ! " % ! % ! ! ) * % " ! " ! + ' " % % ' " ,, & % - ) + % % ) !% " & % )! % % & " . ) & + % " ! ! % !% " & - + + "* %% " + + % . & ) ' (
% #+ ! ) , " +
! " #$ %& ' ( '
( ' % ( ) *+ % , . 332 − 512 1500×1500×500mm3 82mm . // % 01 ' ( ' // ' % % 2 σstart = 100mm ' = 0.9 - " 01 $ % ,- . ' 3 % ' ' . ) ' % -44
. % ' ) % ' // 01 ) ** ) &
' % 5 . 6 ' 3 % *+ . % 3 ' // ' 01 30% . 7 % // "75.04mm$ 01 "109.05mm$ % ) *- . 5 01 !& ' % // . ) *8 % . ' ) ** .
) ' // ' 01
¯ − 3λpvp M ¯ + 3λpvp M
!" #
!" %
$
! " ! ! # "$ " # " $ ! # ! ! " # #%
& # " # ! " # ! ! " & # " # " # ## # mm
41.47 ± 6.42 38.25 ± 5.18 87.73 ± 11.10 109.05 ± 35.14
31.08 ± 15.01 29.34 ± 12.68 77.83 ± 31.09 75.04 ± 25.36
& "'" ()) # # ! " # ! ! " & # " # " # ## # mm
45.95 ± 2.52
33.82 ± 5.47
!" # # $ % ! ! " # # &' ( )* +, ) - . ! / % 255 × 255 × 105 . 0 0.94mm × 0.94mm × 1.50mm . N = 20 1 . # 0 20mm × 20mm × 40mm ! # 22
. 34 5 . ! / 61 # # ! 7) 0 # . /# ! 899 # ! ! σstart = 4mm = 0.85 39 1:7) ! 8 7) 7) 89 ! ! #
/ ! ! / 61 /
61 7) &! / / 7) / # 0 # 8 0 # n / n = 5 n = 10 n = 18 !
n = 5! n = 10 n = 18 " " " mm
0.634 ± 0.090 4.478 ± 0.927
0.512 ± 0.083 2.929 ± 0.576
0.623 ± 0.099 4.449 ± 0.909
0.490 ± 0.088 2.496 ± 0.445
0.610 ± 0.089 4.388 ± 0.930
0.471 ± 0.076 2.559 ± 0.563
# "
$% " $& "
! "
0.515 ± 0.117
0.463 ± 0.052
! " # " # " # " $# # % " # 0.471 # " # 0.610mm # " " # &' " " $# ( % " % % " " " # " " # n = 5 " # n = 18 5% " 8% " ! " % " "
" )
$ %" " % "
" " $ " * ( " " 40% + 2mm, " " * ( # " " " " $ " $
" " " # " $ - $ " $ - # " " # "
! "
# $ "% "% & M¯ − 3λv1,2
M¯ + 3λv1,2 ' ( # ) # ) !
! " #!$!#!
%&
!
'
& &!
( )&
!
'
! *
+,
- !
& !
.) &
! " / +, ! ! & & ! & !
0 &
12 ) 3 4!5! " & !
& 6 " ) 480 × 720 × 260mm3 !
! " "
386
642
! "
29.3mm!
" & "
' + 7 + %
)& !
"
, & & ! " 3
648!
8 6
σstart = 15 − 50mm
19 : +,
2 !
8
= 0.7 − 0.9
+,
+, 29 ! &! "
200
+,
446
& !
8 4!# &
! " # " $
% &'( ) * +
, n
n = 0 n = 13 %
* 500 &
* &'( *
-./ -.. * 0 &'( , * -.1 &'( * 2
, ! " 0.92 #! " = 0.88$ &'( #! " = 0.86$ + ) & )
,
2
3 3
4 5 , 6 %
, , )
!"
#$# #$# %&& # !" σ ' #
! " ! # # " ! ! !$
! "##$% & ' (()**)
)+,**) - ./0**) 1 (()**) ./0**) - )+,**) ) (()**) ./0**) ./0**) - 2 2 ./0**) - 3 (()**) 4 " 2 " ./0**) (()**) -
$ (()**)
' 5 (()**) ' 5 53, 78mm - 37% ' 5 ./0**) 483, 87mm
. (()**)
2 (()**) ' 6 ωkp Sk vp
mm mm mm mm mm
15.75 ± 2.28 26.35 ± 12.78 36.23 ± 4.60 83.87 ± 54.58
16.48 ± 3.24 17.81 ± 2.75 53.78 ± 7.33 43.81 ± 8.41
! " #$ %&&' ( " ) * (
!* " ! + ! Sk * ! ωk = (ωk1 , ωk2 , ..., ωkn ) , %-- ( ( ( . . + %/0 1 ! 2 #2 -3-' 0.95 , ( ( . . .( . * + -4 ! %5 ! (ωk1, ωk2) %% 1 ( * * + 6 ( " ) 1 ! ωk = (ωk1, ωk2, ..., ωkn ) , * ( ( / + -7 ! %5 ! (ωk1, ωk2)0 +
1 ! * ( ( 8 ( * ! ! ( + ( / 0 ! !
2.5
Second Eigenmode
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5 −3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
First Eigenmode
ωk1, ωk2
Seconod Eigenmode
40
30
20
10
0
−10
−20 −50
−40
−30
−20
−10
0
10
20
First Eigenmode
ωk1, ωk2 !
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δ (φ(x))|∇φ(x)|dx X
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δepsilon ! (φ + ηα) + EΘ (φ + ηα) = log pΘ δ (φ) + ηδ (φ)α |∇φ + η∇α|dx. X
T
∇α 3 # |∇φ + η∇α| = |∇φ| + η ∇φ|∇φ| + O(η 2 ) + # $-4' + + EΘ (φ + ηα) EΘ (φ) 0 ∇φT ∇α 2 EΘ (φ + ηα) = log pΘ δ (φ) + ηδ (φ)α |∇φ| + η $**' + O(η ) |∇φ| X ∇φT ∇α + O(η 2 ). = EΘ (φ) + η log pΘ δ · α|∇φ| + η log pΘ δ (φ) |∇φ| X X
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& τ + 2 ( - "& −τ log pΘ # ) " # $ σΘ 0 3 4" # α = 1- β = 0 τ = 0.1 / " & % & 4 " % & / " σΘ % & 4 5mm / # " τ σΘ 5 # 0 3 # % & " # " σΘ 3 τ τ " σΘ # # / σΘ # τ "" 0# - 0 3 τ
Hausdorff Distance in mm
13
Pat 1 Pat 2 Pat 3 Pat 4 Pat 5 Pat 6
12 11 10 9 8 7 6 5 4 1
2
3
4
5
6
7
8
9
10
11
sigma in mm Hausdorff Distance in mm
10
9
Pat 2, tau=0.1 " , tau=0.2 Pat 4, tau=0.1 " , tau=0.2
8 7 6 5 4 1
2
3
4
5
6
7
8
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10
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sigma in mm
α = 1 β = 0 τ = 0.1 σΘ
τ σΘ
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A ∈ Rm×n A = U ΣV T
U U ∈ Rm×m V T V ∈ Rn×n Σ Σ ∈ Rm×n σi σ1 ≥ σ2 ≥ ... ≥ σmin(m,n) Σ min(m, n) n < m !"# $ ˜Σ ˜ V˜ T A=U
U˜ ∈ Rm×n V˜ ∈ Rn×n Σ˜ ∈ Rn×n u v A Avi = σi ui
AT ui = σi vi .
% & V i vi i U ' n > m Avi = 0 i > m( σi A AT A
A ∈ Rn×n A A = U SU T H. Hufnagel, A Probabilistic Framework for Point-Based Shape Modeling in Medical Image Analysis, DOI 10.1007/978-3-8348-8600-2, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011
S ∈ Rn×n A U ∈ Rn×n
A A
A+
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A ∈ Rn×n
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U1 , U2 , ..., Un # S0 , S1, ... $ S0 = A Sk = UkT Sk−1 Uk .
# U1 , U2 , ..., Un lim Sk = S = diag(λ1 , λ2 , ..., λn )
k→∞
λ1, λ2 , ..., λn A T Sn = UnT Un−1 ...U1T AU1 U2 ...Un .
% Uk T A = U1 U2 ...Un Sn UnT Un−1 ...U1T .
n → ∞ Sn = S U = U1U2 ...Un A
A = U SU T .
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(
' A+
A A+ = U S + U T .
S + '( S
! EHij
=
exp(−μ(si , T mj )) k exp(−μ(si , T mk )) 1 . 1 + k =j exp(μ(si , T mj ) − μ(si , T mk ))
" # $ % & σ ! " =
2
EHij =
1+
k =j
lim rijk =
σ2 →0
exp
σ2 →0
=
1+
1
k =j rijk
.
0 if (si − T mj )2 < (si − T mk )2 . +∞ if (si − T mj )2 > (si − T mk )2
lim EHij =
1 (si −T mj )2 −(si −T mk )2 2σ2
1 if (si − T mj )2 < (si − T mk )2 0 if (si − T mj )2 > (si − T mk )2
s m 1 m s m k = j
0 ! i
j
i
j
k
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Nm j=1
ski − Tk mkj 2 exp − 2σ 2
x
∂ξ(x) ∂x
= log(u(x))
1 ∂u(x) , u(x) ∂x Nm ski − Tk mkj 2 exp − . u(x) = 2σ 2 =
j=1
∂u(x) ∂x
=
Nm
exp(f (x))
j=1
f (x) = −
∂f (x) , ∂x
ski − Tk mkj 2 . 2σ 2
∂ (ski − Tk mkj )T (ski − Tk mkj ) ∂x 2σ 2 (ski − Tk mkj )T ∂(ski − Tk mkj ) = − σ2 ∂x
∂f (x) ∂x
= −
s −T m 2 Nm exp − ki 2σk 2 kj (s − Tk mkj )T ∂(ski − Tk mkj ) ∂ξ ki =− . 2 s −T m N m ∂x σ2 ∂x exp − ki k kl j=1
2σ2
l=1
γijk
s −T m 2 exp − ki 2σk 2 kj = ski −Tk mkl 2 Nm exp − 2 l=1 2σ
m (ski − Tk mkj )T ∂ξ γkij =− ∂x σ2
N
j=1
∂(ski − Tk mkj ) . ∂x
!
Ck (Qk , Θ)
Ak
f (x) Ak ∂Ck (Qk , Θ) ∂Ak
∂ siki − Ak mkj 2 = ∂Ak = =
= −
Nk Nm
γkij
i=1 j=1
2
∂ ski − Ak mkj ∂Ak 2σ 2
∂ (s − Ak mkj )T (ski − Ak mkj ) ∂Ak ki ∂ (s T s − skiT Ak mkj − (Ak mkj )T ski − (Ak mkj )T Ak mkj ) ∂Ak ki ki T ∂ (s T s − skiT Ak mkj − skiT Ak mkj + mkj ATk Ak mkj ). ∂Ak ki ki
∂Ck (Qk , Θ) =0 ∂Ak Nk Nk Nm Nm T γkij mkj mkjT = γkij ski mkj ⇔ Ak i=1 j=1
i=1 j=1
⇔ Ak Υk = Ψk , Υk , Ψk ∈ R3×3 .
! " # $ !% ∂Ck (Qk , Θ) ∂ωkp
k m ωkp (ski − Tk mkj )T + γkij 2 λp σ2
N
=
N
i=1 j=1
=
ωkp + λ2p
& mkj = m¯ j +
Nk Nm
γkij
i=1 j=1
n
q=1 ωkq vqj
∂(ski − tk − Ak mkj ) ∂ωkp
(ski − Tk mkj )T σ2
∂(ski − Tk mkj ) ∂ωkp ∂(ski − tk − Ak mkj ) . ∂ωkp
∂ (ski − tk − Ak (m ¯j + ωkq vqj )) ∂ωkp q=1 n
=
= −Ak vpj .
Nk Nm ωkp 1 ∂Ck (Qk , Θ) = 2 − 2 γkij (ski − T mkj )T Ak vpj . ∂ωkp λp σ i=1 j=1
∂Ck (Qk ,Θ) ∂ωkp
= 0 m k σ2 ω − γkij (ski − tk − Ak m ¯ j )T Ak vpj kp λ2p
N
0 =
N
i=1 j=1
+
n
ωkq
q=1
Nk Nm
T T γkij vqj Ak Ak vpj .
i=1 j=1
ωkp
! div(V ) " V # $
div(V ) =
∂Vx ∂Vy ∂Vz + + . ∂x ∂y ∂z
% &
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div(g · V ) =
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!
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