Handbook of computer vision algorithms in image algebra

H A N D B O OK OF

Computer Vision Algorithms in Image Algebra second edition

H A N D B O OK OF

Computer Vision Algorithms in Image Algebra second edition

Gerhard X. Ritter Joseph N. Wilson

CRC Press Boca Raton London New York Washington, D.C.

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Library of Congress Cataloging-in-Publication Data Ritter, G. X. Handbook of computer vision algorithms in image algebra / Gerhard X. Ritter, Joseph N. Wilson.--2nd ed. p. cm. Includes bibliographical references and index. ISBN 0-8493-0075-4 (alk. paper) 1. Computer vision--Mathematics. 2. Image processing--Mathematics. 3. Computer algorithms. I. Wilson, Joseph N. II. Title. TA1634 .R58 2000 006.4′2--dc21

00-062122

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

© 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0075-4 Library of Congress Card Number 00-062122 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

Preface The present edition differs from the first in several significant aspects. Typographical errors as well as several mathematical errors have been removed. In a number of places the text has been revised to enhance clarity. Several additional algorithms have been included as well as an entire new chapter on geometric image transformations. By popular demand, and in order to provide a better understanding of image algebra, numerous exercises have been added at the end of each chapter. Starred exercises at the end of a chapter depend on knowledge of material from subsequent chapters. As with the first edition, the principal aim of this book is to acquaint engineers, scientists, and students with the basic concepts of image algebra and its use in the concise representation of computer vision algorithms. In order to achieve this goal we provide a brief survey of commonly used computer vision algorithms that we believe represents a core of knowledge that all computer vision practitioners should have. This survey is not meant to be an encyclopedic summary of computer vision techniques as it is impossible to do justice to the scope and depth of the rapidly expanding field of computer vision. The arrangement of the book is such that it can serve as a reference for computer vision algorithm developers in general as well as for algorithm developers using the image algebra C++ object library, iac++.1 The techniques and algorithms presented in a given chapter follow a progression of increasing abstractness. Each technique is introduced by way of a brief discussion of its purpose and methodology. Since the intent of this text is to train the practitioner in formulating his algorithms and ideas in the succinct mathematical language provided by image algebra, an effort has been made to provide the precise mathematical formulation of each methodology. Thus, we suspect that practicing engineers and scientists will find this presentation somewhat more practical and perhaps a bit less esoteric than those found in research publications or various textbooks paraphrasing these publications. Chapter 1 provides a short introduction to the field of image algebra. Chapters 2–12 are devoted to particular techniques commonly used in computer vision algorithm development, ranging from early processing techniques to such higher level topics as image descriptors and artificial neural networks. Although the chapters on techniques are most naturally studied in succession, they are not tightly interdependent and can be studied according to the reader’s particular interest. In the Appendix we present iac++ computer programs of some of the techniques surveyed in this book. These programs reflect the image algebra pseudocode presented in the chapters and serve as examples of how image algebra pseudocode can be converted into efficient computer programs.

1

The iac++ library supports the use of image algebra in the C++ programming language and is available via anonymous ftp from ftp://ftp.cise.ufl.edu/pub/src/ia/.

© 2001 by CRC Press LLC

Acknowledgments We wish to take this opportunity to express our thanks to our current and former students who have, in various ways, assisted in the preparation of this text. In particular, we wish to extend our appreciation to Dr. Paul Gader, Dr. Jennifer Davidson, Dr. Hongchi Shi, Ms. Brigitte Pracht, Dr. Mark Schmalz, Mr. Venugopal Subramaniam, Mr. Mike Rowlee, Dr. Dong Li, Dr. Huixia Zhu, Ms. Chuanxue Wang, Dr. Jaime Zapata, and Mr. Liang-Ming Chen. We are most deeply indebted to Dr. David Patching who assisted in the preparation of the text and contributed to the material by developing examples that enhanced the algorithmic exposition. Special thanks are due to Mr. Ralph Jackson, who skillfully implemented many of the algorithms herein, and to Mr. Robert Forsman, the primary implementor of the iac++ library. We also wish to thank Mr. Jeffrey Palm for preparing the fractal and iterated function system images. We wish to express our gratitude to those at Wright Laboratory for their encouragement and continuous support of image algebra research and development. This book would not have been written without the vision and support provided by numerous scientists at the Wright Laboratory at Eglin Air Force Base in Florida. These supporters include Dr. Lawrence Ankeney who started it all, Dr. Sam Lambert who championed the image algebra project since its inception, Mr. Neil Urquhart our first program manager, Ms. Karen Norris, and most especially Dr. Patrick Coffield who persuaded us to turn a technical report on computer vision algorithms in image algebra into this book. Last but not least we would like to thank Dr. Robert Lyjack of ERIM and Dr. Jasper Lupo of DARPA for their friendship and enthusiastic support during the formative stages of Image Algebra.

© 2001 by CRC Press LLC

Notation The tables presented here provide a brief explantation of the notation used throughout this document. The reader is referred to Ritter [1] for a comprehensive treatise covering the mathematics of image algebra.

Sets Theoretic Notation and Operations

Symbol



 



 

 

%
%'&
%)( %+* %'*& %)2 * >
>?&
>@(
>BADC E

Explanation Uppercase characters represent arbitrary sets. Lowercase characters represent elements of an arbitrary set. Bold, uppercase characters are used to represent point sets. Bold, lowercase characters are used to represent points, i.e., elements of point sets.





!"!#! $ .

The set

The set of integers, positive integers, and negative integers, respectively.

%* ,  -
"
"!#!!#
.0/1"$ . %'*& ,   

"!#!!#
.+$ . The set %'2 * 3   /4.6578
#!"!"!
/98

#8
#!!#!
:.;/<=$ . The set The set

The set of real numbers, positive real numbers, negative real numbers, and positive real numbers including 0, respectively.

F

The set of complex numbers.

F(G F?2 G

The set

=J K KL M

The empty set (the set that has no elements).

F?G I

© 2001 by CRC Press LLC

An arbitrary set of values.

F

H $ . F / H $. The set unioned with F / H
H $. The set unioned with unioned with

The power set of

(the set of all subsets of

"is an element of." "is not an element of." "is a subset of."

).

Symbol

NPORQ

O N `8a-b ` Oq N r rts)u Oz N r rts)u NPRQ  N `8a-b ` q N r rts)u z N r rts)u NŠ‰[Q ’q N r rts)u ’z N r rts)u ˜q Nœ›@Q N{Ÿ   ¡–¢-£ ¤Œ N{   ¥§¦ ~  #¨ Œ¤N{

Explanation

NTSUQPV3WX6YX6Z[N]\^X6Z[Q0_ . W N _ Let set c . O ` N ` `8a-V3b Wbed[Y8a d[family Z[N `fofe \^?sets g-hjilk"indexed g-m hB\nk4byopZ anc indexing _. `8a-b qO N r VwN u SUN0x?Sy#y"y S0N q. rvs)u Oz N r V3W#d{YdUZ[N r+e \^|m\}0k4~|Z{'€@_ . rvs)u Intersection NT‚UQPV3WX6YX6Z[Nƒg-n…„0X†Z{Q0_ . W N _ Let c .  ` N ` `8a-V3b Wbed[Y8a d[family Z[N ` ofe \^sets g=iliindexed opZ c _ . by an indexing set `8a-b q N VwN ‚UN x ‚y#y"y ‚0N q. rvs)u r u z N r V3W#d{YdUZ[N r e \^?g=ili~‡Zˆ'€?_ . rvs)u Cartesian product N‹‰[QPVW8Œd)Ž:8‘YdUZ{N{Ž6Z[Q0_ . ’ q N r V3W-Œ“d u Žd…x=Ž#y"y#yŽ:d q ‘Yd r Z”N r _ . rvs)u ’ z N r V3W-Œ“d u Žd…x=Ž:d–•—y"y"y @Yd r Z{N r _ . rvs)u ˜ ˜ qšV ’ q ˜ . The Cartesian product of ™ copies of , i.e., rts)u Set difference N Q Let NT›@QPand V3WdUZ[beNŠsubsets YdZ[ž Q0of_ . some universal set  , Union

N”Ÿ'V  ›@N

Complement , where



is the universal set that contains

N The cardinality of the set

.

A function that randomly selects an element from the set

Point and Point Set Operations Symbol

©«ªR¬ ©[¯¬

© 2001 by CRC Press LLC

N

Explanation

V Œd u ª  u Ž)y#y"y§Žd q ª  q  . © Ž ¬ pZ ­‡q , then ©0ª®¬ , Ž pZ ­‡q , then ©U¯¬ , V Œd u ¯  u Ž)y#y"y§Žd q ¯  q  . If © ¬ If

N

.

.

Symbol

°{±² °À¿=² °«ÁU² °«ÂU² °'ò É Ãj°

°À˲ °ÅU² ° Ñ ² Б° Õ °Ö × °Ø Ù °§Ú ÛjÜ ¸Ý° ¾ Þ° ß° Á|° Â|° á° áÎ á° áº á ° á#ç è–éÝê ¸° ¾ ì Ìí ì Јí ì Ìø © 2001 by CRC Press LLC

Explanation

°?³²´pµ‡¶ , then °[± ²{·3¸¹+º¼»8º-³'½"½#½§³…¹ ¶ » ¶…¾ . If °?³²´pµ‡¶ , then °À¿=²{·,¸¹'º ¿»8º#³)½#½"½§³¹ ¶ ¿8» ¶¾ . If °?³²´pµ‡¶ , then °0ÁU²ˆ·¸ ¹+º+Á[»8º-³'½#½"½'³¹ ¶ ÁU» ¶…¾ . If °?³²´pµ‡¶ , then °0ÂU²ˆ·¸ ¹ º Â[» º ³'½#½"½'³¹ ¶ ÂU» ¶ ¾ . In general, if °?³²´ˆµ|¶ , and Ã{ĵ®Å{µˆÆǵ , then °'òȷ3¸¹ º Å» º ³'½"½"½…³§¹ ¶ Å» ¶ ¾ . É ´ˆµ °´{µ|¶ Ã{Ä8µÊÅ[µÆǵ , then If É Ã°ˆ·3¸ ,É Ã§¹ º ³'½#½",½§and ³ É Ã…¹ ¶ ¾ . If °?³²´pµ‡¶ , then °À˲{·¹'º »8ºÍÌʹ§Î"»-Î?̃±"±#±'Ìʹ ¶ » ¶ . If °?³²´pµ‡¶ , then °Å«²ˆ·3¸¹ Î »"ϑÐʹjÏ#» Î ³¹jÏ-» º Ðȹ º »"ϖ³¹ º » Πй Î » º ¾ . If °´{µ|¶ and ²´pµ‡Ò , then °:Ñ ²ˆ·,¸¹'º"³À½"½#½§³¹ ¶ ³»8º³'½"½#½'³» ÒÓ¾ . If °´{µ|¶ , then Б°p·Ô¸:Ð4¹ º ³À½"½"½…³§Ð4¹ ¶ ¾ . Õ Õ Õ If °´{µ|¶ , then °ÖÓ·3¸ ¹'º¤Öj³À½"½#½§³ ¹ ¶ Ö ¾ . × × × If °´{µ|¶ , then °ØÓ·3¸ ¹ º Øj³À½"½#½§³ ¹ ¶ Ø ¾ . Ù Ù Ù If °´{µ|¶ , then °§ÚD·3¸ ¹ º Ú ³'½#½"½§³ ¹ ¶ Ú ¾ . If °{·¸ ¹ º ³¹ Î ³"½"½#½³:¹ ¶ ¾ ´pµ‡¶ , then ÛjÜ ¸Ý° ¾ ·1¹ Ü . Þ If °´{µ|¶ , then °”·¹'º|Ìʹ§Î?̃±"±#±'Ìʹ ¶ . ß If °´{µ|¶ , then °[·7¹+º¼¹§Î6±#±"±–¹ ¶ . If °´{µ|¶ , then Á|°”·¹+º+Á[¹§Î?Áà±#±"±…Á¹ ¶ . If °´{µ|¶ , then Â|°”·¹ º Â[¹ Î Âà±#±"±…¹ ¶ . Î Î á á If °´{µ|¶ , then ° Î ·àâ ¹ º Ìã±#±"±…Ìä¹ ¶ . á á If °´{µ|¶ , then ° º ·æå ¹'ºåÌ7å ¹§Îå ̃±"±"±DÌãå ¹ ¶ å . á á#ç ·æå ¹+º=åÁå ¹…Î8å Áà±#±"±…Áæå ¹ ¶ å . If °´{µ|¶ , then ° è–é¤ê ¸¤° ¾ ·ë . If °´{µ|¶ , then ì ì ìœö ë è ²´”í÷ . If ³ íïîðµ|¶ , then Ìíñ·3ò#°0Ì®²óÄô°õ´ ì ì ìœö ë è ²´”í÷ . If ³ íïîðµ|¶ , then Ð{íñ·3ò#°UвóÄô°õ´ ì î1µ|¶ and Û ´{µ|¶ , then ì Ìøˆ·3ò#°0Ì®øœÄô°{´ ì ÷ . If If

Symbol

Explanation

ùûúˆü ù ù! ù!+,

ù]ý1þ|ÿ and 
þ|ÿ , then ùæú [úü ˆ ü 
ù . ùïý ðþ|ÿ , then ù 
ù 
 . If ùïý ðþ|ÿ , then ù!"#$ 
ù"%'&)(
*  . If ùïý ðþ|ÿ If ù-+.#$/ , then
ù.%'&)(
* ù0 . ùïý ðþ|ÿ , then ù1.#4356798! 
ù %:&)(;7
 . If ù]ý1þ|ÿ , then ú‘"ù 8ú<=  If
ù> . ? #  þA@,%'&)(!* ù ù]ý1þ|ÿ , then ù" If

. ù]ý1þ|ÿ B$C 3Ý-ù 86 supremum of ù . If If ù"ED$F9, G4then HHHIF ÿ 4 then the B$C 3Ý-ù 86JEDLK9G6KH$HHMK, ÿ . ù For a N point set order O , LP< ù QR O with 9PSUTVtotal 
ù 9P . ù]ý1þ|ÿ WY&)XZù-3 86 ù If ù" D F , G then H$HHF ÿ , thentheWY&)XZùinfimum -3 86J ofDL[  .G\If[ H$HH [  ÿ . ù For a ] point set total order O ,  P  ù QR)^ Owith _ UTV
ù P . ù]ý1þ|ÿ , then If `a W `b 3-ù 8
ùc3d%'&)(:efhg `a  Bb & b f b e b &ZiI8 . ù]ý1þ|ÿ , then ` %'¤ù(j-3 8Eki aVb` %'('WY&)%'fhWlUùiIgmX . In If ù  D nnn ÿ , then ` %'¤ù(j!3 8Eo& . particular, if If

ù21, ú‘ù ù? BC 3Ý-ù 8 N ù W5&)X)ù!3 8 ] ù `a  W $` b 3¤!ù 8 ` %'¤ù(j!3 8 Morphology

In the following table, Symbol

pt pyx pyz

p 1-q p‚ q

p q,Mr and s

Explanation

þ‡ÿ .

 3wvjIvj$HHHMv8
þ‡ÿ . u # p x 
þ|! ÿ /*
p . The complement of p ; i.e.,  pyz ${}|~€ {
p . Minkowski addition is defined as p 1 q#${m|~€ {
p F~
qy (Section 7.2). qƒ3 pyx 1 q t 8 x Minkowski subtraction is defined as p,‚ The reflection of

(Section 7.2). © 2001 by CRC Press LLC

denote subsets of

p

across the origin

Symbol

Explanation

„ƒ…<†

„ † is denoted „…<† and is defined by „…<†ƒ‡ˆd„,‰4†‹ŠŒ-by† (Section 7.3). „ † is denoted „Ž<† and is defined by The closing „Ž<†ƒ ‡ˆd„ofŒ†‹Šby‰S† (Section 7.3). ’ ‘ # ‡ w ˆ  “  ”  • Š Let be an ordered pair of structuring elements. „ is given by The hit-and-miss „J‘’‡#–—/˜4“™!transform š›„œžVŸof• the ™,šset„¡5¢ (Section 7.5). The opening of

„ƒŽ<† „ƒ/‘

Functions and Scalar Operations

Symbol

£ ˜ ¤¥§¦ ©'ª«,¬'­d® ˆ £ Š ¯ ¬'®±°:² ˆ £ Š

A£ ¶A· ¦‹¸ £6º » £Eº ½ ° …£ £‹É ° £Ê ° Ë Ê$£ º £Eº © 2001 by CRC Press LLC

£

is a function from

¤

Explanation into

¦<¨

£ ˜ ¤¥§¦ is the set ¤ ¨ £ ˜¤¥§¦ is the set The of the function £– ˆU³±range Š˜:³´!¤µ¢ . £ The inverse of the function . £ ´-¦‹¸ , then ¤ ¦ The £ ˜¹¤.set¥§of¦all¨ functions from into , i.e., if £ ˜S¤.¥§¦ and a subset ¼ š›¤ , the Given a function £ £Lº » ˜ ¥§¦ , is defined by to ¼ ¬ , restriction £Lº » ˆ ¬ ŠE‡ £ ofˆ ¬ Š for ´ ¼ .¼ £ ˜ ¼ ¥§¦ and ° ˜4¾¥§¦ , the extension of £ to ° is Given ŠÁÀÃÂ\³´ ¼ £Eº ½ ˆ³±ŠE‡=¿ °£ ˆwˆ³±³VŠÄ ÀÃÂ\³´¾/Å ¼ ¨ defined by £ ˜¤Æ¥§¦ and ° ˜4¦¥RÇ , the Given two functions ° £ …  ˜  ¤ ¥RÇ is defined by composition ˆ ° … £ ŠÈˆw³±ŠE‡ ° ˆ £ ˆ³VŠFŠ , for every ³´ ¤ . £ ° Let ° and be real or° complex-valued functions, then ˆ £‹É ŠMˆ³±ŠL‡ £ ˆU³±Š É ˆ³VŠ . £ ° be real or complex-valued functions, then Let ˆ £Ê ° ŠMand ˆd³VŠE‡ £ ˆ³±Š Ê ° ˆd³VŠ . £ Let be a real or complex-valued £ ´ÍÏÎ , function, ˆ Ë Ê£ ŠÐˆd³VŠEand ‡ Ë Ì Ê beˆ £ ˆa³±Šreal Š. or complex number, then º £Eº ˆw³±Š6‡ º £ ˆw³±Š º , where £ is a real (or complex)-valued º £ ˆw³±Š º denotes the absolute value (or function, and £ ˆ³VŠ . magnitude) of The domain of the function

Symbol

ÑÐÒ

Ñ ÒÓÔÆÕÖÔ is given by Ñ Òm×ØÚÙ6ÛØ . Ý onto the ã th coordinate is defined The projection functionØ Ü4ÙL d × Ø Ø ÛØ Ý . 4 Ü Ý Ý  â  ä  å $ å M å ä  ä $ å  å M å ä by Þ Ô The cardinality of the set . The identity function

ÜSÝ Ó!àhß áAÞ â Ô à Õ§Ô Ý æç'èé ×YÔ Ù æê±ëì5æ$í ×YÔ Ù Ømî!ï Ømò!ï ó Øô õ Øö ÷ Øùø Øú ëé ï 


Explanation

×dØVÙ

Ô

A function which randomly selects an element from the set .

Ø ä mï ðñ , Øî!ï is the maximum of Ø and ï . Ø ïmðñ , Øò!ï is the minimun of Ø and ï . For ä Øð ñ the ceiling function ó ØSô returns the smallest For Ø integer that is greater than or equal to . Øð ñ the floor function õ ØSö returns the largest integer For Ø that is less than or equal to . Øð ñ the round function returns the nearest integer to Ø . For For

If there are two such integers it yields the integer with greater magnitude.

Ø ä mï ðû , Øú ëé ïmÛ è if there exists ü ä è ð û mè ý ï such that ØÛï üÿþ è .

For

Ñ Øð ×UرÙEÛ 



The characteristic function   




 




with

is defined by

Images and Image Operations Symbol

 ä <  ä

ð !#  " ð !# %  ð !# &  ' ( © 2001 by CRC Press LLC

Explanation Bold, lowercase characters are used to represent images. Image variables will usually be chosen from the beginning of the alphabet.

!

The image  is an -valued image on $ . The set the value set of  and $ the spatial domain of  .

!

Ñ

Ñ

Let be a set with unit . Then whose pixel values are .

!



Let be a set with zero . Then whose pixel values are .

Û

×

The domain restriction of +* )-, defined by ' (

 ð"!# ! Ù

%

!

is called

&

denotes an image, all of denotes an image, all of

to a subset .! () of . Thus, ' ( .

ð

$

is

Symbol

Explanation

/1032

The range restriction of /54.687 to the subset 9;: 6 is defined by /80 2=< /?>A@CBED 9GF . The double-bar notation is used to focus attention on the fact that the restriction is applied to the second coordinate of / : BHDI6 . Thus if J
/T UWVYX 2[Z

If /A4"67 , \]: B , and 9^: 6 , then the restriction of and 9 is defined as /T UWVYX 2[Z < /+>_@ \ D 9GF . Thus if J
Let B and f extension of

/T e @r/T gsFut @v/OwxT /y[TuzRz3zT /{ F }~ / € ‚ g „ C@ / F

ƒ

/ ŠŽ„ /Ž” g ˜ ”‰/ /e ¡W¢’£ / e /©¨ ª /

/

to

be subsets of the same topological space. The

/ 4.67 to g 4.61h is defined by 5 O / B /T eO@ M F <ji @ @ M M Fkml MM 4I 4 g Fnkml fpo B.q Row concatenation of images / and g , respectively the row concatenation of images / w t / y tu|R|3|t / { . Column concatenation of images / and g . „

Nx6†…ˆ‡ , then the image If /A4"67 „‹and Š / given by , i.e., „ „

„ C@ / 4.‡‰7 F

is

@C/ F L < K @ M tŒ @ M FF N Œ @ M F < @C/O@ M FFut M 4IB P . „ ŠŽ„ 4"6h If N f …B and /A4"67 , the induced image / Ž Š „ „ <K @v‘ t /O@ @C‘ FF F N’‘.4 f“P . defined by /

is

is a binary operation on 6 , then an induced operation on can be defined. Let / t g 4.68• ; the induced operation is given by /Ž” g
67

”

˜

Let 4"6 , /A467 , and ” be a binary operation on induced operation on images is defined by ˜ ”™/ <šK scalar @ M tŒ @ M FF N Œ @ M F < ˜ ”™/O@ M F3t M 4IB P .

6

. An

UŸ /œe <ž @ M t Œ @ M F F N Œ @ M F < /O@ M F e Z t M 4AB.  / g 4†@ ›¤ F 7 ; Let ¡W¢3£ / t <j ¥ @ M tŒ @ M FF N Œ @ M F < ¡c¢3£ e UŸ Z /O@ – Ft M 4AB§¦ . e If /A4"67 and 6 has a conjugation operation *, then the ¨ pointwise conjugate of image / , /d¨@ M F < @r/O@ M F F . ª / reduction by a generic reduce operation ª Nx6denotes 7-…ˆ6 (Section 1.4). Let

/ t g 4"›7

;

The following four items are specific examples of the global reduce operation. Each assumes /A4"›7 and B <K3M w t M y t3|R|3|t M { P . © 2001 by CRC Press LLC

\

.

Symbol

Explanation

«­¬

«E¬¯®°¬O±v²³´Oµ^¬O±v²1¶3´Oµš·R·3·µ^¬O±r²O¸©´

¹ ¬

¹ ¬¯®°¬O±v²³´Q·¬O±C²1¶3´·x·3·R·3·¬a±v²O¸©´

º ¬

º ¬¯®»¬O±C²³¼´O½5¬O±r²1¶R´O½†·3·R·½5¬O±r²O¸©´

¾ ¬

¾ ¬¯®»¬O±C² ³ ´O¿5¬O±r² ¶ ´O¿†·3·R·¿5¬O±r² ¸ ´

¬?ÀbÁ ¬É ¬©Ê ¬Ë

Dot product,

.

.

. .

¬+ÀbÁ.®Ã‰±r¬=·ÁQ´®Ä« ±v¬a±È²1´·uÁb±v²1´ ´ . ÅÆÇ ¬

Complementation of a set-valued image .

¬

Complementation of a Boolean image .

¬

Transpose of image .

Templates and Template Operations

Symbol

ÌÎÍ ÏÍCÐ ÏÒÑÔÓCÕ ÇbÖR×

Ï3Ü à ± ÏRÜ ´ à8å ± Ï Ü ´ àsç å ± Ï Ü ´ à8ê å ± 3Ï Ü ´ © 2001 by CRC Press LLC

Explanation Bold, lowercase characters are used to represent templates. Usually characters from the middle of the alphabet are used as template variables. A template is an image whose pixel values are images. In particular, an Õ -valued template from Ø to Ù is a function Ï+Ú ØEÛ Õ Ç . Thus, ÏÒÑ Ó Õ Ç Ö × and Ï is an Õ Ç -valued image on Ø .

Ï Ñ Ó Õ Ç Ö × . For each Ý Ñ Ø , Ï Ü ® Ï ± Ý ´ . The image Ò Ï3ܯÑ"Õ Ç is given by ÏRÜ ®Þαv² ÍÏRÜ v± ²1´´ Ú ² Ñ † Ù ß. Þá Í â ÏÒÑ Ó Õ Ç Ö × , then the support of Ï is If Õ;Ñ ´Ü à ß ± Ï and denoted and±r²1is´Òdefined by ®Ã ã ä à ± Ï Ü ´Q®LbyÞ² Ñ Ù ÚÏ Ü ß. ±r²1´Ò®Ã ã æ á Çå Ö × àOå ± Ï Ü ´Q®LÞ² Ñ If ÏÒÑ Ó , then Ù ÚÏ Ü ß. ±v²1´è® ã é™æ á Çç å Ö × à ç å ± ÏRÜ ´Q®LÞ² Ñ If ÏÒÑ Ó , then ٠ڒÏRÜ ß ±v²1´è®Ã ã ëÒæ á Çê å ÖR× à ê å ± Ï Ü ´Q®LÞ² Ñ Ù Ú’Ï Ü ß If ÏÒÑ-Ó , then Let

. .

Symbol

Explanation

ìÎíïîdð

A parameterized ñ -valued template from ò to ó with ì+õ ôšöø÷CñùŽúRû . parameters in ô is a function of the form

ì3ü

ìÒý ÷ ñsù ú û . ì3ü þdív
ÿ°ì ð  í 1 ð .

Let

The transpose

ì3üOý ÷ ñ ú ù û

is defined as

Image-Template Operations In the table below,

ó

is a finite subset of

Symbol

  ì

ì  



.

Explanation

í

 ð

ý

ì+ý

÷Èñsì ùŽúRû , then ñù , Let ñ be a semiring and  the generic right convolution product of  with is defined  ìsívÿ ð í1 ð  ì  í1 ð
 írÿððŽõÿ†ý ò  írÿ as    ìÒý

÷ ñ û ú ù , the With the conditions above, except that now ì  generic left convolution product of with is defined as ì   ! írÿ

' & ì

ð í%1 ð  ì þ ívÿð  " íWÿð ðbõÿ;ý ò  rí ÿ
# $  
*  ý ý +.- 0/ Let ò)( , ñù , and ìÒý ÷ ñsù ú û , where ñ ,

.

The right linear convolution product is defined as

èívÿ ' & ì21

ì & 

ð  í%1 EDð ì  í%1 Fð G 3 íWÿð ðbõÿ“ý ò  ívÿ465 þ87 ù:98ìÒ;=ý A@CB ÷Cñ û ú ù , the left With the conditions above, except that

linear convolution product is defined as

PQ O R

ì  &  )HIJ rí ÿ "  íWÿð ðŽõ’ÿ.ý ò $ íWÿð 5  í%1 EDð ì þ írÿELð M þ87 í ð N ù:98; >K@ R For S TVU
WXY and T[ZU
WXY]\^ , the right morphological

max convolution product is defined by

R `_ S

SP`  R

S` _ RbadcfeAgh"i:egkjlj:mng Tpo hiqe%g
j4s8a t R

r S e%}kj~RC€e%}kjCƒ‚ W:u8v8w8x y?zA{.|

For S TVU
WXY and T Z U ^XY \ W , the left morphological max convolution product is defined by

R ` _ S a…† ‡„ e%gqh3iqeˆg4j"j mCg Tpo hi:eˆg
ja r eA}kj~GR s eg4j=Š ‹ ‚ e S j Œ s8t W:u8v wnx z‰{ R For S'TVU
WXY and T[ZUŽWXkY]\$^ , the right morphological min

convolution product is defined by

S
`  Rba c eAg:h"iqeˆg4j"j:mng To hiqeˆg
js8a t

© 2001 by CRC Press LLC

 S eA}kj~f“R  eA}kj  ‚ Wqu’‘ x y?z { |

Symbol

” – • —

—'¾ ½ ”

Explanation

— ˜V™
š›œ

” ˜[™
ž›œ]Ÿ š

For and , the left morphological min convolution product is defined by

” –¡ • —’¢…¤ ¥€£ ¦%§q¨A©q¦%§4ª"ª «C§ ˜ ¬ ¨$©:¦ˆ§
ª ¢ ­ — ¦¶kªk·f¸ ” ® ¦§
ª=¹ º ¦%³3´ ª »G¼ ®8¯ š °8±=² µ ž — ˜  ™0¿ÁÀœ Ÿ š and ” ˜#  ™q¿Àœ Ÿ š à , the right For

multiplicative max convolution product is defined by

” ¾½ —

— ¾ ½ ” ¢dÄ ¦%§q¨3©q¦ˆ§
ª3ª «C§ ˜f¬ ¨l©q¦§4ª Æ ¢ Å ³ — ¦%¶kªÊÉ ”$Ë ¦%¶kª$Ì ®8¯ šq°’±8Ç µCÈ ¼ —'˜ÍA™qœ… ¿ÁÀ Ÿ š and ” ˜  A™q¿À œÎž Ÿ à š , the left multiplicative For

max convolution product is defined by

— •¾ ”

” ¾ ½ —’¢…¤ ¥€£ ¦%§q¨3©q¦ˆ§
ª3ª «C§ ˜f¬ ¨©q¦%§4ª ¢ Å ³ — ¦%¶kª É ” ® ¦§4ªE¹ º ¦ ´ª »Ï¼ ®8¯ š:°8± µ ž — ˜  ™q¿Àœ Ÿ š and ” ˜#  ™q¿Àœ Ÿ š à , the right For

multiplicative min convolution product is defined by

” •¾ —

— •¾ ” ¢dÄ ¦%§q¨3©q¦ˆ§4ª"ª «C§ ˜p¬ ¨l©q¦%§
ª ¢ ­ ³ — ¦%¶kªqÉ
¸ ”Ë ¦A¶kªÌ ®8¯ : ¼ š °8± ² Ç µCÈ š —'˜  ™q¿ÁÀœ Ÿ š and ” ˜   ™q¿Àœ Ÿ ž à , the left multiplicative For

min convolution product is defined by

” •¾ —=¢…¤ ¥€£ ¦§:¨"©:¦ˆ§
ª3ª «n§ ˜p¬ ¨©q¦%§
ª ¢ ­ %¦ ³ — ¦Aª ¶kªqÉ4¸ ” ® ¦§4ªE¹ º ´ »Ï¼ ®8¯ š:°8± ² µ

Neighborhoods and Neighborhood Operations

Symbol

Ð ¨lÑ

Ñ ˜[Ònš Ÿ ž Ñ,¦?ØÙª Ñ ¸ Ñ¡äå¾ · Ñdæ

© 2001 by CRC Press LLC

Explanation Italic uppercase characters are used to denote neighborhoods. A neighborhood is an image whose pixel values are sets of ¬ points. InÑÔ particular, « ¬ÖÕ×ÒCš a neighborhood from to Ó is a function . A parameterized neighborhood Ñ) from « in Ú is a function of the form Ú

¬ Õ

Ó with parameters  ÒCš Ÿ ž .

to

Ñ ˜  ÒCš Ÿ ž Ñ'¸ ˜  ÒCž Ÿ š , the transpose § C « ¶ Ü Ñ % ¦ 4 § Þ ª Ý ¢…Û ˜¬ ˜ , that is, ¶ ˜ Ñ,¦%§
ª€ßáà'§ ˜ãâ ¸¦%¶kª . Ñ]ä Ñdæ The of by is defined by Ñ,¦§4dilation ª ¢ ç è ¯=é
ê3ë?ì=íEî3ï]ð$î%ñ
òó[îAô'õñ4ò3ò . Let Ñ'¸¦¶kª

is defined as

Image-Neighborhood Operations

ö

In the table below, Symbol

ý

!#  ý

>=# %# A

ý

ý

ú

ýÿ
þ

If

, then

ù
ú

.

Explanation

    þ ù
ú  þ  .  ù
ú þ  ú   , and reduce operation Given þ  $% ù
&ú ' ù ,, ýÆ the generic right reduction of  with ý is !#        $ ,

    ,

1     $.0/ defined as ( "!# ý 4 ( 56 ý*(  )+)7( )  )  þ8 6 (  )  $   ( 23. ) . Thus, ú  With the conditions above, except that now ýÖþ  , the  with is defined as generic left reduction of 9 !#  )  (  !# ý;:<) . A( ý  Given  þã' ÷  , and the image average function ? ÷  ÷ , the right reduction of with ý yields the neighborhood averaging operation, ( @=# ý*)+(  )  ?  ,0  .   Given  C the image median function  B  ÷  þã' ÷ ÷ , ,and the right reduction of with ý yields the neighborhood median filtered image, (  # A ý)D(  )  B  ,-  .

ù
úÊû ü

"!#

÷ø

is a finite subset of

Matrix and Vector Operations In the table below,

E

and

F

represent matrices.

Symbol

EHG EI: E JF , EHF K E LMF E>#PN O F E># N O: F

Explanation The conjugate of matrix The transpose of matrix

E

E

. .

The matrix product of matrices The tensor product of matrices The p-product of matrices

E># N O: F  F&:3# N O EQ:  :

E

E

E

and

The dual p-product of matrices .

E

F

and and . and

F

F

.

F

.

, defined by

References [1] G. Ritter, “Image algebra.” Unpublished manuscript, available via anonymous ftp from ftp://ftp.cise.ufl.edu/pub/src/ia/documents, 1994. © 2001 by CRC Press LLC

To our brothers, Friedrich Karl and Scott Winfield

© 2001 by CRC Press LLC

Contents 1.

IMAGE ALGEBRA 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10.

2.

IMAGE ENHANCEMENT TECHNIQUES 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12. 2.13. 2.14. 2.15. 2.16. 2.17.

3.

Introduction Point Sets Value Sets Images Templates Recursive Templates Neighborhoods The p-Product Exercises References

Introduction Averaging of Multiple Images Local Averaging Variable Local Averaging Iterative Conditional Local Averaging Gaussian Smoothing Max-Min Sharpening Transform Smoothing Binary Images by Association Median Filter Unsharp Masking Local Area Contrast Enhancement Histogram Equalization Histogram Modification Lowpass Filtering Highpass Filtering Exercises References

EDGE DETECTION AND BOUNDARY FINDING TECHNIQUES 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8.

Introduction Binary Image Boundaries Edge Enhancement by Discrete Differencing Roberts Edge Detector Prewitt Edge Detector Sobel Edge Detector Wallis Logarithmic Edge Detection Frei-Chen Edge and Line Detection

© 2001 by CRC Press LLC

3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17. 3.18. 3.19. 3.20. 3.21. 3.22.

4.

THRESHOLDING TECHNIQUES 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10.

5.

Introduction Global Thresholding Semithresholding Multilevel Thresholding Variable Thresholding Threshold Selection Using Mean and Standard Deviation Threshold Selection by Maximizing Between-Class Variance Threshold Selection Using a Simple Image Statistic Exercises References

THINNING AND SKELETONIZING 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9.

6.

Kirsch Edge Detector Directional Edge Detection Product of the Difference of Averages Canny Edge Detection Crack Edge Detection Marr-Hildreth Edge Detection Local Edge Detection in Three-Dimensional Images Hierarchical Edge Detection Edge Detection Using K-Forms Hueckel Edge Operator Divide-and-Conquer Boundary Detection Edge Following as Dynamic Programming Exercises References

Introduction Pavlidis Thinning Algorithm Medial Axis Transform (MAT) Distance Transforms Zhang-Suen Skeletonizing Zhang-Suen Transform — Modified to Preserve Homotopy Thinning Edge Magnitude Images Exercises References

CONNECTED COMPONENT ALGORITHMS 6.1. 6.2. 6.3. 6.4.

Introduction Component Labeling for Binary Images Labeling Components with Sequential Labels Counting Connected Components by Shrinking

© 2001 by CRC Press LLC

6.5. 6.6. 6.7. 6.8.

7.

MORPHOLOGICAL TRANSFORMS AND TECHNIQUES 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9.

8.

Introduction Basic Morphological Operations: Boolean Dilations and Erosions Opening and Closing Salt and Pepper Noise Removal The Hit-and-Miss Transform Gray Value Dilations, Erosions, Openings, and Closings The Rolling Ball Algorithm Exercises References

LINEAR IMAGE TRANSFORMS 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10.

9.

Pruning of Connected Components Hole Filling Exercises References

Introduction Fourier Transform Centering the Fourier Transform Fast Fourier Transform Discrete Cosine Transform Walsh Transform The Haar Wavelet Transform Daubechies Wavelet Transforms Exercises References

PATTERN MATCHING AND SHAPE DETECTION 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8. 9.9. 9.10.

Introduction Pattern Matching Using Correlation Pattern Matching in the Frequency Domain Rotation Invariant Pattern Matching Rotation and Scale Invariant Pattern Matching Line Detection Using the Hough Transform Detecting Ellipses Using the Hough Transform Generalized Hough Algorithm for Shape Detection Exercises References

10. IMAGE FEATURES AND DESCRIPTORS 10.1. 10.2.

Introduction Area and Perimeter

© 2001 by CRC Press LLC

10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9. 10.10. 10.11. 10.12. 10.13. 10.14.

Euler Number Chain Code Extraction and Correlation Region Adjacency Inclusion Relation Quadtree Extraction Position, Orientation, and Symmetry Region Description Using Moments Histogram Cumulative Histogram Texture Descriptors: Spatial Gray Level Dependence Statistics Exercises References

11. GEOMETRIC IMAGE TRANSFORMATIONS 11.1. 11.2. 11.3. 11.4. 11.5.

Introduction Image Reflection and Magnification Nearest Neighbor Image Rotation Image Rotation using Bilinear Interpolation Application of Image Rotation to the Computation of Directional Edge Templates 11.6. General Affine Transforms 11.7. Fractal Constructs 11.8. Iterated Function Systems 11.9. Exercises 11.10. References

12. NEURAL NETWORKS AND CELLULAR AUTOMATA 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8. 12.9. 12.10.

Introduction Hopfield Neural Network Bidirectional Associative Memory (BAM) Hamming Net Single-Layer Perceptron (SLP) Multilayer Perceptron (MLP) Cellular Automata and Life Solving Mazes Using Cellular Automata Exercises References

APPENDIX. THE IMAGE ALGEBRA C++ LIBRARY

© 2001 by CRC Press LLC

CHAPTER 1 IMAGE ALGEBRA

1.1.

Introduction

Since the field of image algebra is a recent development it will be instructive to provide some background information. In the broad sense, image algebra is a mathematical theory concerned with the transformation and analysis of images. Although much emphasis is focused on the analysis and transformation of digital images, the main goal is the establishment of a comprehensive and unifying theory of image transformations, image analysis, and image understanding in the discrete as well as the continuous domain [1]. The idea of establishing a unifying theory for the various concepts and operations encountered in image and signal processing is not new. Over thirty years ago, Unger proposed that many algorithms for image processing and image analysis could be implemented in parallel using cellular array computers [2]. These cellular array computers were inspired by the work of von Neumann in the 1950s [3, 4]. Realization of von Neumann’s cellular array machines was made possible with the advent of VLSI technology. NASA’s massively parallel processor or MPP and the CLIP series of computers developed by Duff and his colleagues represent the classic embodiment of von Neumann’s original automaton [5, 6, 7, 8, 9]. A more general class of cellular array computers are pyramids and Thinking Machines Corporation’s Connection Machines [10, 11, 12]. In an abstract sense, the various versions of Connection Machines are universal cellular automatons with an additional mechanism added for nonlocal communication. Many operations performed by these cellular array machines can be expressed in terms of simple elementary operations. These elementary operations create a mathematical basis for the theoretical formalism capable of expressing a large number of algorithms for image processing and analysis. In fact, a common thread among designers of parallel image processing architectures is the belief that large classes of image transformations can be described by a small set of standard rules that induce these architectures. This belief led to the creation of mathematical formalisms that were used to aid in the design of special-purpose parallel architectures. Matheron and Serra’s Texture Analyzer [13], ERIM’s (Environmental Research Institute of Michigan) Cytocomputer [14, 15, 16], Martin Marietta’s GAPP [17, 18, 19], and Lockheed Martin’s PAL processor [20] are examples of this approach. The formalism associated with these cellular architectures is that of pixel neighborhood arithmetic and mathematical morphology. Mathematical morphology is the part of image processing concerned with image filtering and analysis by structuring elements. It grew out of the early work of Minkowski and Hadwiger [21, 22, 23], and entered the modern era through the work of Matheron and Serra of the Ecole des Mines in Fontainebleau, France [24, 25, 26, 27]. Matheron and Serra not only formulated the modern concepts of morphological image transformations, but also designed and built the Texture Analyzer System. Since those early days, morphological operations have been applied from lowlevel, to intermediate, to high-level vision problems. Among some recent research papers on morphological image processing are Crimmins and Brown [28], Haralick et al. [29, 30], Maragos and Schafer [31, 32, 33], Davidson [34, 35, 36], Dougherty [37, 38], Goutsias [39, 40], Koskinen and Astola [41], and Sivakumar and Goutsias [42].

© 2001 by CRC Press LLC

2

CHAPTER 1.

IMAGE ALGEBRA

Serra and Sternberg were the first to unify morphological concepts and methods into a coherent algebraic theory specifically designed for image processing and image analysis. Sternberg was also the first to use the term “image algebra” [43, 44]. In the mid 1980s, Maragos introduced a new theory unifying a large class of linear and nonlinear systems under the theory of mathematical morphology [45]. More recently, Davidson completed the mathematical foundation of mathematical morphology by formulating its embedding into the lattice algebra known as Mini-Max algebra [46, 47, 48]. However, despite these profound accomplishments, morphological methods have some well-known limitations. For example, such fairly common image processing techniques as feature extraction based on convolution, Fourier-like transformations, chain coding, histogram equalization transforms, image rotation, and image registration and rectification are — with the exception of a few simple cases — either extremely difficult or impossible to express in terms of morphological operations. The failure of a morphologically based image algebra to express a fairly straightforward U.S. government-furnished FLIR (forward-looking infrared) algorithm was demonstrated by Miller of Perkin-Elmer [49]. The failure of an image algebra based solely on morphological operations to provide a universal image processing algebra is due to its set-theoretic formulation, which rests on the Minkowski addition and subtraction of sets [23]. These operations ignore the linear domain, transformations between different domains (spaces of different sizes and dimensionality), and transformations between different value sets (algebraic structures), e.g., sets consisting of real-, complex-, or vector-valued numbers. The image algebra discussed in this text includes these concepts and extends the morphological operations [1]. The development of image algebra grew out of a need, by the U.S. Air Force Systems Command, for a common image-processing language. Defense contractors do not use a standardized, mathematically rigorous and efficient structure that is specifically designed for image manipulation. Documentation by contractors of algorithms for image processing and rationale underlying algorithm design is often accomplished via word description or analogies that are extremely cumbersome and often ambiguous. The result of these ad hoc approaches has been a proliferation of nonstandard notation and increased research and development cost. In response to this chaotic situation, the Air Force Armament Laboratory (AFATL — now known as Wright Laboratory MNGA) of the Air Force Systems Command, in conjunction with the Defense Advanced Research Project Agency (DARPA), supported the early development of image algebra with the intent that the fully developed structure would subsequently form the basis of a common imageprocessing language. The goal of AFATL was the development of a complete, unified algebraic structure that provides a common mathematical environment for image-processing algorithm development, optimization, comparison, coding, and performance evaluation. The development of this structure proved highly successful, capable of fulfilling the tasks set forth by the government, and is now commonly known as image algebra. Because of the goals set by the government, the theory of image algebra provides for a language which, if properly implemented as a standard image processing environment, can greatly reduce research and development costs. Since the foundation of this language is purely mathematical and independent of any future computer architecture or language, the longevity of an image algebra standard is assured. Furthermore, savings due to commonality of language and increased productivity could dwarf any reasonable initial investment for adapting image algebra as a standard environment for image processing. Although commonality of language and cost savings are two major reasons for considering image algebra as a standard language for image processing, there exists a multitude of other reasons for desiring the broad acceptance of image algebra as a component of all image processing development systems. Premier among these is the

© 2001 by CRC Press LLC

1. 1 Introduction

3

predictable influence of an image algebra standard on future image processing technology. In this, it can be compared to the influence on scientific reasoning and the advancement of science due to the replacement of the myriad of different number systems (e.g., Roman, Syrian, Hebrew, Egyptian, Chinese, etc.) by the now common Indo-Arabic notation. Additional benefits provided by the use of image algebra are • The elemental image algebra operations are small in number, translucent, simple, and provide a method of transforming images that is easily learned and used; • Image algebra operations and operands provide the capability of expressing all image-to-image transformations; • Theorems governing image algebra make computer programs based on image algebra notation amenable to both machine dependent and machine independent optimization techniques; • The algebraic notation provides a deeper understanding of image manipulation operations due to conciseness and brevity of code and is capable of suggesting new techniques; • The notational adaptability to programming languages allows the substitution of extremely short and concise image algebra expressions for equivalent blocks of code, and therefore increases programmer productivity; • Image algebra provides a rich mathematical structure that can be exploited to relate image processing problems to other mathematical areas; • Without image algebra, a programmer will never benefit from the bridge that exists between an image algebra programming language and the multitude of mathematical structures, theorems, and identities that are related to image algebra; •

There is no competing notation that adequately provides all these benefits.

The role of image algebra in computer vision and image processing tasks and theory should not be confused with the government’s Ada programming language effort. The goal of the development of the Ada programming language was to provide a single highorder language in which to implement embedded systems. The special architectures being developed nowadays for image processing applications are not often capable of directly executing Ada language programs, often due to support of parallel processing models not accommodated by Ada’s tasking mechanism. Hence, most applications designed for such processors are still written in special assembly or microcode languages. Image algebra, on the other hand, provides a level of specification, directly derived from the underlying mathematics on which image processing is based and that is compatible with both sequential and parallel architectures. Enthusiasm for image algebra must be tempered by the knowledge that image algebra, like any other field of mathematics, will never be a finished product but remain a continuously evolving mathematical theory concerned with the unification of image processing and computer vision tasks. Much of the mathematics associated with image algebra and its implication to computer vision remains largely unchartered territory which awaits discovery. For example, very little work has been done in relating image algebra to computer vision techniques which employ tools from such diverse areas as knowledge representation, graph theory, and surface representation.

© 2001 by CRC Press LLC

4

CHAPTER 1.

IMAGE ALGEBRA

Several image algebra programming languages have been developed. These include image algebra Fortran (IAF) [50], an image algebra Ada (IAA) translator [51], image algebra Connection Machine *Lisp [52, 53], an image algebra language (IAL) implementation on transputers [54, 55], and an image algebra C++ class library (iac++) [56, 57]. Unfortunately, there is often a tendency among engineers to confuse or equate these languages with image algebra. An image algebra programming language is not image algebra, which is a mathematical theory. An image algebra-based programming language typically implements a particular subalgebra of the full image algebra. In addition, simplistic implementations can result in poor computational performance. Restrictions and limitations in implementation are usually due to a combination of factors, the most pertinent being development costs and hardware and software environment constraints. They are not limitations of image algebra, and they should not be confused with the capability of image algebra as a mathematical tool for image manipulation. Image algebra is a heterogeneous or many-valued algebra in the sense of Birkhoff and Lipson [58, 1], with multiple sets of operands and operators. Manipulation of images for purposes of image enhancement, analysis, and understanding involves operations not only on images, but also on different types of values and quantities associated with these images. Thus, the basic operands of image algebra are images and the values and quantities associated with these images. Roughly speaking, an image consists of two things, a collection of points and a set of values associated with these points. Images are therefore endowed with two types of information, namely the spatial relationship of the points, and also some type of numeric or other descriptive information associated with these points. Consequently, the field of image algebra bridges two broad mathematical areas, the theory of point sets and the algebra of value sets, and investigates their interrelationship. In the sections that follow we discuss point and value sets as well as images, templates, and neighborhoods that characterize some of their interrelationships.

1.2.

Point Sets

A point set is simply a topological space. Thus, a point set consists of two things, a collection of objects called points and a topology which provides for such notions as nearness of two points, the connectivity of a subset of the point set, the neighborhood of a point, boundary points, and curves and arcs. Point sets are typically denoted by capital bold letters from the end of the alphabet, i.e., W, X, Y, and Z.

RTS0UVS0WYX*Z R%X\[^] mQ_oniSfpfSjjkjeSfq b0r

Points (elements of point sets) are typically denoted by lower case bold letters . Note also that if , then x is of from the end of the alphabet, namely , where for each , denotes a real number form called the ith coordinate of x.

RM_a`cbedSfbhgiSjjkjeShb ]fl

[s]

q_KntS^pfS

The most common point sets occurring in image processing are discrete subsets of with or 3 together with the discrete topology. n–dimensional Euclidean space However, other topologies such as the von Neumann topology and the odd-even product topology are also commonly used topologies in computer vision [1].

[^]

used There is no restriction on the shape of the discrete subsets of in applications of image algebra to solve vision problems. Point sets can assume arbitrary shapes. In particular, shapes can be rectangular, circular, or snake-like. (here we view Some of the more pertinent point sets are the set of integer points ), the n–dimensional lattice (i.e., ) with , and or rectangular subsets of . Two of the most often encountered rectangular point sets are

u d uw v [ e u w ] v ^ [ ] e u x ] _  u @ y uzy|{k{{8y}u_ ~ RX*[^€ HR*_"`7b d SjkjjSfb STb r X\u8‚eƒ„Hm3_>niS3jjkjeSfq†… ‡ q ˆ _ p q_ˆ‰ ug ]l

© 2001 by CRC Press LLC

1. 2 Point Sets

5

Š

‹"Œe}Ž*Œe‹4‘“’c”–•˜—f”e™›š5œ*Œ ™%Hž Ÿ ”–• Ÿ¢¡z£
¤ — ž Ÿ e” ™ Ÿ¢¥¦£
¤0§ — Š or Ÿ ¥ §© ‹ Œ¨ Ž*Œ¨ ‹ ‘ ’”•—f”e™›š5œ*Œ ™% ¤&Ÿ ”• ŸC¡ — ¤IŸ ”e™ C Š We follow standard practice and represent these rectangular point sets by listing the points in matrix form. Figure 1.2.1 provides a graphical representation of the point set ‹ Œ ¨ Ž,Œ ¨ . of form

1

­n

2

1

... ...

2

... ... .. . .. .

¬m

.. . .. .

«y

.. . .. . ... ...

ªx Figure 1.2.1. The rectangular point set

®°¯|±³| ² ´ ±µ²

.

Point Operations

¶µ

As mentioned, some of the more pertinent point sets are discrete subsets of the . These point sets inherit the usual elementary vector space operations. vector space (or ) and Thus, for example, if , then the sum of the points x and y is defined as

®

®¸· ± µ

®¸· ¶ µ

¹¯Kºc»–¼˜½¾¾k¾e½h» µf¿ ½ÁÀ\¯KºcÂi¼½3¾¾k¾e½f µh¿ÄÃ

¹ÅMÀ*¯Æºc»–¼ÅMÂi¼½¾¾k¾½f» µ ÅM µh¿ ½ while the multiplication and addition of a scalar Ç Ã ± (or Ç Ã ¶ Ǡț¹8¯>ºÇ È» ¼ ½3¾¾k¾e½fǐț» µ ¿

) and a point x is given by

ÇÉÅʹ*¯"º7ÇHÅM» ¼ ½¾¾¾h½eÇÉÅÊ» µ ¿ ½

and

respectively. Point subtraction is also defined in the usual way. In addition to these standard vector space operations, image algebra also incorporates three basic types of point multiplication. These are the Hadamard product, the cross (or ), and the dot product which are defined by product (or vector product) for points in

±eË ¶^Ë ¹*țÀ*¯>º» ¼ È ¼ ½†¾k¾¾h½f» µ Èk µ ¿ ½ ¹ ´ À*¯Æºc»eÌÈk ËÍ » Ë ÈkÂÌi½f» Ë Èk ¼ Í » ¼ È›Â Ë ½h» ¼ È ÂÌ Í »hÌ5È ¼ ¿ ½

and respectively.

© 2001 by CRC Press LLC

¹3ÎÏÀ\¯ »–¼TÈkÂi¼Å » Ì ÈkÂ Ì ÅÐÈkÈÈhÅ » µ È µ ½

6

CHAPTER 1.

IMAGE ALGEBRA

Note that the sum of two points, the Hadamard product, and the cross product are binary operations that take as input two points and produce another point. Therefore, these whenever X is closed under these operations can be viewed as mappings operations. In contrast, the binary operation of dot product is a scalar and not another vector. , where denotes the appropriate field This provides an example of a mapping of scalars. Another such mapping, associated with metric spaces, is the distance function which assigns to each pair of points x and y the distance from x to y. The most common distance functions occurring in image processing are the Euclidean distance, the city block or diamond distance, and the chessboard distance which are defined by

Ñ Ò\шÔxÑ Ó ÑKÒHÑÕÔ°Ö

ÑÐÒѸԈ×

Ö

ØÁÙÛÚTÜÏÝÞ^ßáà ã â Ùcç äèMéä ÞëêÏìHîí Ü äå†æ ï ÙÚTÜÏÝÞ^ß ã â ç ä èMé ä Ü äå†æñð ð ò ÙÛÚTÜÏÝÞßCó8ôç–õ ç ä èMé ä Ü ð ð0öÁ÷QøCù¦ø¢ú†û

and

respectively. Distances can be conveniently computed in terms of the norm of a point. The norms three norms of interest here are derived from the standard

The

üý

æ þÚ þ Ó ß ÿãâ ç ý ý ý å†æ ð ð


ü







norm is given by

þÚ þ ß â ç Ü å†æ–ð ð 



â æ ç ߈óYôç–õ ç æ Ü eÜ ç . Specifically, the Euclidean norm is given by å † ð ØÁð ÙÛÚTâ3ÜÏð Ý3û ÞQß þ Ú è Ý þ þ Ú þ ê ß ð ç êæ ð ç ê ð . Thus, city block distance ÙÚTÜ7Ý3ÞHß þ Ú ê .è Similarly, Ý þ æ and thethechessboard ï â ò Ù ÚTÜ7Ý3ސusing distance can be computed the formulation ß þÚ è Ý þ . by using that the p-norm of a point x is a unary operation, namely a function þ|þ шNote Ô × . Another assemblage of functions Ñ Ô × which play a major role in ý ö then ith projection on various applications functions. Given Ñ
× Ùâ Ú3, ÞV ç denotes õ Ü are hÜ the, isprojection ßKç the, where denoted by and defined by X, where ÷ ú†û the ith coordinate of x. where







  













Characteristic functions and neighborhood functions are two of the most frequently occurring unary operations in image processing. In order to define these operations, we need to recall the notion of a power set of a set. The power set of a set S is defined as the set of all subsets of S and is denoted by  . Thus, if Z is a point set, then    . Given !"# (i.e., $ ), then the characteristic function associated with %'& X is the function

ß õÑ Ñ ö Ñ û

Ñ

õ fÜ ö Ô ÷ û 

© 2001 by CRC Press LLC

(

1. 2 Point Sets

7 .<;>=

)+*-,/.102436587:9

defined by

.A;> @ =4B

7:9 ?

For a pair of point sets X and Z, a neighborhood system for X in Z, or equivalently, =GFIHJKB a neighborhood function from X to Z, is a function CE, D 0ONQP

;M=

,

C

0

C

L L It follows that for each point L , . The set is called a neighborhood for x. =UFVH#WYX There are two neighborhood functions on subsets of RS which are of particular importance in image processing. These are the von Neumann neighborhood and the Moore CTD , 02[Z \ \62[,^]`_acbedf] 0gh-\i2j mr;sZ d d by neighborhood. The von Neumann neighborhood ,k]`_ dl] anmo0pdisb#dqdefined 5ett C ,]_ dq] 2Q L

D 0u;>= N ,

where L

02QZ \ RS

S v

L

S \i2[,^]

? HeW X =wFI S

asb#dl]

dD

axmy0 db#dlmz;MZ v

, while the_ Moore neighborhood D

5 tt

B

is defined by

? S

Figure 1.2.2 provides a pictorial representation of these two neighborhood functions; the hashed center area represents the point x and the adjacent cells represent the adjacent points. The von Neumann and Moore neighborhoods are also called the four neighborhood and eight neighborhood, respectively. They are local neighborhoods since they only include the directly adjacent points of a given point.

{

N(x) =

M(x) =

Figure 1.2.2. The von Neumann neighborhood |M}/~+ and the Moore neighborhood €}/~+ of a point x. There are many other point operations that are useful in expressing computer vision algorithms in succinct algebraic form. For instance, in certain interpolation schemes it becomes necessary to switch from points with real-valued coordinates (floating point coordinates) to corresponding integer-valued coordinate points. One such method uses the induced floor operation ‚qƒ>„f…+†>‡Vˆ† defined by ‚‰~lƒ<Š‹}‚‰Œ`:ƒqŽ‚/ŒfƒqŽ  qŽu‚‰Œ ƒy , where † ~>ŠQ}ŒŽlŒqeŽ+  fŽlŒ ‘A…'† and ‚‰Œo’“ƒK‘6ˆ denotes the largest integer less than or equal to † Œo’ (i.e., ‚:Œo’”ƒ–•4Œl’ and if —M‘˜ˆ with —c•[Œo’ , then —M•™‚‰Œo’“ƒ ).

Summary of Point Operations We summarize some of the more pertinent point operations. Some image algebra implementations such as iac++ provide many additional point operations [59].

© 2001 by CRC Press LLC

8

CHAPTER 1.

Binary operations. ­

Let

›®^¯   ¡f¯ ¢ ¡£ £ £f¡q¯Y°±¥>«4¬

šœ›ž^Ÿ`  ¡qŸo¢#¡£ £ £¡lŸo¤l¥ ¡y¦œ›§¨© ª¡l¨¢©¡+£ £ £q¡l¨ ¤q¥j«œ¬ °

addition subtraction multiplication division supremum infimum dot product cross product ^Á>› concatenation scalar operations

¤

, and

.

š³²´¦n›‹^Ÿ   ²´¨   ¡£ £ £q¡lŸ ¤

²n¨ ¤ ¥

š>µs¦n›‹^Ÿ   µs¨   ¡£ £ £q¡lŸ ¤

µc¨ ¤ ¥

šA¶ ¦˜›·^Ÿ   ¨   ¡+£ £ £f¡lŸ ¤ ¨ ¤ ¥ š+¸¦n›·^Ÿ   ¸©¨   ¡£ £ £f¡qŸ ¤ ¸#¨ ¤ ¥ ¹ º©»/š¡¼¦+¥-›½^Ÿ  +¾

¨   ¡£ £ £¡lŸ ¤¿¾

¨ ¤ ¥

À:ÁÂ`š¡¼¦+¥-›¼Ÿ  +Ã

¨   ¡£ £ £¡lŸ ¤¿Ã

¨ ¤ ¥

š+ÄŦc›Ÿ   ¨   ²xŸ ¢ ¨ ¢ ²Æ¶ ¶ ¶²˜Ÿ ¤ ¨ ¤ Ç¥

š³È¿¦A›[^Ÿ ¢ ¨ÉʵcŸoÉ ¨ ¢ ¡fŸlÉ ¨   µnŸ   ¨ Éy¡lŸ   ¨ ¢ µnŸ ¢ ¨   ¥ ­ šÌË ›‹Ÿ ¡£ £ £f¡qŸo¤¡l¯e ¡£ £ £¡l¯ ° ¥ ÍyÎlšn›·^ÍÏÎqŸ ¡+£ £ £q¡lÍyÎfŸl¤q¥Y¡

where Unary operations.

IMAGE ALGEBRA

In the following let

Îs«xЪ²K¡µ¿¡lÑl¡ ¾

¡ ÃuÒ

šc›½kŸ   ¡lŸ ¢ ¡£ £ £q¡fŸ ¤ ¥O«n¬

¤

.

µ-šn›wÅµÓŸ   ¡£ £ £¡fµÓŸ ¤ ¥

negation ceiling floor rounding projection sum product maximum minimum Euclidean norm è   norm èê norm dimension neighborhood

ԓšlÕ>›wԉŸ   Õl¡£ £ £¡Ô:Ÿ ¤ Õ#¥ ֓šl×>›w։Ÿ   ×l¡£ £ £¡Ö:Ÿ ¤ ×#¥ Ø šfÙڛÛØ Ÿ   Ù¡£ £ £¡Øܟ ¤ ÙÝ¥ »lޝš+¥ß›ŸlÞ àušc›‹Ÿ   ²áŸ ¢ ²â¶ ¶ ¶+²ãŸ ¤ ä-šx›wŸ   Ÿ ¢

¶ ¶ ¶yŸ ¤

¾ šs›åŸ   ¾ à šs›åŸ`  à æۚæ ¢ æۚæ   æۚæ ê

ݍ

Ÿ ¢ ¾

¶ ¶ ¶ ¾

›é Ÿ` eéY²4é Ÿf¢yép²Æ¶ ¶ ¶²Æé Ÿo¤+é ›Gé Ÿ` ªé ¾

é Ÿf¢1é ¾

ëyÀ:ì6:š+¥¿›Á ¤ íM:š+¥ïîå¬

characteristic function

ð+ñu

Ÿ ¤

Ÿq¢ à ¶ ¶ ¶ à Ÿ ¤ o ¢ ¢ Ÿ   ²¶ ¶ ¶²nŸ ¤

­

¥O›Uò6õ ó

À/Â À/Â

¶ ¶ ¶ ¾

é Ÿl¤+é

­

«>ô ­Aö «>ô

It is important to note that several of the above unary operations are special instances of spatial transformations ô÷Iø . Spatial transforms play a vital role in many image processing and computer vision tasks. In the above summary we only considered points with real- or integer-valued coordinates. Points of other spaces have their own induced operations. For example, ¤ typical operations on points of ôù›Vú¢ ¥ (i.e., Boolean-valued points) are the usual logical operations of ûyüý , þ#ÿ , lþ©ÿ , and complementation.

Point Set Operations




  

 !#"%$&'()  *+,*+!#"%$&'() -./+ 0((1)  *2-.3*' 0((1)

Point arithmetic leads in a natural way to the notion of set arithmetic. Given a (i.e., ) and an arbitrary point we vector space Z, then for ôi¡oøœ« define the following arithmetic operations: addition subtraction point addition point subtraction

© 2001 by CRC Press LLC

1. 2 Point Sets

9

465

are the usual set operations of union, intersection, Another set of operations on set difference (or relative complement), symmetric difference, and Cartesian product as defined below.

7839;:=<.>?0>@37 A6B>@9C 7D,9E:F<>?>@37HGJI%K0>L@(9'C 7MN9E:=<.>?0>L@(7 GJI%K0>(@3O 9C 7P,9Q:<>?0>@7R8,9QGJI%K0>(@3O 7RD/9C 7S39:<TVUXWZY\[3?U@7!G#I%K&Y'@(9C

union intersection set difference symmetric difference Cartesian product

45

Note that with the exception of the Cartesian product, the set obtained for each of the . above operations is again an element of , Another common set theoretic operation is set complementation. For . the complement of X is denoted by , and defined as In contrast to the binary set operations defined above, set complementation is a unary operation. However, complementation can be computed in terms of the binary operation . of set difference by observing that In addition to complementation there are various other common unary operations which play a major role in algorithm development using image algebra. Among these is the cardinality of a set which, when applied to a finite point set, yields the number of elements in the set, and the choice function which, when applied to a set, selects a randomly chosen point from the set. The cardinality of a set X will be denoted by card(X). Note that

^7

7]@4 5 ` _7 :]?b>@cdGJI%K0>1@O 71C

^7 :QcXM67 

e GJB6K?4 5/fhg`ikjmlonqparsrJtvu wsxzy{y|r}y|~y.uJxzXlNj 4 5€ W

.e ‚ A6ƒ e.„ ?4 5 f c… That is, e GJB6K T†73[‡@ g and e.‚ A6ƒ e.„ TV7[X:ˆU , where x is some randomly chosen element of X. while

As was the case for operations on points, algebraic operations on point sets are too numerous to discuss at length in a short treatise as this. Therefore, we again only summarize some of the more frequently occurring unary operations.

Summary of Unary Point Set Operations In the following

7Š‰Q‹Œ .

Ž{7E:!<6Ž{U=?0U'@37(C _7 :=?b>@(‹ŒGJ„I%K0>(@3O 7(„ C o‘%TV7[TZ’vA6B“’vƒVI%ƒk” „ ‘A6ƒmI•”– „ ”q73[ ƒme.I%‚ ’—TZe.7„ [TZ’vA6B“’vƒVI%ƒs” ‘A6ƒmI•”– e.”q‚ 73[„ „ „ „ [{@3‚7 „œe TZB6GJI%K#Aa˜3™†š A6 I ™ ˜ I•”›[ e GJB6A6K•ƒ T†7Tm73[{: ” GJB6KJƒVI%G#™kƒs”šA6’17 The interpretation of a‘—Tž73[ is as follows. Suppose X is finite, say 7Ÿ: <.U 6WJU\¡6W£¢¢a¢WJU–¤JC . Then o‘%TV7[:¥a‘—Tv¢a¢¢Jo‘%TZo‘%TZo‘%TZU 6WzU–¡|[.W¦U%§[.W¦U%¨©[|W%¢¢a¢WzU–¤6[ , where a‘—TmU%ª«WzU ¬6[ denotes the binary operation of the supremum of two points defined earlier. For example, if U ª : TZ­ ª Wš ª [ for ƒ®: ¯W%¢a¢¢Wz° , then o‘%TV7[ : TZ­  –± ­ ¡²±R³a³³± ­ ¤ W š  X± š ¡X±R³a³a³v± š ¤ [ . More generally, o‘%TV7[ is defined to be the negation complementation supremum infimum choice function cardinality

least upper bound of X (if it exists). The infimum of X is interpreted in a similar fashion. If X is finite and has a total order, then we also define the maximum and minimum and , respectively, as follows. Suppose of X, denoted by and , where the symbol denotes the particular total order on X.

´7 µ+7 U  ¸· U ¡¹·Š³a³³0· U ¤

© 2001 by CRC Press LLC

·

7H:¶<U   W•U ¡ W%¢a¢a¢vW•U ¤ C

10

CHAPTER 1.

º2»h¼-½–¾

IMAGE ALGEBRA

¿»h¼=½XÀ

Á%Â

and . The most commonly used order for a subset X of Then is the row scanning order. Note also that in contrast to the supremum or infimum, the maximum and minimum of a (finite totally ordered) set is always a member of the set.

1.3.

Value Sets

A heterogeneous algebra is a collection of nonempty sets of possibly different types of elements together with a set of finitary operations which provide the rules of combining various elements in order to form a new element. For a precise definition of a heterogeneous algebra we refer the reader to Ritter [1]. Note that the collection of point sets, points, and scalars together with the operations described in the previous section form a heterogeneous algebra. A homogeneous algebra is a heterogeneous algebra with only one set of operands. In other words, a homogeneous algebra is simply a set together with a finite number of operations. Homogeneous algebras will be referred to as value sets and will be denoted , and . There are several value sets that by capital blackboard font letters, e.g., occur more often than others in digital image processing. These are the set of integers, real numbers (floating point numbers), the complex numbers, binary numbers of fixed length k, and/or ), and the extended the extended real numbers (which include the symbols , , non–negative real numbers. We denote these sets by , , and , respectively, denotes the set of positive real numbers. where the symbol

Ã{ÄLÅ

Ê²Ò Î ¼RÊ1Ï(ÐoÉbÈÑ Í Ê Ó Î ¼RÊ1Ï(Ð6Ç0ÈÔāÉbÈˆÑ Ê

Æ

ÇœÈ ÉbÈ Á ćÊ{ÄqË0Ä‡Í Á ÂÌ Ê\ͣΠ¼®Ê3Ï1ЩÇ0ÈˆÑ Ê‡ÎÕvÖ ¼Ê ÏÐ©×•Ä ÇØÈÙÑ

Operations on Value Sets

Å

ÅÛÚÐ6ÁÄ–Ê{Ä%Á Â Ì Ñ

Å

are the usual The operations on and between elements of a given value set , then the binary elementary operations associated with . Thus, if operations are the usual arithmetic and logic operations of addition, multiplication, and maximum, and the complementary operations of subtraction, division, and minimum. If , then the binary operations are addition, subtraction, multiplication, and division. Similarly, we allow the usual elementary unary operations associated with these sets such as the absolute value, conjugation, as well as trigonometric, logarithmic and exponential functions as these are available in all higher-level scientific programming languages. we need to extend the arithmetic and logic operations of as For the set follows:

ż-Ë

ÊÓÎ Ê Ü ÇݦÉvÈÞ\¼¶ÝzÉvÈÞ£Ç Ü ¼ÉvÈ Ü Ú(Ê‡Ò Î Ü ÇßÈà¼ÔÈQÇ Ü ¼È Ü ÚáÊ Î ¼ÈÇÙâÝzÉvÜ ÈÞXÜ ¼¶ÉvÈ ÜݦÉvâ ÈßݦÞ%ÉvÇÈÈh Ü ÚáÊ Ó Î ÞX¼®Ý¦ÉvÈÞ ¼ â Note that the element ÉbÈ acts as a null elementâ in the system ݞÊXÓ Î Ä Ä Ç0Þ if we view the operation + as multiplication and the operation as addition. The same cannot be said about the element È in the system ÝZÊ Ó Î3āã{Ä Ç0Þ since ÝzÉvÈÞ.ǓÈä¼ȈÇLÝ«ÉvÈßÞ²¼® â ÉvÈ . In order to remedy this situation we define the dual structure ÝZÊ Ó Î Ä ãåÄ Çbæ†Þ of ÝZÊ Ó Î Ä Ä Ç0Þ as follows: Ü Ç æJç ¼ Ü Ç ç Ü Ä ç ÚáÊ Ü Ç æ ÝzÉvÈލ¼®Ý¦ÉvÈßÞ£Ç æ Ü ¼¶ÉvÈ Ü Ú(Ê‡Ò Î Ü Ç æ Èh¼ÈQÇ æ Ü ¼È Ü Ú(ÊXÎ ¼ÔÜ ÈÛÇ Ü æ ÝzÉvÈލ¼ÔÈ ÜݝÉvã3ÈÈhÞ–Ç ¼ˆæ Èà Ü ÚáÊ Ó Î Èã ¼ © 2001 by CRC Press LLC

1. 3 Value Sets

11

èœé êzëì–í3îï{î èðòñ|ó è èbð ôbé è0éó êžë ì£í î õ{î ï{î èLîvèð†ñ ö¹÷dë ì£í öaø ö6ø0ùàôåö 0ô ê«ôbéßñ‡ùRé ú.ûJü úýoþoüvþÿ ý ú
ü úý |ü þ ÿ ý þ   þ and úý  aü þ ÿ ý þ  þ .

acts as a null element in the system Observe, however, Now the element and introduce an asymmetry between and The that the dual additions is known as a bounded lattice ordered group [1]. resultant structure we Dual structures provide for the notion of dual elements. For each by , where . The following duality define its dual or conjugate laws are a direct consequence of this definition:

žëí



î õ{î ïåî îð

ð

Closely related to the additive bounded lattice ordered group described above is . Here the dual of the multiplicative bounded lattice ordered group ordinary multiplication is defined as

  ð 

    î 

ù

÷áë



ùÔë

with both multiplicative operations extended as follows:

 3éhùé  ùé   ð éhùé ð  ùé !  éàùÔé"#œù$ ! ð éhùé% ð Ø ùé

è0é 1ö ÷ ë í +'



ë)í *'

÷(ë í ÷áë í 

ë&í ('

î ï{î  ð 

î õ{î



ö6ø

Hence, the element 0 acts as a null element in the system and the element acts as a null element in the system . The conjugate of an element of this value set is defined by

ö 465%ø7oî õ{î ï{î9èL8 î-è 8 ;ð :

. ö0/21 -ø , 

èØé

ö&÷'ë  öùÔè0é ó ö3 ù 

if if if

Another algebraic structure with duality which is of interest in image algebra is the , where . value set The logical operations and are the usual binary operations of max (or) and min (and), are defined by the tables shown respectively, while the dual additive operations and in Figure 1.3.1.

õ

5%ø70ùê<5 7 ñ ì£í = ù 5 7 >éî|ôåé?Øù@•î Aîôbéî éB

ï

ôbé é 0 1 é ôåé 0 é ôåé 1 0 1 ôåé é é é é é ôbé ôåé é åô é

è8

è8 ð

è8

è8 ð

ôbé é 0 1 é ôbé 0 é ôbé 1 0 1 ôbé é é é ôbé é ôåé ôbé ôbé é Figure 1.3.1. The dual additive operations è 8 and è 8 ð . ù 0 î AD Note that the addition è 8 (as well as è 8 ð ) restricted to 5 7 C 0

1

0

1

is the complement of the exclusive-or operation, xor, and computes the values for the truth table (i.e., p if and only if q). of the biconditional statement

EFG

© 2001 by CRC Press LLC

12

CHAPTER 1.

IMAGE ALGEBRA

The operations on the value set H IJ can be easily generalized to its k-fold Cartesian H IJLKNM H IJ>O H IJPORQDQDQSO H IJ . Specifically, if T MVU TW6X2Y6YDY*X(T#Z0[]\^ k i _ H IJLM K W Z _ _

X D Y 6 Y * Y + X _ ` [ a \ H c T L b + X * _  b d \ H J e

X D Y 6 Y

Y ( X h j T , where for , then I I J  M U K g M f l k T W i _ W XmYDY6Y*X(T Z k i _ ZDn . k Z The addition i should not l bek confused with thek usual addition To0prq on H J K . k In fact, for TPX+_`\sH J K TBi _ M U T W _ W [utvX Y6YDY+X U T Z _ Z [ut n , where k o0p~q Mf U T b k _ b [ t Mxwcy f e{e{zz UU TT bb k __ bb [|[|T} T}o0p~q M y)€ Many point sets are also value sets. For example, the point set  Mƒ‚m„ is a metric space as well as a vector space with the usual operation of vector addition. Thus, Uv‚„ X k [ , where the symbol “k ” denotes vector addition, will at various times be used both product and

as a point set and as a value set. Confusion as to usage will not arise as usage should be clear from the discussion.

Summary of Pertinent Numeric Value Sets In order to focus attention on the value sets most often used in this treatise we provide a listing of their algebraic structures: (a) (b) (c) (d) (e) (f) (g)

U<‚ (X …9X(†-X k X Q [
In contrast to structure c, the addition and multiplication in structure d is addition . and multiplication These listed structures represent the pertinent global structures. In various applications only certain subalgebras of these algebras are used. For example, the subalgeand of play special roles in morbras of , where phological processing. Similarly, the subalgebra , is the only pertinent applicable algebra in certain cases. The complementary binary operations, whenever they exist, are assumed to be part of the structures. Thus, for example, subtraction and division which can be defined in . terms of addition and multiplication, respectively, are assumed to be part of

Uv‚) ‰ (X …-X k [ Uv‚  ‰ X+†-X k t [ ‘jM@’ y X f X((q X YDYDY+XL_X Y6YDY”“

Uv‚ ˆm‰ X(…-X*†-X k X k k t [ Uv‘ X+…-X(†-X [

U HX(…-X(†-X k X Q [

U<‚ X(…-X(†9X k X Q [

Value Set Operators

•

q—–

are again the usual As for point sets, given a value set , the operations on is a lattice, then operations of union, intersection, set difference, etc. If, in addition, the operations of infimum and supremum are also included. A brief summary of value set operators is given below.

© 2001 by CRC Press LLC

•

1. 4 Images

13

˜ ™+ša›œrž for some value set Ÿ ! ˜¡ Pšx¢¤£6¥>¦§¥¨›˜3©0ª¨¥¨›š¬« ˜¡­Pšx¢a£¥P¦®¥9›˜3¯~° ±®¥¨›š²« ˜§³šN¢a£¥¦®¥¨›#˜$¯~° ±®¥!›P ´ šc« ˜§µš¢@£0¥P¦®¥9›#˜¶ š=¯° ±¥®› ´ ˜¡­š!« ˜·Pšx¢@£r¸v¯+™”¹»º¦§¯¬›˜3¯~° ±®¹-›Pš¬« ¥»¼*©0½{¥»¾—¸{˜º!›¶˜ ¥»¯~ª0±¿¸À˜§ºÁ¢@¥»¯~ª0±~½{° ¯~ÂÀ½ÄÃÆÅÇ©0Ș É6ÊË ¸Ì˜§º¨¢ÉDÊrË*ª0¾0ÍÇÊ*ÍΩ0Ș ½{° Èϸv˜§º¨¢N½Ì° È*½{Í}Ê+ÍЩ0È>˜

For the following operations assume that union intersection set difference symmetric difference Cartesian product choice function cardinality supremum infimum

1.4.

.

Images

The primary operands in image algebra are images, templates, and neighborhoods. Of these three classes of operands, images are the most fundamental since templates and neighborhoods can be viewed as special cases of the general concept of an image. In order to provide a mathematically rigorous definition of an image that covers the plethora of objects called an “image” in signal processing and image understanding, we define an image in to general terms, with a minimum of specification. In the following we use the notation (i.e., ). denote the set of all functions

˜ÁÑ ˜¨Ñ¡¢j£0È=¦®ÈǽÌɯ®È+Ê*° ¥”Ãƽ̩0°È+ª0©0ÍӚBÃÆ©¨˜®«

š=ÒИ

Definition: Let Ô be a value set and X a point set. An Ô -valued image on X is any element of Ô)Õ . Given an Ô –valued image ÖØ×ÓÔ)Õ (i.e., ÖÙ ÚÜÛÝÔ ), then Ô is called the set of possible range values of a and X the spatial domain of a.

Þ›PŸ2ß

represent a. The graph It is often convenient to let the graph of an image of an image is also referred to as the data structure representation of the image. Given , then an element of the data structure representation the data structure is called a picture element or pixel. The first coordinate x of a pixel is called the pixel location or image point, and the second coordinate a(x) is called the pixel value of a at location x. The above definition of an image covers all mathematical images on topological spaces with range in an algebraic system. Requiring X to be a topological space provides us with the notion of nearness of pixels. Since X is not directly specified we may substitute any space required for the analysis of an image or imposed by a particular sensor and scene. with of form , where For example, X could be a subset of denote spatial location and t a time variable. the first coordinates or Similarly, replacing the unspecified value set with provides us with digital integer-valued and digital vector-valued images, respectively. An implication of these observations is that our image definition also characterizes any type of discrete or continuous physical image.

àÇá]ârã{ä)å|àÏãÀäçæ|æ}è-äéPêPë

ã<ò2åLóræ

ì*íîrïcðí

ã{ä å”à ã{ämæ”æ

äñé`ê õ

äáxãvòmå|ó¿åvôÆæ

ìmö”÷

õáøã<ì öL÷)å+ì öúù å+ì öLû æ

Induced Operations on Images

õ

Operations on and between -valued images are the natural induced operations of the algebraic system . For example, if is a binary operation on , then induces a defined as follows: binary operation — again denoted by — on

õ

© 2001 by CRC Press LLC

ü

ü

õý

õ

ü

14

CHAPTER 1.

IMAGE ALGEBRA



þmÿ
 . Then þ   ÿ 
  $þ  úÿ !#"$  ÿ %'& and our value set is the algebraic structure of the real For example, suppose þm ÿ )S+ÿ *;ÿ ,-ÿ - . Replacing  by the binary operations )!.ÿ */ÿ , , and - we obtain numbers (&9 Let

the basic binary operations

þ0) 1   ÿ  
  $þ /)2. »ÿ1!#"~ÿ þ3*41   ÿ  
  $þ *4. 5»ÿ16!7"ÿ þ8,91   ÿ 
  $þ /,:. »ÿ1!#"~ÿ and

þ8-:

ÿ

$þ (/-:. úÿ16!7"

;   5<= 

on real-valued images. Obviously, all four operations are commutative and associative. In addition to the binary operation between images, the binary operation  on also induces the following scalar operations on images:  For >?@@ACBED F  , 

¤þ >( þ G 5 ÿH 98  %>(þ (úÿ16!7"

and

þI>9 %  ÿ :8 5 $þ 
E>ÿ16!7"J$

Thus, for images:

>:

&

, we obtain the following scalar multiplication and addition of real-valued

úþ

ÿ

»þ Dÿ

>K* 9 G 5 H 98  %>K* / 5 ##!#"

and

þ9 ; 5 ÿH <=  %>)ñþ/ 5Dÿ##!#";$

>=)

It follows from the commutativity of real numbers that,

»þ Bþ3*L>MANBED?>=)þ9 BþO)?>/$

>K* 9

Although much of image processing is accomplished using real-, integer-, binary-, or complex-valued images, many higher-level vision tasks require manipulation of vectorand set-valued images. A set-valued image is of form !QPSRNT . Here the underlying value set is U(R;T V /W CX+Y , where the tilde symbol denotes complementation. Hence, the operations on set-valued images are those induced by the Boolean algebra of the value set.  For example, if <ZU(R;TNY , then

þ

¿ÿ -ÿ -ÿ þ2ÿ þ8V: þ8W:

and

where

ÿ

$þ V:. [Dÿ##!7" ÿ ÿ 5< 5 $þ W:. [Dÿ##!7" ÿ

;   5< 5 ; 

þ3X ]\  ÿ 5< ^ `þ/_ 5Dÿ þ/_ 59 bc—þ/  .

© 2001 by CRC Press LLC

!

!ÿ a

1. 4 Images

15

The operation of complementation is, of course, a unary operation. A particularly useful unary operation on images which is induced by a binary operation on a value set is known as the global reduce operation. More precisely, if d is an associative and commutative binary operation on e and X is finite, say fhgjiLklGm
called the global rreduce byy/d } , which y r operation y/} r  induced y/} y/} is definedy/as } g zC{|

k~g

e

y g%‡

Thus, for example, if Š

kƒG~g

€<‚ d

and

k l ~d

k n ~„dM…L…C…d

k p ~/† r

g%‰

is the operation of addition (dgFˆ ), then

and

Š g

‹Œ;3Ž‘’“Ž/”4‘/•2Ž/–C‘/•˜—L—C—E•2Ž/5/™E‘4š

In all, the value set provides for four basic global reduce operations, namely ¡ (›œ¤ •Kœ—ž. œŸœ ‘ , and Žœ:¢@Ž+œ9£ Ž

Ž

Induced Unary Operations and Functional Composition 

induced In the previous section we discussed unary operations on elements of ¥ by a binary operation ¦ on ¥ . Typically, however, unary image operations are induced directly by unary operations on ¥ . Given a unary operation § ¨¥2©ª¥ , then the induced   ©«¥ is again denoted by f and is defined by unary operation ¥ §

¨

ŽE‘’¬;5œ<­‘<‘

­‘’

§

Ž/‘<‘®œ1¯6°7±Jš 

Note that in this definition we view the composition §6² as a unary operation on ¥ Ž with operand a. This subtle distinction has the important consequence that f is viewed as   to ¥ — and a as an argument of f. a unary operation — namely a function from ¥ © for f, we obtain the For example, substituting  for ¥  and the sine function ³C´µ ¨ › › , where induced operation ³C´µ]¨ › © ›

³L´5µ

›

ŽE‘’¬;5œH­‘<‘

¨

­‘’

³L´5µ

Ž/‘‘4œ1¯°#±š

As another example, consider the characteristic function ¶K·;¸

ÅÇÆ;È
ÉÊ4ËÌ <¹G‘º’]»3¼¾½À¿ Á=ÂÄÃ

Ê ½ÎÍ



š

Then for any , ¶K·N¸ is the Boolean (two-valued) image on X with value 1 at ŽE‘ 2 Ž ' ¯ › and value 0 if . An obvious application of this operation location x if Ž‘ of an image. Given Ž/a ‘ÒÑ is the thresholding floating ÂÐÏ Ï point image a and using the characteristic function ¶KÓ Ô5Õ Ö×5Ø(ÙGÚ^ÛÝÜçÇ Þ¾è ßáà8 ãå ã=ï æ é<êâK ë4ì í äÄë ßáî

then the image b in the image algebra expression ðMñ ۓòôó®õ8ö Ô5Õ Ö5×ØòEÚ

© 2001 by CRC Press LLC

16

CHAPTER 1.

IMAGE ALGEBRA

is given by ÷

ø%ùúûü
÷ú5ûý<ýþ.÷.ú[ûýø“ÿ/úûý
?ÿ/úûý ü  ÷ú[ûý^ø ÿ!"

The unary operations on an image discussed thus far have resulted either reduction operation, or another þ&% -valued in a scalar (an element of ) by useÿ9ofø theúÿEglobal ý ('  # $ # . More generally, given a function # , image by use of the composition ÿ provides for a unary operation which changes an -valued image then the' composition #)$ úÿE ý . Taking the same viewpoint, but using a function f between into a -valued image # spatial domains instead, provides a scheme for realizingþ+naturally operations for ÿ/0 " *,%.- induced and , then we spatial manipulation of image data. In particular, if # ÿ 2 by define the induced image $1# ÿ

$#

øùGú43.ü<ÿ/ú

#

ú53ý<ý<ýþ637*&8

Thus, the operation defined by the above equation transforms an -valued image defined over the space X into an -valued image defined over the space Y. Examples of spatial based image transformations ÿ/09 " -;:=<+> are affine and perspective trans, where is a rectangular ?A@CB array. If forms. For instance, suppose þG-H%ID = E(F > and # is defined as

#

úJüLKJý^øNM

ú5JüOK;ý

P Q0J

ü

ú5R TSUJ+üOK;ý.PJWV

J3øY

ÿ

then $X# is a one sided reflection of a across the line . Further examples are provided by several of the algorithms presented in this text. Simple shifts of an image can ÿE be

achieved a spatial transformation :=using <+> either 300 <+> " -;by , , and , we define a shift or point addition. In particular, given of a by y as ÿ1Z73 ÿZE3

øùGú5[Eü
÷.ú\[Jý<ýþ ÷.ú4[ýøFÿú4[1S&3ºý®ü[1SC3&!-]

Note that is an image on for the equivalent formulation ÿ1Z73

-^Z_3

since

[1S737&-a`

[W!-NZE3

, which provides

øùGú5[Eü
÷.ú\[Jý<ýþ ÷.ú4[ýøFÿú4[1S&3ºý®ü[Qb-cZE3Q þ-dZY3^%e-

# Ofú4[course, one could just as well define a spatial transformation by ÿZE3 ø ÿ ýø][1S73 $6# . # in order to obtain the identical shifted image Another simple unary image operation that can be defined in terms of a spatial ÿ!gfihkjlfnm a, denoted mapÿo is image transposition. Given an image ÿoqp“ÿ ü þ<+r < %(< F , then <Xr the transposeúJ+of üKýø%ú5KüOJEý is given by # . by , is defined as $+# where # @ F @ Binary Operations Induced by Unary Operations

þst%u

Various unary operations image operations induced by functions # " can be generalized to binary operations on . As a simple illustration, consider the þ9)v+w1%x9 úyGý øcynz defined by # , where k denotes some nonexponentiation function # negative real number. Then f induces the exponentiation operation ÿ

© 2001 by CRC Press LLC

z c ø {ú5ûü<÷.ú[ûýý:þ=÷ú[ûý^øc| ÿ/ú5ûý~} z üû&b-&ü

1. 4 Images

17

where a is a non-negative real-valued image on X. We may extend this operation to a binary image operation as follows: if €
‚Yƒ^„†…ˆ‡Š‰‹Œ , then

€XŽ’‘“4”kL•–“5”
——˜™•–“5”
—k/€š“†”
— Žœ›žŸ ”&ƒ C¡¢ The notion of exponentiation can be extended to negative valued images as long as we follow the rules of arithmetic and restrict this binary operation to those pairs of real-valued images for which €+“4”
— Žœ›£Ÿ ƒ0…¥¤X”0ƒ0  . This avoids creation of complex, « undefined, and indeterminate pixel values such as “¦¨§©—–ª e¬ « N­¯®X°(±‰ , respectively. ‰ However, there is one exception to these rules of standard arithmetic. The algebra of images provides for the existence of pseudo inverses. For €²ƒU… Œ , the pseudo inverse of a, which for reason of simplicity is denoted by €+³ ¬ is defined as

€ ³ ¬ ^´)“~”k‚)“\”
—O—W˜™‚8“~”
—

§ €+“5”
—™¸ · ±&¹nº»X¼©½¾ ¼‚)“4”
—0±qÀÁ¢ €+“5”
—²µ¶ µž¿

Note that if some pixel values of a are zero, then €¨ÂO€ ³ ¬ Ä · à , where 1 denotes unit image all of whose pixel values are 1. However, the equality €1ÂL€+³ ¬ ÂL€¸€ always holds. Hence the name “pseudo inverse.” The inverse of exponentiation is defined in the usual way by taking logarithms. ÅÇÆ¯È ÅÇÆ¯È Specifically,

€PcÉl“4”kO•–“~”
——˜™•–“5”
—k Žœ›žŸ €+“5”
—”Cƒ UÊl¢ \Å Ž Æ¯È As for real numbers, Ž € is defined only for positive images; i.e., €
–‚Eƒ7“†…Ë̗ Œ .

Another set of examples of binary operations induced by unary operations are the characteristic functions for comparing two images. For €
–‚/ƒU… Œ we define

ͨÎÐÏ “~€q—kYÑ “4”kO•–“4”
—O—)˜1•–“~”
—k’§ €+“5”
—Òӂ)“~”
—©¹ºL»¼©½L¾ ¼Õ•“4”
—kY±–Ö ¶ Ú+Ù5Þ
ÛåÓæ)Ù~Þ
Û©ßçèLéê¯ëìµÔãîíL¿ êÕàÙ4Þ
ÛkÜYï–ð ͨ×ÐØlÙ~ÚqÛkÜYÝ Ù4Þ8ßLà–Ù4Þ
ÛOÛ)á1à–Ù~Þ
Ûkܒâ8ãµäk ñ1ò Ø Ù5ÚqÛÜÄÝÙ~ÞkßàÙ4Þ
ÛÛÁá1à–Ù4Þ
ÛÜcâ)ãäkÚ+Ù4Þ
Ûkܸæ)Ù4Þ
Ûßçèélê©ëìãžíê™à–Ù~Þ
ÛkÜï ð ñ¨óÐØlÙ~ÚqÛkÜYÝ Ù4ÞkßOà–Ù4Þ
ÛOÛ)á1à–Ù~Þ
Ûkܒâ8ãäkÚ+Ù5Þ
ÛôÓæ)Ù~Þ
Û©ßçèLéê©ëLìãÔíLêÕàÙ4Þ
ÛkÜYï–ð ñ¨õ Ø Ù~ÚqÛkÜYÝ Ù4ÞkßOà–Ù4Þ
ÛOÛ)á1à–Ù~Þ
Ûkܒâ8ãäkÚ+Ù5Þ
ÛöÓæ)Ù~Þ
Û©ßçèLéê©ëLìãÔíLêÕàÙ4Þ
ÛkÜYï–ð ñPò ÷ ؖÙ5ÚqÛÜÄÝÙ~ÞkßàÙ4Þ
ÛÛÁá1à–Ù4Þ
ÛÜcâ)ãäkÚ+Ù4Þ
ܸۙ ø æ)Ù4Þ
Ûßçèélê©ëìãžíê™à–Ù~Þ
ÛkÜï ð–ù

Functional Specification of Image Operations The basic concepts of elementary function theory provide the underlying foundation of a functional specification of image processing techniques. This is a direct consequence of viewing images as functions. The most elementary concepts of function theory are the notions of domain, range, restriction, and extension of a function. Image restrictions and extensions are used to restrict images to regions of particular interest and to embed images into larger images, respectively. Employing standard mathematical notation, the restriction of Ú_ú0ûü to a subset Z of X is denoted by ÚÌý þ , and defined by

ÚÌý þ0ÿ Ú
ÙbûÛkÜ]ÝnÙ5ÞkßLÚ+Ù4Þ
ÛLÛPá™Þ&ú8ð–ù þ

Thus, ÚÌý þEúÓû . In practice, the user may specify Z explicitly by providing bounds for the coordinates of the points of Z.

© 2001 by CRC Press LLC

18

CHAPTER 1.

IMAGE ALGEBRA

There is nothing magical about restricting a to a subset Z of its domain X. We can just as well define restrictions of images to subsets of the range values. Specifically, if and , then the restriction of a to S is denoted by and defined as

 

 

   " !    )-, *+&  /. . The In terms of the pixel representation of   we have  #%$ &(') *+& double-bar notation is used to attention on the fact that the restriction is applied to  focus 0 . the second coordinate of

Image restrictions in terms of subsets of the value set is an extremely useful

concept in computer vision as many image processing tasks are restricted to image domains over which the image values satisfy certain properties. Of course, one can always write this type of restriction in terms of a first coordinate restriction by setting so that . However, writing a program statement such is of little value since Z is implicitly specified in terms of S; i.e., Z as .” Thus, Z would have to be must be determined in terms of the property “ precomputed, adding to the computational overhead as well as increased code. In contrast, direct restriction of the second coordinate values to an explicitly specified set S avoids these problems and provides for easier implementation. As mentioned, restrictions to the range set provide a useful tool for expressing and S is the interval , various algorithmic procedures. For instance, if denotes the image a restricted where k denotes some given threshold value, then to all those points of X where a(x) exceeds the value k. In order to reduce notation, we . Similarly, define

$ & - , , " 2&  /. 6 # 4 5

 3# 4 5

1 #

*+&  

- 87  <;>=@? ACB

29"'): / 7

 ED =  F  ;G=H? ACB @I"=  @J =H? ACB 'K EL"=   ;MNA
? =OB 'P @=  @Q =R '8S@TVU HWN=   ;M"A
? =YX ! As in the case of characteristic functions, a more general restriction  form of range  a  c

b & d . is given when S corresponds to a set-valued image [Z]\_^H` ; i.e., 2&

In this case we define

@ #[$ +&('Y "+& Y-, "+&  we define For example, for 'f6g [7 , "+& kj 6/G& O. '-  LVh F Wih  $ +&('Y "+& YY F Iih  $ +&('Y "+& -, "+& kn 6/G& O. '-  D h F h  $ &(') *+& P, *+&  # 6/G& . '- F_q p h 

  2 & .e! , "2&   $  &(') "& )) $ &(') "& P-, , "2s&  r $ &(') *+& *+& #

l 6/+& m. ' o 6/+& m. ' 6/G& .t!

-  1  

Combining the concepts of first and second coordinate (domain and range) , , and restrictions provides the general definition of an image restriction. If , then the restriction of a to Z and S is defined as

uv

 4 ; 5 ?  B # Y1w ("! $ &(') "& )-, & x1ySHTiUd *]&   . , 4 ;  ?  B #   , and It follows that 4 ; 5 ?  B # 4 ; 5 ? ^ B # 4 5 .

 to 6 {z on Y, where X and Y are subsets of the The extension of d h same topological space, is denoted by E4 and defined by ~}€ - E4 h +&  #v| "6/G& & }€ & & -‚8ƒE ! In actual practice, the user will have to specify the function b.

© 2001 by CRC Press LLC

1. 4 Images

19

Two of the most important concepts associated with a function are its domain and range. In the field of image understanding, it is convenient to view these concepts as functions that map images to sets associated with certain image properties. Specifically, we view the concept of range as a function

E„ …t†V‡‰ˆ‹Š_ŒŽH‘ defined by „E…t†V‡‰ˆH’+“"”–•˜—E„s™8ŒšŠC„–•›“"’2œ”(VžE„ Ÿ<žE¡Kˆ œ™-¢8£ .

Similarly, the concept of domain is viewed as the function

¤ žE¡K…t¥+†Š_Œ §¦ ¨G©2ª/«i©­¬¯® Ž° s±

where

Œ{ ¦ ¨²© ª «i© ¬ ® •y³H´šŠs´x•“ ¦ ¨>µV¶ ·t® ± “-™¸Œ ±/¹ ™xH ±(º ™8H‘N»

and domain is defined by

¤ žE¡P…t¥†F’+´(”•y³Hœ™¼¢½Š¾“ ¦ ¨GµV¶ ·t® ’+œ” •´/’Gœ”(•u„–VžE„ Ÿ<žE¡Kˆ-„s™xŒ »À¿

These mappings can be used to extract point sets and value sets from regions of images of particular interest. For example, the statement

ŸCŠÁ• ¤ žE¡P…t¥†F’2“ÂEÃ"ÄE”

—œ™-¢½Š–“"’2œ”/ÆÇV£

yields the set of all points (pixel locations) where a(x) exceeds k, namely . The statement

ŸÅ•

Ÿ‹Š •›„E…‰†V‡tˆE’2“ ÃNÄ ” È

on the other hand, results in a subset of instead of X. Closely related to spatial transformations and functional composition is the notion of image concatenation. Concatenation serves as a tool for simplifying algorithm code, adding translucency to code, and to provide a link to the usual block notion used in matrix and , then the row-order concatenation of a with algebra. Given and is defined as b is denoted by

“-™xŒFÉEÊ « ÉmË ’“ ¦ ´(”

Note that

´d™¸Œ(ÉÌÊ « É@Í

’+“ ¦ ´”-™[Œ{ÉEÊ « ÉEÍYÒ‰Ó .

’+“ ¦ ´(”Γ ¦ ÏtÐ G¨ ÑO¶ Ä ® ¿

Assuming the correct dimensionality in the first coordinate, concatenation of any so number of images is defined inductively using the formula that in general we have

ÔÕcÖ ×dÖØ_٠ژÔYÔ+՚Ö/×Ù@ÖØ_Ù

ÔÕ"Û_Ö ÕÀÜtÖVÝ<Ý@ÝÖ ÕÀÞGÙ(Ú%Ô)Ô+Õ*ÛHÖ ÕiÜ_ÖVÝ@Ý@ÝÖ ÕVހßÛÙ<Ö ÕiÞ²Ù"à Column-order concatenation can be defined in a similar manner or by simple transposition; i.e.,

áââ

åæ ââ "Õ ä Û ææ ââ ææ ââã Õ ä Ü ææ æ Ú[ÔÕ"ÛHÖ ÕiÜtÖVÝ@Ý<Ý"Ö ÕiÞGÙ2èéà .. . ä ç ÕVÞ

© 2001 by CRC Press LLC

20

CHAPTER 1.

IMAGE ALGEBRA

Multi-Valued Image Operations Although general image operations described in the previous sections apply to both single and multi-valued images as long as there is no specific value type associated with the generic value set , there exist a large number of multi-valued image operations that are quite distinct from single-valued image operations. As the general theory of multivalued image operations is beyond the scope of this treatise, we shall restrict our attention to some specific operations on vector-valued images while referring the reader interested in more intricate details to Ritter [1]. However, it is important to realize that vector-valued images are a special case of multi-valued images. and , then a(x) is a vector of form If where for each , . Thus, an image is of form and with each vector value a(x) there are associated n real values . . Real-valued image operations generalize to the usual vector operations on , then In particular, if

ê

ê ëuìí x îPïxêFð ÷xëùø_õ"ö@ö<öVõú îiû­ñ+òóïì îýë[ñî ô õ"ö<ö@ö"õtî í ó

îõ2þvïÿñ]ìíió ð î
þxë%ñ+î ô þ ô õ"ö<ö@öiõYî í dþ í ó îOþxë%ñ+î"ôOþ/ô<õö@ö<öVõYî í Oþ í ó î
Kþxë%ñ+î*ôKþeô@õö@ö<öVõ)î í  þ í ó î
Kþxë%ñ+î ô Kþ ô õö@ö<öVõ)î í  þ í ó  If 8ë3ñ  ô õ"ö<ö@öVõ  ó
ï ìí , then we also have í

 îPëuñHô î"ô@õö@ö<öVõ  í î í ó"õ

îPë%ñ ô î ô õ"ö<ö@ö"õ  í î í ó"õ etc. In the special case where

î*ñ+òóë%ñ+î"ô@ñ+òó@õ"ö@ö<öVõÀî í 2ñ òóYó îüïñ]ì{íió ð î û ñ+òó ñ]ìíÀó ð

ë%ñEõEõVö<ö@öVõ @ó , we simply use the scalar sï8ì

and define , and so on. As before, binary operations on multi-valued images are induced by the correon the value set . It turns out to be useful sponding binary operation   "! to generalize this concept by replacing the binary operation  by a sequence of binary ! , where ' , and defining operations $#% &

î ›îõEî Eî

_ìí §ìí 0ìí ìí iì{í ìí ì Pë ø_õ"ö@ö<öVõú î(Àþ)[ñî(Àôþ õYî(+*Oþkõ"ö@ö@öiõYî( í þó, For example, if -#.ì{í.xì{í/!½ì is defined by ñ10 ô õ"ö@ö<ö"õ0 í ó2 # ñ 3 ô õ"ö@ö@öiõ3 í óë54%67098$0Àû:%3 # ýø<;÷=;>'(?{õ then for îFõ)þdïdñ2ìíÀó ð and @Cë›î(Àþ , the components of @tñ2òó(ëÿñ @ ô ñ2òóOõ"ö<ö@ö"õ @ ñ+òó)ó have í values @A#éñ+òó(ë›î"ñ2òó -#Eþ/ñGòó(ë54B670C8@î û ñòóDKî+#éñ2òó"ýø;÷=;E'F? for 'wë øéõ"ö<ö@ö"õ¯ú . * * ! ì As another example, suppose  ô and  * are two binary operations ì ¼ì  defined by

ñ0 ô õ0 * ó  ô ñ3 ô õ3 * ó(ëG0 ô 3 ôIH 0 * 3 * and

© 2001 by CRC Press LLC

ñ 0 ô õ 0 * ó  * ñ 3 ô õ3 * óëJ0 ô 3 * E0 * 3 ô õ

1. 4 Images

21

respectively. Now if KDL2M)NO P=Q R-S represent two complex-valued images, then the product TBU KWV(M represents pointwise complex multiplication, namely

T7XZYD[IU\X KC] X^YD[ M_] XZYD[=` K ZX YD[ M ^X YD[ WL KC] X^YD[ M XZYD[Da K ZX YD[ M ] X^YD[[b Q Q Q Q

Basic operations on single and multi-valued images can be combined to form image processing operations of arbitrary complexity. Two such operations that have proven to be extremely useful in processing real vector-valued images are the winner take all jthcoordinate maximum and minimum of two images. Specifically, if KDLM>N X P=c [ S , then the jth-coordinate maximum of a and b is defined as

Kd)e fgM U\h$XZY L TiX^YD[[BjT7X^YD[kU K XZYD[=lnm K+f X^YD[o MCf ^X YD[ =L p+qrWsut2v xl w s T7XZYD[IU M XyYD[Az L while the jth-coordinate minimum is defined as

K{)e f-M U\h$XZY L TiX^YD[[BjT7X^YD[kU K XZYD[=lnm K+f X^YD[| MCf ^X YD[ =L p+qrWs tv }l w s T7XZYD[IU M XyYD[AzFb Unary operations on vector-valued images are defined in a similar componentwise fashion. Given a function ~ j PE€P , then f induces a function P=c%P=c , again denoted by f, which is defined by

~ X ‚ ] L ‚ Q CL ƒuƒ-ƒCL ‚ c [I„…X ~ X ‚ ] [ L(~ X ‚ Q [ LDƒ-ƒ-ƒ:L(~ X ‚ c [2[ b These functions provide for one type of unary operations on vector-valued images. In particular, if K U†X K ] L7K LCƒ-ƒuƒCL7K [ N X P=c [ S , then

Q

c U X ~ X K ] [ (L ~ X K [ L,ƒuƒ-ƒ:L(~ X K [2[ b ~ X K [I„ ~‡_K … Q c Thus, if ~ U‰ˆ-Š ‹\j PŒP , then ˆ-Š^‹,X K [IUŽX1ˆ-ŠZ‹,X KC] [ L,ƒuƒ-ƒ:L ˆ-ŠZ‹,X K [[Cb c Similarly, if ~ U‘F’ , then

 F‘ ’ X K [IU O  ‘+’ X K ] [ L,ƒuƒ-ƒ9L  ‘+’ X K [ R b c Any function ~ j P=c“P”c gives rise to a sequence of functions ~ f UG• f ‡~ j — LCƒuƒ-ƒCL ‹ . Conversely, given a sequence of functions ~f j P=c%€P , P=c%€P ,U˜where – U — ‹ where – LCƒ-ƒuƒCL , then we can define a function ~ j P”c™€P”c by ~ XYD[k„JX ~ ] X YD[ L(~ Q X^YD[ LCƒuƒ-ƒ9L:~ c XZYD[[ L where YU\X ‚ ] L,ƒuƒ-ƒ9L ‚ [ NP=c . Such functions provide for a more complex type of unary c image operations since by definition

~ X K [IU\X ~ ] X K [ LCƒuƒ-ƒ9L ~-š X K [[IU…h$X^Y L2M X^YD[[%j M X^YD[IU…X ~ ] X K X^YD[[ LCƒ-ƒuƒ9L~-š X K X^YD[2[[z L which means that the construction of each new coordinate depends on all the original coordinates. To provide a specific example, define ~ ] j PIQ“P by ~ ] X‚ L› [
Q

© 2001 by CRC Press LLC

Q

22

¡E¢_£¤=¥A¦u§©¨ª£¤I¥A¦-§

given by

¡)«˜¬¡®­$¯ ¡ ¥A° .

in the image

CHAPTER 1.

IMAGE ALGEBRA

Applying f to an image

±>² £ ¤I¥A¦u§

results

¡¬ ± ° \ « ³®¬^´I¯2µ¬y´ °° ¢¶µ¬Z´ ° \ « ¬ ·$¸^¹,¬ ± u­ ¬Z´ ° °Cº¼» ½ ·$¾¬ ± ¥ ¬^´ °2° ¯ » ½ ·$¬ ± ­A¬^´ °° º ·u¸Z¹C¾C¬ ± ¥ ¬^´ °°° =¯ ´ ²%¿\À7Á ¤I¥ and a denotes a complex-valued Thus, if we ¡ represent complex numbers as points in ¬ image, then ± ° is a pointwise application of the complex sine function. Global reduce operations are also applied componentwise. For example, if ±G² ¬¤” ° § , and à « »AÄFÅ$Æ ¬ ¿ ° , then Ç ± «…¬ Ç ± ­$¯CÈuÈ-È9¯ Ç ± Â9°

«ÊÉËÍÎ Ì ± ­ ¬^´ Ï ° ¯CÈ-È-È:¯Î Ì ±  ¬^´ Ï °1ÑÒ ² ¤  Á Ï-Ð ­ Ï-Ð ­ Ó « Ó Â ­uÕ Ô ¬ ± ° ² ¤ § since each ± Ô ² ¤ § In contrast, the summation ÔyÐ ­ ± Ô ÔyÐ ¬¤” ° § ¨Ö¤ § . Ô projection function Õ is a unary operation × «…¬ × ­-¯DÈ-ÈuÈ9¯ × ¯ Similarly, ± ± ± Â(° Ø ± «…¬ Ø ± ­ ¯DÈ-ÈuÈ9¯ Ø ±  ° ¯ and Ù «\¬ Ù ­$¯DÈ-È-È:¯ Ù ± ± ± Â(° Á

. Note that the

Summary of Image Operations The lists below summarize some of the more significant image operations.

Binary image operations. It is assumed that only appropriately valued images are employed for the operations listed below. Thus, the operations of maximum and minimum apply to real- or integer-valued images but not complex-valued images. Similarly, union and intersection apply only to set-valued images. generic ±(Ú µ¼µ/«Û«…³$¬Z³+´I¬Z´I¯ Üi¯ ¬^Üi´ ¬^°´ ° ¢¢ÝÜ7¬^Ü7´ ¬ °´ « « ± ¬Z´ ¬ °´ Ú µ_¬^´ µ_° ¬Z¯=´ ´ ¯=²%´ ¿¶À addition ± º µ/«…³+¬Z´I¯ Üi¬^´ °° ¢ÝÜ7¬ ´ ° « ± ¬Z´ °Cºµ_¬^´ ¯=° ´ ²&¿ÞÀ ±"× ß µ.«Û³$¬Z´I¯ Üi¬^´ °° ¢Üi¬^´ ° « ± ¬Z´ ° ß × µ_¬y° ´ ¯=´ ²%¿À multiplication ± Ø µ.«Û³$¬Z´I¯ Üi¬^´ °° ¢Üi¬^´ ° « ± ¬Z´ ° Ø µ_¬y´ ° ¯=´ ²%¿À maximum °° ° ± ° ° ²%¿À ± minimum à º ± «Û«Û³+³$¬Z´I¬ ´I¯ Üi¯ Üi¬^´ ¬^´ °2° ¢ ¢ ÜiÜi¬´ ¬^´ ° « « à º ¬Z´± ¬^´ ¯=° ´ ¯=´ ²%¿À scalar addition ÃàßA± «\³®¬^â9¯µ_¬yâ °2° ¢Ýµ_¬yâ ° « Ãß ¬^± âã ° ¯=²% ¿À scalar multiplication á «Û³$¬Z´I¯ Üi¬^´ °° ¢Üi¬^´ ° « ± ¬Z´ á=µ_° ¬y´ â ¯=²%´ ¿ º>á À ± º>µ. point addition union ±ä µ.«Û³$¬Z´I¯ Üi¬^´ °° ¢Üi¬^´ ° « ± ¬Z´ ° ä µ_¬y´ ° ¯=´ ²%¿À intersection ±å «Œç=¬^´I¯Ü7¬Z´ °¢Ý° Ü7¬ ´ « ° ¬ ´ ± ° å ¯=´ ° ²%¿À °° ° ± ° æFè}é+ê ²¿.ë ±(æ exponentiation logarithm

© 2001 by CRC Press LLC

ì ½gí æ ± «îi¬^´I¯Ü7¬Z´ °° ¢Üi¬^´ ° « ì ½-í Fæ èïé+ê ± ¬ ´ ° ¯=´ ²%¿/ð

1. 5 Templates concatenation concatenation

ñ^ òIó ôI õIöJò=ó ÷7øCùyúAû ü ýþIòÞÿ
ò öñ^òó ôIõ  ô

  þIôEÿ 

23

 ;< ñ^ò9õIö®ñIþ7ñ Dõõ"!iñDõIö$#&%('òCñ Dõ&)ôñ Dõ þ+*-,/.10324"%65/07WñDõIö98-: BA  ñ^ò9õIö® ñI þ7 ñ D õõ"!i ñD õIö$#&%=I' òCñ D õ& ö98-: I  ñ^ò:õIöC+ ñ I þDi ñD õ2õ"!Bi ñD õIöE#&%6I' òCñD õkö> ô_ôñFñ DD õ-õ þ+þ+*<*-,?,/.1.103032/2?4"4"%@%@5?5/0707(W ññDD õIõIöG 8H: ñ^ò9õIö® ñI þ7 ñ D õõ"!i ñD õIö$#&%=I' òKJ LNM&OQPRJ LNM3S+T-U/V1W3XY"Z6[/W7\JLNM]9^-_ `ab JcdM]e-JLS\J LNM/M"f\HJLNM]$g&Z(hRcKJ LNM&iQPRJ LNM3S+T-U/V1W3X?Y"Z@[/W7\JLNM]9^-_ `kl j b JcmM]Ce<J LSD\HJLNM?M"fB\HJLNM]Eg&Z6hcKJLNMo]G n P&JFLNMpS+T
Unary image operations. As in the case of binary operations, we again assume that only appropriately valued images are employed for the operations listed below. value transform spatial transform domain restriction range restriction extension domain range generic reduction image sum image product image maximum image minimum image complement pseudo inverse image transpose

1.5.

ƒˆ ‰Šc‹]ŒˆJcmM&]e-JLS\JLNM?MŽf\J LNM]ŒˆJcKJLNM?M3S+LwvŽx_ ck‰ˆ‘]’e<J“&S/cJ‡ˆJ‡“+M/M?M‹f”“wv•w_ c—– ˜s]™e<J LS/cKJ LNMM‹f”Lšv‘›R_ cNœ $]™eJLS/cJLNM/Mžf”cKJ LNM&v Ÿ¡_ Z6h¦L
vŽx c– {‹]£¢oJLS\JLNM?Mf†\JLNM&]£¤ cKP&J JFLNLNM¥ M§Z6h¦L
vŽ•¨x© ª}«‹¬­®¯JcmM&]$e LwvŽx°fB±"²v³Ž´<µ ¶ µ·¯¸¹Nº»9¼½ ¼¾¿dÀÂÁø‡·dºR»Åļ”ƏǣÈÉ&¹wƎÊE´<µ ¶ µN¼»G·¸¹Nº ½ ˔·
»$·K¸¹+Ì3ºÍ1·K¸¹NÎpºÍÐÏpÏpÏ<Í1·¸‡¹KÑdº ÒÓ·Ž»°ÔÒ ÕÖ ·¸¹Nº"»×·K¸¹ Ì º¯Ø‘·K¸¹ Î ºKØÙÏpÏ ÏKØÚ·K¸¹ Ñ º Û·»ÜÔÛ ÕÖ ·K¸¹NºR»C·K¸¹+Ì3ºÏ ·K¸¹NÎ ºÝÏÏpÏ Ï+Ïp·K¸¹KÑdº ÞÚ·y» ÔÞ ÕÖ ·K¸¹NºR»9·¸¹ Ì ºNߞ·K¸¹ Î º¯ßEÏpÏpÏàߞ·K¸¹ Ñ º áÚ·y» Ôá ÕÖ ·K¸¹NºR»9·¸¹+Ì3ºN➷K¸¹NÎ º¯âEÏpÏpÏà➷K¸¹KÑmº ·ã »å䯸 ¹Ræ/çH¸¹Nºº‹ÈŽç¸¹Nº»éKè· ¸ ¹Nº3æ+¹wÆyÊwê Ì ·Kë Ì »r슸 ¹æí&¸¹Nº/ºŽÈí&¸¹Nº»sì õ î ïðÔ<ñ°ò6ö-ó¦÷/ø1·Kùp¸ú/¹Nû"º”ò@ü/»Gô ù õ ý ·1þK»Ä-¸/¸‡ÿ¯
æ º3æ?·1þ ¸‡ÿ¯æ <º/ºŽÈ&·1þ ¸ÿNæ <º+»×·K
¸ 1æÿmº æN
¸ æDÿdºÝƏʏ½

Templates

Templates are images whose values are images. The notion of a template, as used in image algebra, unifies and generalizes the usual concepts of templates, masks, windows, and neighborhood functions into one general mathematical entity. In addition, templates generalize the notion of structuring elements as used in mathematical morphology [27, 60].

© 2001 by CRC Press LLC

24

CHAPTER 1.

IMAGE ALGEBRA

Definition. A template is an image whose pixel values are images (functions). In particular, an  -valued template from Y to X is a function      . Thus, and t is an  -valued image on Y. 
    . The image For notational convenience we define has representation !#"$
%&'()%* %+ ,.-0/

The pixel values at point y.

  1%2

of this image are called the weights of the template

If t is a real- or complex-valued template from Y to X, then the support of is denoted by 687:9<;>= and is defined as ?A@

@

9 ; =CBED'F!GIHKJL9 ;

354

F=LO MB NQPSR

More generally, if 9LG+T1UVXW5Y and U ?Zis@ an algebraic structure with a zero element 0, then @ MB N^P . the support of 9 ; will be defined as 9 ; =B[D5F\G\H JL9 ; F=L] For extended real-valued templates we also define the following supports at infinity: ?`_ @

and

?edf_ @

@

9 ; =CBED5F*G\HaJL9 ;

9 ; =gBED5F!GIH JL9 ;

F=L] MB bcP

@

F=Li MB h0bjPSR @ that HmlkL= is

If X is a space with an operation k such a group, then a template is said to be translation invariant (with respect to the operation k ) if and only @ @ if for each triple FCl)pSlQqrGmH we have that 9 ; F=BE9 ;tsvu Fkwqx= . Templates that are not translation invariant are called translation variant or, simply, variant templates. A large class of translation invariant templates with finite support have the nice property that they can be defined pictorially. For example, let HyB]z`{ and y = (x,y) be an arbitrary point of @n} @n} @†} X. Set Fg|SB l~Lh€<='lgF B k]^l~^='lC‚ƒv„F…0B k+tl~Xh+5= . Define 9‡G T1ˆ V W V by @ { @ @ @ @ defining the weights 9 ; p‰=B[tlA9 ; FgŠ‹=BŒ$l9 ; F =BŽl9 ; F`…5=0BEl‚ƒ„O9 ; F=0BN { whenever?Zx@ is not an element of D5pAl
Fg|l)F l
F`…^P . Note that it follows from the definition { of t that 9<;>=ZBD'pSl)F | l
F l
F … P . Thus, at any arbitrary point y, the configuration of the { support and weights of 9 ; is as shown in Figure 1.5.1. The shaded cell in the pictorial representation of 9 ; indicates the location of the point y. 9LGcTnUoVW V

‘

’

’

x

x +1

’

y -1

‘

y

3

1

4

2

‘

y

x

Figure 1.5.1. Pictorial representation of a translation invariant template.

© 2001 by CRC Press LLC

1. 5 Templates

25

There are certain collections of templates that can be defined explicitly in terms of parameters. These parameterized templates are of great practical importance.

Definition. A parameterized “ -valued template from Y to X with parameters in P is a function of form ”I•`–˜—š™)“›‡œž The set P is called the set of parameters and each Ÿ+ +– is called a parameter of t.

regular

Thus, a parameterized ¡ -valued template from Y to X gives rise to a family of -valued templates from Y to X, namely ¢`£¤¦¥$§A¨c©1ªg«¬5­®¯r°\±³² . ¡

Image-Template Products The definition of an image-template product provides the rules for combining images with templates and templates with templates. The definition of this product includes the usual correlation and convolution products used in digital image processing. Suppose ª is a value set with two binary operations ´ and µ , where ´ distributes over µ , and µ is associative and commutative. If ¶°©1ªg«A¬5­ , then for each ·c°!¸ , ¶'¹*°wªo« . Thus, if º º º °+ªg« , where X is finite, then ´»¶ ¹ °€ªg« and ¼¾½ ´¿¶ ¹$À °cª . It follows that the binary operations ´ and µ induce a binary operation µ ´Á ®^ª «Ã Æ!Ç

© ª « ¬ ­ÃÄ

where

º

ª ­Å

µ ¶L°Èª ­ ´.

is defined by Æ

Ç

¼ ½ º

½É· À

Ç ¼ Í ½ ÊÌË'

´¿¶<¹ À

º

½1Î À ´¿¶<¹½ÉÎ ÀÀ`Ï

Ç»Ñ

Therefore, if

Ð

Æ

ÎZÒ Å ÎÓ ÅÔ<Ô5ÔvÅ ÎÕvÖ Ç

½É· À

½

º

½1Î Ò À ´˜¶5¹v½ÉÎ Ò À)À µ`½

, then º

½ÉÎ Ó À ´˜¶5¹v½:Î Ó À
À µ×<×5×^µ`½

º

½1Î Õ À ´¿¶<¹v½:Î Õ ÀÀ`Ï

µ ¶ is called the right convolution product of a with t. Note that The expression º ´. µ ¶ is an image on Y. Thus, templates allow for the while a is an image on X, the product º ´. transformation of an image from one type of domain Æc to Ç an entirely different domain type. º µ ´.¶ into Replacing ½
ª Å µ Å ´ À by ½nØ Å$Ù³Å × À changes ÆmÇ º

´ Ù ¶ Å

the linear image-template product or simply the convolution of Æ

Ç+Ú ½É· À

º

°Øg« Ç ÝÌÞ¹ ½:Î À

, and

¶°

© Øg« ¬ ­

© 2001 by CRC Press LLC

½

º

with ¶ , where

½1Î À ×'¶<¹v½:Î ÀÀ Å

«

.

Every template Ý]° . Obviously, ½1Ý'Þ

Ý Û ½:· À

Û^Ü

º

© ª Ç­ ¬ « Ý À Þ

has a transpose ÝÌÞ\° © ªg« ¬ ­ which is defined and ÝÌÞ reverses the mapping order from Ð Ä ª ­

26 to

CHAPTER 1. ßáà âgã

äîé€ë1âgï0í ã

IMAGE ALGEBRA

. By definition, ä'åæç»è.éâgã and êrënäÌåæ‡çìèí é€â , whenever èé€âoã and . Hence the binary operations ç and ð induce another product operation ð çÁ ñ ë â ï í ã]ò

àóâ ïô

â ã

õ!ö

where

ð èréÈâ ï ä çw

is defined by õA÷Éø‰ùgö

ö

÷

ê ë ä åæ‡çìè í

ê ü úÌû'

÷:ýù ä åæ

÷:ýùù`þ ç¿è

ð è is called the left convolution product of a with s. The expression ä ç. ð è , it is not necessary to use the transpose When computing ä ç ÷ ê ü ú'û'

÷:ýù ä åæ

÷:ýùùgö

÷

÷:ø‰ù

ê ü äÌÿ ú'û' õ+ö

ç˜è

÷ ê ü ú<û'

since

ð è as ä ç. ÷:ýù
ùLþ

÷Éø‰ù ä ÿ

ä å

çìè

This allows us to redefine the transformation õA÷¦ø‰ùCö

÷1ýù
ùLþ

ç¿è ÷

ù




For the remainder of this section, we assume that â ô ð is a monoid and let 0 denote the zero of â under the operation ð . Suppose è+éâ ã and !éyë1â í ï , where X and Z are subsets of the same space. Since â is a monoid, the operator ç ð can be extended to a mapping ò

ð çÁ ñtâ ã õö



è çð

where

is defined by

  

õA÷:øgù‰ö

÷

ê ü ú<û'

÷:ýù è




ë â

í ï

àáâ ïô



 

 

÷:ýùù

æ

ç

ö

mö

+þ

ð è is defined in a similar fashion. Subsequent examples will The left convolution product ä ç demonstrate that the ability of replacing X with Z greatly simplifies the issue of template implementation and the use of templates in algorithm development. Significant reduction in theù number of computations involving the image-template ÷ product can be achieved if â ô ð ô ç ø is a commutative semiring. Recall that if é ë â í ï , éß with ÷ ù ð is defined as ÷ ù 'ý ÷:ýù a ö point then theö support of t at ÷:ýù ö respect to the ý operation æ , we have that æ é ñ ý . Since ÷ ù æ whenever ÷Éýæ ù ÷Éýù é ö æ æ è ç whenever é and, therefore,



 !



   " 



÷

 



÷1ýù




÷:ýùùgö

÷

÷1ýù

÷Éýù
ù`þ

 $#%'&'($) ,+.-0value /$1 2"b(y)   does not depend on the size of It follows that the computation of the new pixel  . Therefore, if * , then the computation of X, but on the size of b(y) requires a total of 34*5,6 operations of type and . As pointed out earlier, substitution of different value sets and specific binary op><@ image transforms. 7G@FH Our prime exerations for and 87 results in a wide variety of=7?different 7J><@ :9 <; and the value sets BA DC B9 E9 and BA EC amples are the Iring EA EC B9 E9 provides for two lattice products: The structure KLA MJN ê ü ú<û'

è

÷

æ

ç

ê ú'û'ü

ù

ö

è

÷

÷

æ

ç ÷

÷

ù

ç

÷

ô ³ô

ô ô ô ³ô

å

ù

ô ô ô ³ô

ù

õmö

è

© 2001 by CRC Press LLC

ù)ù

æ

ð

ð

æ

ç

å

ë

ô ô ô ò ô ò å í

þ

1. 5 Templates

27

OQP2RTS?U

V PcghSjikmljP2ghS2n:o WX4YQZ[]\0^`_'acb$de f and OpU r kGo fKq where OGPsRTSJU t P8ghSjiwv]k l PxghS=n0y WX$YQZ[$^u_'acb$d e f r In order to distinguish between these two types of lattice transforms, we call the operator z r Pxk l SJU, min convolution the morphological max convolution operator and the morphological l SJvalue U, , thenPxk the operator. It follows from our earlier discussion thatq if {"|~}?j€ OGPxisR?S~‚uU ƒ , the zero of „†…h€ under the operation of z . Similarly, if {‡|ˆ}j€ , of b(y) ƒ . then where

The left morphological max and min operations are defined by

k r UŠ‹ Œ‰ PxRGoIOGPsR?SIS~wOQP2RTSJU V k PxRTSji P2ghS2n:o?R‘“’ ” W z f • WXŽYGZ0[]\4^_'acb$de f and

kr – U ‹ ‰Œ P2RQoDOGPsRTSIS~OGPsR?SJU t k P2RTSji v PcghS8n0o?R—~‘“’ ” o W • WX$YQZ[ ^ _'a b d e q f f

respectively. The relationship between the morphological max and min is given in terms of lattice duality by

r kuU˜Pxk$™ r z ™S ™ o ˆf q ™ PxghSšU f PxghS8n ™ ™ where image is definedk ™ by k › the Ÿf› Y e f , andk ™ Pxthe R?SKU conjugate k l P2ghS=n ™ (or dual) of k ™ „ PxY…hRT€S¡œU ž isk vlthef P2ghS template . It follows „ ž…<€ œ defined by W e ‚ . › that W o o o¥`¦4§?¨'© provides for two lattice products. SpecifThe value set „G€ ¢¤£ z q ª«also ¬~­ ically, we have ®°¯ ¦ ªJ±c²h³J« ´ ¬j±2¿h³ ±x¿h³= where § ¯ÁÀ ¦ µ¶$·Qª¸:¹«º'»2¼Ž¬~½à ¾ and ®¯ ¦ ªG±s²T³J« Ä ¬Æ±8¿h³ ±x¿h³= where § ¨ ¯ÁÀ ­ Ç ªG±x²?³«ÍÌ µ¶$·Q¸¹4Å`º'» ¼ ½¾ ªG±2²T³`«Ò ± ³`«"Ñ ± ³`«Ñ Here 0 ¯Àis the zero of ÈGË ÉjÊ under the operation of , so¯À that whenever ÎÏÐ ÎÓÏÔÐ Ë . Similarly, whenever . © 2001 by CRC Press LLC

28

ÖÕ

Ö×

CHAPTER 1.

IMAGE ALGEBRA

and are called the multiplicative maximum and The lattice products multiplicative minimum, respectively. The left multiplicative max and left multiplicative min are defined as

Ø Ö—Õ ÙÛÚ–Þ ÜÝ

ö

ðØ ç æ Ù Ý÷ ÝßàcáJâDãGàxáhäEäÛåãGàsá?ä çè$éQêë]ì ï àxáTäGñ à2òhä=ó0â?áôõ ø Ý àxíIî ä and ö Þ ðØ ç Ý÷ Ú ù Ù Ø Ö—× ÙÛÚ ÜÝ Ýß à2áQâDãGàsáTäIä~åãGàsá?ä çè$éQêë$ì ï àxáTäGñ†ú à2òhä2ó:â?ápô~õ ø Ý â respectively. The duality relation between the multiplicative max and min is given by Ù Ö × Ø Ú Ø$û Ö—Õ àxÙ íIûî ä û Ù û Ú Ù û and Ø çû Ú ð Øü à û . Hereä ý û â denotes the conjugate of r in þGÿ
. where àxòhä à à2òhäEä àxá?ä à2òhä=ó Summary of Image-Template Products Ù  é and Ø In the following list of pertinent image-template products  ? é  . Again, for each operation we assume the appropriate value set  ô . ô right generic convolution product  Ù Ö Ø

Ú  Ú   Ù Ö Øü  àxáGâIãGàsáTäDä~product åãGàsáTä à àxòhä à2òhäIäjâ?á°ô õ right linear convolution Ú

   "!$#&%'!)(*(,+-%.!/($10   '7)(98:;'7)(<(;#/!>=@?BA 24365 right morphological max convolution product

 D C E  F!G#&%G.!/(<(+ %'!H(/JI  '7)(  :LF7)("M4#/!N=O? A 24365@K right morphological min convolution product

 D P   '!#<%.!H(&(+%F!H(/RQ  F7)(  T  : '7)(UMV#/!S=@? A 24365SK right multiplicative max convolution product

 C   F!G#&%.!H(<(+ %'!/($  W

I  '7)(YXO : '7)(ZMV#/!N=? A 42 365 K

right multiplicative min convolution product

P   '!#<%.!/(<(+-%.!H($  W

Q  "7)(X T  : "7)(FMV#/!N=? A 42 365@K

right xor max convolution product

BD [C   F!G#&%.!H(<(+-%.!/($

I  F7)( [  : '7)(&^H#/!S=? A 42 365]\

right xor min convolution product

 D [P _ "!G#&%.!H(<(`+-%.!H($

© 2001 by CRC Press LLC

Q  "7)(  [ T :L'7)( ^ #/!S=O?BA 42 365 \

1. 5 Templates

29

In the next set of operations, ab_cZd/e fg . left generic convolution product

iNj,k lnm'op
ij,k€ m"op
m

m'o/r$ƒ*jzm"|)r
ay

bO}B„

g

left morphological max convolution product

‡ j k € mFoGp&qGm.o/r
mFoHr

ay



jzm"|)rFŠ4p/o

bO} „

g@‰

left morphological min convolution product

‡ j,kŒ€m'op
ay

m"oHr

 

j;m'|)rUŠVp/o

b}B„

g ‰

left multiplicative max convolution product

ij,k € mFoGp&qm.oHr
ay

m'o/r,jzm"|)rZŠ‘p/o

b} „

g>‰

left multiplicative min convolution product

iNj,k_€ m'op
ay

m'oHr z j m"|)rFŠVp/o  b}B„

g ‰

left xor max convolution product

‡ j,k€ mFoGp&qm.oHr
ˆ 4y ‚

ay

mFoHr ; j m'|)r&”Hp/o b}•„ ’

g]“

left xor min convolution product

‡ j,k_€ m"oGp&qm–o/r
© 2001 by CRC Press LLC

Ž y4‚

ay g “

m'o/r

’ 

jzm'|)r ” p/o

bO}•„

30

CHAPTER 1.

IMAGE ALGEBRA

Binary and Unary Template Operations Since templates are images, all unary and binary image operations discussed earlier apply to templates as well. Any binary operation — on ˜ induces a binary operation (again denoted by — ) on ™"˜/š†›œ as follows: for each pair 6ž ŸN¡¢™"˜/š†›œ the induced operation £—¤Ÿ is defined in terms of the induced binary image operation on ˜/š , namely ¥ § § ª¬« ¡®­ . Thus, if ˜°¯©± , ²ž4ŸŒ¡ ™ ±/š › œ , and —³¯©´ , then ¥ —¤Ÿ6¦"§{¨© § —¤Ÿ § $´NŸ6¦ § ¯1 ´SŸ , where  § ´SŸ § denotes the pointwise sum of the two images  § ¡@±/š and Ÿ § ¡µ±/š . The unary template operations of prime importance are the global reduce operations. Suppose Y is a finite point set, say ­¶¯ · «/¸ ž «)¹ žzºººzž «z»;¼ , and Ÿ½¡ ™ ˜/š › œ . Any binary semigroup operation — on ˜ induces a global reduce operation

¾¿

™ ˜ š › œÁÀ

˜ š

which is defined by

¾

¾ ¾Å Ÿ¯ ÂÃÄ Ÿ § ¯ Æ&Ç¤È Ÿ § Æ ¯ÁŸ § È —¤Ÿ §4É —BÊÊʲ—¤Ÿ § Ì Å Ë ¾ Thus, for example, if ˜N¯± and — is the operation of addition (—¯Á´ ), then ¯Í and Î Î Ÿ¯ §4ÏÑÐ,ÒÓ ÔÕÒÓ²ÖH×NÒÓÑØ/×¢ÙÙÙ¬×WÒÓ²Ú¬Û Therefore, Ü is an image, namely the sum of a finite number of images. Ò In all, the value set Ý"Þ†ß ß ß¤à†ß¤á$â provides for four basic global reduce oper× çÙ . ations, namely Ü ßäã ßæå , and Ò Ò Ò Ò If the value set è has two binary operations é and ê so that Ý<è†ß&é)ßxêNâ is a ring (or Ð semiring), then under the induced operations ë¤ì"è/í†î ß&é)ßxêï is also a ring (or semiring). Analogous to the image-template product, the binary operations ê and é induce a template convolution product

é ð ì è$ò î íó ì è í î êñ

Ð1ô Ð

ì è$ò î

Ð

defined as follows. Suppose õæö÷ìZè ò î í , ÷ ö ì"è/í†î , and X a finite point set. Then the Ò Ð é õ ê , where ø ö ìZè ò î , is defined as template product ø

Ô Ò ø FÝ ùVâ û£üý "Ý õþLÝFùVâ)ê Ý"ÿ)â<â 
>ö Lù ö Ôú ÒÓ Û Ó Ð õ ê is given by the formula Thus, if õ ö ì Þ ò î í and ö ì Þ í î , then ø Ô × Ò Ò  ø Ý'ùVâ õ þ ÝFùVâ Ý'ÿ)â Ó Ô þ Ù£ÒÓ Û í  The lattice product ø õ à is defined in a similar manner. For õæö1ì"Þ  ò  îí Ð Ô Ò and ö ì Þ í î , the product template r is given by Ò   ø Ý'ù4â õ þ Ý'ù4â Ý"ÿ)â Ó Ô þ ×ÒÓ Û í

The following example provides a specific instance of the above product formulation.

© 2001 by CRC Press LLC

1. 5 Templates

31

Example: Suppose  "!$#&%('*) %' are the following translation invariant templates:

1

, s- = y

1

+

2

. ty =

1

3 -1

687

Then the template product /103254

1

9r: = y

;

3 -1

is the template defined by

2

1

6

3

-2

-1

If <>=*?A@CBED&FHJG I K FG are defined as above with values LNM outside the U support, then the template product OQPR
2

VrW = y X Z

4 0

Y

3 5

X

2

1

Z

4 0

The template t is not an []`\_^ -valued template. To provide an example of c8d the template product a5b , we redefine t as

1

e W ty =

3 1

© 2001 by CRC Press LLC

32

CHAPTER 1.

lnm

Then fhgjiQk

IMAGE ALGEBRA

is given by

+

1

r-y =

o

3

+

1

2

1

6

3

2

1

The utility of template products stems from the fact that in semirings the equation

prsh q tvu sx q wzy|{}t p~s q uy s€ q w holds [1]. This equation can be utilized in order to reduce the computational burden associated with typical convolution problems. For example, if N‚„ƒz…‡†‰ˆ‹Š Œ  ‚ Ž , then

;

4

9r: = y

;



6 -4

6

-4

9

-6

-6

4

† ˆ is defined by

” ’ œz|—}™ ‘ž” ” ’ œ|Ÿ ‘“”– ’ •˜— ‘”š ’ ™v› x ’ › x where

¢

¡ sW = y

2

3

  W ty =

-2

2 3 -2

¨x©

requires nine multiplications and The construction of the new image £¤ ¥8¦“§ eight additions per pixel (if we ignore boundary pixels). In contrast, the computation of the ¨–­® ¨–¯ § requires only six multiplications and four additions per pixel. For image £–¤ª¥}«¬¦§ large images (e.g., size °z±³²´Qµ¶°±·²´ ) this amounts to significant savings in computation.

Summary of Unary and Binary Template Operations In the following

­·¸¯T¹»ºv¼J½r¾¿

generic binary operation template sum max of two templates

© 2001 by CRC Press LLC

¼

and

­

­‹À¯ §

­~ē¯

¯

denotes the appropriate value set.

¤ ¤

¤

«

­*À¯³®ÂÁÃ"­ Á À¯ Á ­ ¯³®ÂÁÄ­ Á ¯Á « § § Á Á ­|ÄůƮ Á Ä­ Ä5¯ «

1. 6 Recursive Templates

33

Ç~ȓÉ5Ê Ë$Ç|ÈÅÉÆÌÎÍžÏ„Ç Í È5É Í × Í ÐÑÉÒÏ8ÓÔ*Ð Õ É Ö Ù$Ú>Ð Ø Û É Í Ù Ö É Í Û$Ü É Í·Ý Ü–ÞÞÞ·Ü É Í generic reduce operation Ø ßàÉáÏâß É Í×Ö É Í Ûå É Í Ý å ÞÞzÞ å É Í sum reduce Íãä Ø product reduce æçɘÏèæ É Í Ö É Í Û‡Þ É Í Ý Þ|ÞzÞÞJÞ É Í Íã³ä Ø max reduce éêÉ˜Ï é É Í Ö É Í Ûë É Í Ý ë„ÞÞzÞÑë É Í Íã³ä Ø min reduce ìêÉ˜Ï ì É Í Ö É Í Û È5É Í Ý È ÞÞzÞ È5É Í Íã³ä Ø ä In the next list, ÇNínîÂï|ðñò , ÉNíçîÂï ò ñ , X is a finite point set, and ï denotes min of two templates

the appropriate value set.

generic template product linear template product additive max product additive min product multiplicative max product multiplicative min product

1.6.

ô ó Ö Ç Ü õ É Êöó Í ¬Ë ÷Ì Ö ‹ø Ô*Ð ù ô ó Ö Ç å õ É Êöó Í ÂË ÷(Ì Ö ß ú ã ü ó Ö Ç ë ÉNÊöó Í Ë¬÷(Ì Ö é ú ã ü ó Ö Ç È ÉNÊöó Í Ë¬÷(Ì Ö ì ú ã ô ó Ö Ç ë ÉNÊöó Í ËÂ÷Ì Ö é ú ã ô ó Ö Ç˜È ÉNÊöó Í ËÂ÷Ì Ö ì ú ã

ËÂÇ*úË÷(Ì

Ç ú ˬ÷Ì å É Í ËûÌ ò ò

ò

É Í ËûÌ$Ì

Njúˬ÷Ì Þ É Í Ë¬ûÌ ò

ò

ô

Ç ú ˬ÷Ì å É Í ËûÌ Ç*úˬ÷Ì Þ É Í Ë¬ûÌ Ç ú ˬ÷Ì Þ É Í Ë¬ûÌ

Recursive Templates

In this section we introduce the notions of recursive templates and recursive template operations, which are direct extensions of the notions of templates and the corresponding template operations discussed in the preceding section. A recursive template is defined in terms of a regular template from some point set X to another point set Y with some partial order imposed on Y.



(or poset) is a set P together Definition. A partially ordered set ýÿþ
 with a binary relation , satisfying the following three axioms for arbitrary    þ :

   ý      !"#  ý$%'&)(*+,+-&. 0/  (ii)   1%2    ý3& ($)&4)  (iii) 

(i)

Now suppose that X is a point set, Y is a partially ordered point set with partial

monoid. An 6 -valued recursive template t from Y to X is a function 798;:<75 =?,>7and @BA9CE6 DGa F! H 6BI
JKBL9M , where NPORQESUTVKBW and NXQSUTYKZL , such that [E\]_a ^ `cb NEX b ]Bddfehg'i jk\l)mnfoeqprts _-`fb
order

© 2001 by CRC Press LLC

34

CHAPTER 1.

IMAGE ALGEBRA

Thus, for each zR{-| , }.~?)zB€ is an  -valued image on X and }h‚f)zB€ is an  -valued image on Y. In most applications, the relation ƒ…„V| or ƒ‡†ˆ| usually holds. Also, for consistency of notation and for notational convenience, we define }‰~‹Š1Œ}‰~?
-1

t =

t

1

1

= 1

Figure 1.6.1. An example of an integer-valued invariant recursive template from

«¬

to

«¬ .

If t is an invariant recursive template and has only one pixel defined on the target point of its nonrecursive support ­c®•¯°±² , then t is called a simplified recursive template. Pictorially, a simplified recursive template can be drawn the same way as a nonrecursive template since the recursive part and the nonrecursive part do not overlap. In particular, the recursive template shown in Figure 1.6.1 can be redrawn as illustrated in Figure 1.6.2

-1

1

1

1

t=

Figure 1.6.2. An example of an integer-valued simplified recursive template. The notions of transpose and dual of a recursive template are defined in terms of those for nonrecursive templates. In particular, the transpose ³3´ of a recursive template t is º È À  ¹'¿BÀÁÂtÉÄÅÀÇÆ , then the additive dual of defined as ³´ µ·¶³´¸Z¹³P´ º» . Similarly, if ³
¼¾½—¿ZÁ

t is defined by ³Pɞµ ¶ ³¸É ¹³‰É º» . The multiplicative dual for recursive is defined in a likewise fashion.

¿cÂÊ Ë -valued templates

Operations between Images and Recursive Templates In order to facilitate the discussion on recursive templates operations, we begin Í Ì , the morphological Ï max Î , by extending the notions of the linear convolution product Í Í Ï and multiplicative max Î to the corresponding recursive operations Ì º ¹1Î º , and Í Î º the , respectively. © 2001 by CRC Press LLC

1. 6 Recursive Templates

35

X and Y be finite subsets of ÐZÑ with Y partially ordered by Ò Ø ÔٗР×
Ú Let ÐÛ9ÜÝ ÛßÞ à á , then the recursive linear convolution product Óãä â à Ø

. If ÓÕÔÖÐZ× and is defined by

Óåä â à 9Ø æèçžé)ê Úë éê?ìvìÕíê Ôãî ÚÕë é)êBìBæ ðqñ‹ò ï é<ó•ôõì é Ó é)öìf÷.Ø.ø%ù’é)öìì â ï é éüìf÷.Ø ù'é<ü‹ìvìPýÿþ à úPñqò óû õ á ë Ý ä computes a new pixel value ë éê?ì based The recursive template operation â ) é  ö ì à of the source image and some previously calculated new on both the pixel values Ó ) é  ü ì which are determined by the partial order Ò and the region of support pixel values ë ü Ò ê for every of the participating template. By definition of a recursive template, ù ù ü
— é Ø ì ê   • é Ø ì < é ? ê ì is always recursively computable. Some Ô Ô à and
à . Therefore, ë partial orders that are commonly used in two-dimensional recursive transforms are forward and backward raster scanning and serpentine scanning. ä that the computation of a new pixel ë éê?ì can It follows from the definition of â à be done only after all its predecessors (ordered by Ò ) have been computed. Thus, in contrast to nonrecursive template operations, recursive template operations are not computed in a globally parallel fashion. é•Ø ù ìxæ for all Note that if the recursive template t is defined such that à ê Ôî , then one obtains the usual nonrecursive template operation



Óãä â à 9Ø æ

   é)ê

Úvë éê?ìvìåí ë é)êBìBæ

 



é Ó é•öìB÷.Ø øù é)öìì Ú ê Õ Ô î ð ñqò )é ó ôvõ ì



þ

Hence, recursive template operations are natural extensions of nonrecursive template operations. Recursive morphological max and multiplicative max are defined in a similar Ø and Ô Ù Ð × fashion. Specifically, if ÓªÔªÐ × Ú Ð × Ü Ý × Þ à á , then



ë æ

   Ó  

is defined by



 324 6 5(7 3238:9  B3C(4   7 *C"8"D $ , A @ ) ? ;6!<# %$'>= / "!$#&%(' * )+-,.0/ 1 = X\[  A  KTI J LYX>Z , For 1FEGHKM I0J L6N and EPQO GRHSKUITJ L6NWV GRHSM ] ^ 9   1 is defined by

_3``



"!$#  ) +-,  / 1

32ba 65Q7 *238:9



;6!<# = )c? ,dA /

_*C"_a   7 3C(R8eD

The operations of the recursive morphological min and multiplicative min are defined in the same straightforward fashion.

© 2001 by CRC Press LLC

0g f ]h6i0j ^ f  

36

CHAPTER 1.

IMAGE ALGEBRA

Recursive morphological max and min as well as recursive multiplicative max and min are nonlinear operations. However, the recursive linear convolution remains a linear operation. The basic recursive template operations described above can be easily generalized to the generic recursive image-template convolution product by simple substitution of the and . specific operations, such as multiplication and addition, by the generic operations with identity, then one can define the generic More precisely, given a semiring recursive product

k

mnbo"lo0kFp

l

krl qFs&nutwvrxnutyozn{y|.} {>~ qzu€ nu{

by defining

ƒ‚…„ kUl q† by _mˆ‡`p‚MŠ-‰ ‹Œ m*Ž@‘dp\’ „0m3“pkP†.”e•–m*“p˜—$lFŠ@‰ ‹YŒ *Ž ™ ‘  ’ _mˆš"pk›† q •–m3š(p3—"œ }

Again, in addition to the basic recursive template operations discussed earlier, a wide variety of recursive template operations can be derived from the generalized recursive rule by and . Additionally, parameterized recursive substituting different binary operations for templates are defined in the same manner as parameterized nonrecursive templates; namely as functions

k

l

†Ws$ € x nutWoQnž{ |d} {\~ q0 o where P denotes the set of parameters, and †&m Ÿ–pB‚…¡¢†&m Ÿ–p ” o"†&m Ÿzp q£ qT and †&m Ÿ–p q ¥ x n { | } {>~ .

with

†&m¤Ÿzp ”¦¥ x n t | {

Summary of Recursive Template Operations

list of pertinent recursive image-template products §©¨¦ª« and ¬ ¨r­Rªž«W®QInªu¯\the°d± ¯>following ²´³Tµ . As before, for each operation we assume the appropriate value set ª . recursive generic convolution product

¿RÏ ¸ Ѭ ÐeÒ ¿ÏÂ3Ó ¶ Ö ×Ø ¿ 3 À ˆ ¿ ` À @  ©  à * ¿ u À  À ¹ ½ § * ¿ Ì Ê @ Ë d Í y  ΠÁ ©¨ ÄÅ®BÇ@Æ ÈYÉ ·§ ¸ ¶ ³ ¬\¹»¼º -® Á ¿*Õ" ¸ ¬ Ò ¿*Õ"Â˜Ó Ù × Í 3 Ê Ô Á Î -ÆÇ ÈYÉ ± µ ¼¾ ³

recursive linear convolution product

ÿ ××Ø Ñ à e ø ù Û ÚÜÝßÛ Þà\áãä â××åæçè-é_æ*çêcêÜëyé_æˆçuê á çíì©îÅè ð"ñ(ò ï æ æ*öê÷ à Þ ù æ*öêcê ×× ú¢ñ$òQæ3ûó*ô-õ$ê æ*é_æ3þ(êu÷ æ3þ(êcê Ù ó-ü õ$ý recursive morphological max convolution product ××  Ú Û à.øQù  á 
Ú Þ à\áãä â× " ð < ñ ò  ×å_æ*ç_è@é_æˆç`ê-ê©ë_éBæ3ç`ê ç]ì îÅè  æ*óô@õdê  æ3öê Û à Þ ù æ3öê úYñ$ò  \û é_æ*þ(ê æ*þ"ê ócü õ$ý ××

© 2001 by CRC Press LLC

ï

Ú

Ù

ÿ ××Ø ××

1. 7 Neighborhoods

37

recursive morphological min convolution product



,

7

>=@?

M N

 -.0/1       "!#!%$&'"! )(+*) , 243#56! 7 4 8!:9<;   ?  8!BA  CD. / 1FE  K!:9 ; L K!'A O 2HGI5 J recursive multiplicative max convolution product



U

7

D=W?

 - . / %Q P      '&# "!!R$F "! S(%*T 4 U 243#56! 7 4 8!V   ? '8!4A CD. / E  KX!YV 4 KX!'A 2 GI5 J right multiplicative min convolution product



,

7

 =W?

M P N O



M N

 -. / 1  Q      '&# "!!R$F "! S(%*T , 243#56! 7  8!FVZ;   ? 48!'A  CD. / 1 E  K!FV ; 'KX!'A O 2 GI5 J \ Q [   is also straightThe definition of the left recursive convolution product forward. However, for sake of brevity and since the different left products are not required for the remainder of this text, we dispense with their formulation. Additional facts about recursive convolution products, their properties and applications can be found in [1, 61, 62].

1.7.

Neighborhoods

There are several types of template operations that are more easily implemented in terms of neighborhood operations. Typically, neighborhood operations replace template operations whenever the values in the support of a template consist only of the unit elements with the property of the value set associated with the template. A template , the values in the support of consist only of the unit of is called that for each a unit template.

eS^gf

]%^`_4aZbFcDd

]Dh

a

<] ^jilknmpoHq m o shown in Figure 1.7.1 is a unit rLksut
s:vxw since the value 1 is the unit with respect

For example, the invariant template template with respect to the value set to multiplication.

yt

=

1

1

1

1

1

1

1

1

1

Figure 1.7.1. The unit Moore template for the value set

© 2001 by CRC Press LLC

z4{F|W}~|:€ .

38

CHAPTER 1.

IMAGE ALGEBRA

‚ ƒ…„l†n‡p‰‹ˆ Š~Œ ‡ ˆ shown in Figure 1.7.2 is a unit template % L† ‰:Š%ŽWFŽW<‘ since the value 0 is the unit with respect to

Similarly, the template with respect to the value set the operation .



0

’r =

0

0

0

0

“L”•:–%—W˜F—W™›š . ¬ ­ is the °±£g° If œž Ÿ:¡ is an ¢ž£g¤ array of points, ¥§¦T”Z¨ , and ©~¦«ª>”n¬l­¯® unit template, then the values of the ¢ž£›¤ image b obtained from the statement ²`³ ´ Moore ¥%µ ™ © are computed by using the equation ² “ ¶"š ´ · ¥:“ÁšÃÂl©lÄ:“'Áš ´ · ¥:“'ÁšÅ ÆZÇ ¸¹ ¨&º»¼¾½4¿IÀ ¸¹ ¨ºX»¼¾½4¿IÀ ²›´ ¥+µ ™ © and We need to point out that the difference between the mathematical equality µ È ² x ³ ´ latter the new image is computed the pseudocode statement ¥%™ © is that ´Ï Îin. theObserve ´ ¥:“Áš only that since and for those points y for which Ê œ › É Ã Ë 4 “ D © @ Ä Í š Ì : ¥ 4 “  Á F š W Â Æ Ð “ ¶Zš ´ ËÓ4©>ÄWš , where Ð “¶Zš denotes the Moore neighborhood of y (see Figure 1.2.2), Figure 1.7.2. The unit von Neumann template for the value set

it follows that

² “'¶"š ´

· ¥:“ÁšÓÇ ¸¹ ¨&º@ÑÒ¼ Ä À

This observation leads to the notion of neighborhood reduction. In implementation, neighborhood reduction avoids unnecessary multiplication by the unit element and, as we shall shortly demonstrate, neighborhood reduction also avoids some standard boundary problems associated with image-template products. To precisely define the notion of neighborhood reduction we need a more general , which was defined in terms of a binary notion of the reduce operation operation on . The more general form of is a function

Ø

Õ

Ô F³ Õ × ¨ Ö Õ

Ô

Ô ³ Õ
¨ Ù Ú Ö Õ — Õ ´âá ¥ Ù Ú Ä ³ ¥%¦ Õ ) ¨ —Z¶S¦+ãÈä . where Ûܦ`Ý4ÞI¨FßDà and ¨ Ù Ú ¼ À Õ ´ ”Z¨ , where œåâ:Ÿ ¡ is an ¢æ£)¤ For example, if ¨ ­ Ûç¦ Ý Þ ¨Fß6è , then one such function could be defined as é ³ ” ¨
Ù Ú Öæ”%— where

ê Ý ¥ Ù Ú ¼ Ä À ß ´ ¸¹ ê Ú Ä ¥:“Áš . ¼ À

© 2001 by CRC Press LLC

array of points and

Another example would be to define

Ô ³ ” ¨
Ù Ú Öæ”

1. 7 Neighborhoods as

39

ëRìíÅî ïðòñ ó'ôÍõ ÷4øHùHú ðûö ïFð¾ñXó¾óý þÿ ü ï ðûñXó í




, then

ë

implements the averaging function, which

we shall denote by average. Similarly, for integer-valued images, the median reduction

î ï

  



.

  

í

í





 "!  # # #$

í


 %&('

, where )

õ


*+

ì uô

is a unit template with respect to the Now suppose / 021 , 354 7698 -:
7<=  &> @? 8 is a neighborhood system defined by ; ; of the semiring , ) 6 <
 BA 3 , and 4 6  . It then follows that C  ;D3 is given by

operation
* )

ì íZî ïFð¾ñó'ô õ

     %.'

is defined as   ,  !  #-# #



ñ

õ

í


* C

õ E F(ë G$H&I ð"JLKIó
í


7=

;M3

ñ


=(

õjí õ EFë G$H&NPORQSUTPVWXPY

This observation leads to the following definition of an image-neighborhood convolution product. Given Z\[^] , T`_ba+c , a neighborhood system dfehg\ikjl (i.e., d _nm jl,o-p ), and a reduction function qre a+cts u i a , then the image-neighborhood z convolution product v5e w Tyx d is defined by v

for each z {y_ g . Note that the product in that x is a function z

e a cy„ x

In particular, if ‘

_“’-† (”

 

is the

šœ›

• „

q

V{+X w

,

T…_‡†+c

ˆ

z

m Ts x

c}|

u~€‚ o

m j l o p eŠ‰=‹Œi

i

a p

›

š

ƒ

Y

is the Moore neighborhood, and

jŽ

unit Moore template defined earlier, then •

z

is similar to the image template product

z T—– ‘

w

z T– ˆ

.



w denotes the von Neumann unit template Likewise, T™˜ _ T˜ d , where ’ † ž  Ÿ ” (Figure 1.7.2) and N denotes the von Neumann neighborhood (1.2.2). The latter equality š š › › and ¡ŒeRw T ˜ d , then since € VW=X w¢ for all stems from the fact that if v eRw T ˜ › › Wr_ Z5£¥¤ ž¦Ÿ V €§X and ¤ ž¦Ÿ V €§X w`d V{+X for all points {¨_ ‰ ‹ , we have that v

V{X w

© ª «

´ µ7¶=· ¸¨¹-ºPµ¶=·9»

¼

´ µ¶=·Â»ÄÃŵÆ·PÇ

½¾.¿}À.Á

º

c}|§¬.­®°¯±7²&³

¯

³

Unit templates act like characteristic functions in that they do not weigh a pixel, but simply note which pixels are in their support and which are not. When employed in the image-template operations of their semiring, they only serve to collect a number of values that need to be reduced by the gamma operation. For this reason, unit templates are also referred to as characteristic templates. Now suppose that we wish to describe a translation invariant unit template with a specific support such as the È™É¥È support of the Moore template t shown in Figure 1.7.1. Suppose further that we would like this template to be used with a variety of reduction operations, for instance, summation and maximum. In fact, we cannot describe such an operand without regard of the image-template operation ËÊ by which it will be used. For us to derive the expected results, the template must map Ë all points in its support to the unitary value with respect to the combining operation . Thus, for the reduce operation of summation Ì , the unit values in the support must be 1, while for the maximum reduce operation Í , the values in the support must all be 0. Therefore, we cannot define a single template operand to characterize a neighborhood for

© 2001 by CRC Press LLC

40

CHAPTER 1.

IMAGE ALGEBRA

reduction without regard to the image-template operation to be used to reduce the values within the neighborhood. However, we can capture exactly the information of interest in unit templates with the simple notion of neighborhood function. Thus, for example, the as Moore neighborhood M can be used to add the values in every Î+ÏÐÎ neighborhood as well ÓŒÔ ÑÒ , to find the maximum or minimum in such a neighborhood by using the statements Ô Ô Ö Ö ÑtÕ , and Ñ™× , respectively. This is one advantage for replacing unit templates with neighborhoods. Another advantage of using neighborhoods instead of templates can be seen by considering the simple example of image smoothing by local averaging. Suppose ÑÙØ¥ÚÛ , where ÜMÝßÞ=à is an

áŒÏâ

array of points, and ãäØÄå-Ú9æ(çè

æç

is the Î+ÏÐÎ unit Moore template Ó

ì ï Ñ—Ò with unit values 1. The image b obtained from the statement é¨ê ëíî§ image obtained from a by local averaging since the new pixel value é ïñ é

ïñ ð9ëó ò ô õ ö

Û}÷§ø ùúûü

Ñ ï7ý ð+þã ÿ ïý ð9ë

óò ô õ ö

Ñ ïý ð

ã-ð ð

represents the is given by




Û}÷ øù"ú7ûŽü

Of will be a boundary effect. In particular,  course,

there    , then  &%' ( )# +*,%- ! . */%-0.1 "# -*,%-0.2 . 0 #   ! "#  $

if





which is not the average of four points. One may either ignore this boundary effect (the most common choice), or one may use one of several schemes to prevent it [1]. However, each of these schemes adds to the computational burden. A simpler and more elegant way is to use the Moore neighborhood function = @ M combined with the averaging reduction 3546387(9:3<;9 . The simple statement    %>?A provides for the desired locally averaged image without boundary effect. Neighborhood composition plays an important role in algorithm optimization (FGIand H D B C 5 B E simplification of algebraic expressions. Given two neighborhood functions .KJL ?* B5E , is a neighborhood function , F then the.<J dilation of BDC by B5E , denoted by BDC M GNHO L which is defined as B  PQ    PQ -*[&\^]_P 0 - R B B C S2TUQV WYX(Z &P M*>` 4 "a*b`>caed &PQ f B where B . Just as %g for composition, *? template

 &%g? *

halgorithm *? ?* BDC B5E BDC Bid/ E for optimization can be achieved by use of the equation l j k . For , appropriate neighborhood functions !and neighborhood reduction functions FGHm.(J
 PQ M*w\ is said to be translation invariant if B B for every point .  B z by Given a translation invariant neighborhood N, we define its reflection or conjugate |  ~2 2~2 -))- 2~( cdFG {P  |r *bP |r  !]a}aed |r f Bz B^z , where Bz B and denotes the origin. Conjugate neighborhoods play an important role in morphological image processing. Note also that for a translation invariant neighborhood N, the kth iterate of N can be expressed in terms of the sum of sets  PQ  {|€ # C &P +* B n B n
© 2001 by CRC Press LLC

1. 7 Neighborhoods Furthermore,

since

41 ‚^ƒ)„+…#†‡+ˆŠ‰

 ‚ƒ<„+…#† ‡Qˆ-‰•-–—Œ Ž8‘Y’K“€”

‚e†{‹€ˆ

Œ

 ‚ƒ<„+… †‡Qˆ-‰•€– Ž8(‘Y’K“€”

and

 ‚ƒ„…#†{‹€ˆ-‰œ+– , we have the symmetric relation ˜2™‘›š“1”

‚ ƒ † ‡QˆŒ‚ ƒ<„+… †‹rˆž‰>‚e†&‡ˆ .

Summary of Image-Neighborhood Products and ‚¦ 

In the following list of pertinent image-neigborhood products Ÿ/ /¡¢ , £¤t¥ , –"© . Again, for each operation we assume the appropriate value set ¡ . ”{§!¨ generic neighborhood reduction « Ÿ6ª ‚¬Œt­1†&‡h®¯h†&‡ˆ0ˆ±°c¯²† ‡QˆŒ6³ ”

ŸQ´ ‘Yš“ –h®‡e gµN¶

neigborhood sum ·

« Ÿ ‰ ‚xŒ 

¸¹

†&‡²®º¯h†»‡Qˆºˆ¼°½¯h†&‡ˆŒ

¿ 

¾

™‘Yš8“

¢À

Ÿ'†{Á+ˆ'®‡_ µ¬Â à Ä

neighborhood maximum ·

Æ Ÿ Å D

‚ÇŒ

¸¹

†&‡h®0¯h†»‡ˆ0ˆ¼°¯²† ‡QˆMŒ

¿ 

È ¢²À

™‘›š“)É

†&Á+ˆ-®‡, gµ à Ä

neighborhood minimum ·

Æ ŸÊ

‚¬Œ

¸¹

†&‡h®0¯h†»‡ˆ0ˆ±°c¯²† ‡QˆŒ

¿ 

Ë ¢hÀ

‘›š“

Ÿ-†Á+ˆ-®‡e gµÌ à Ä

™‘›š“

Ÿ-†Á+ˆ-®‡e µ à Ä

Note that ·

«

Ÿ Å ‚ÇŒ

¸¹

† ‡²®º¯²†»‡ˆ0ˆ±°c¯h†&‡QˆŒ

¿ 

È ¢²À

« « Æ Æ and, therefore, Ÿ Å ‚mŒsŸIÅ ‚ . Similarly, Ÿ Ê ‚mŒsŸÊ ‚ . Although we did not address the issues of parameterized neighborhoods and recursive neighborhood operations, it should be clear that these are defined in the usual way by simple substitution of the appropriate neighborhood function for the corresponding Boolean template. For example, a parameterized neighborhood with parameters in the set – ¢ . Thus, for each parameter ϱ Í , ‚e†YÏrˆ is a neighborhood P is a function ‚°(Í6Î ” §!¨ system for X in Z since ÐeÑÓҀԽÕ!ÖØ×ÚÙ(Û . Similarly, a recursive neighborhood system for a partially ordered set Ñ Ö_ÜÞÝßÔ is a function Ьà>ÑÐâáãÜ ÐDäQÔÕÖÇ×æå{Ù!ÛÜ Ù)çè satisfying the ë ÐDäMÑé+Ô , and for each í±êìÐDäžÑ&é+Ô , íÝé . conditions that for each é,ê_Ö , é[êì

© 2001 by CRC Press LLC

42

1.8.

CHAPTER 1.

IMAGE ALGEBRA

The p-Product

It is well known that in the linear domain template convolution products and image-template convolution products are equivalent to matrix products and vector-matrix products, respectively [63, 1]. The notion of a generalized matrix product was developed in order to provide a general matrix theory approach to image-template products and template convolution products in both the linear and nonlinear domains. This generalized matrix or p-product was first defined in Ritter [64]. This new matrix operation includes the matrix and vector products of linear algebra, the matrix product of minimax algebra [65], as well as generalized convolutions as special cases [64]. It provides for a transformation that combines the same or different types of values (or objects) into values of a possibly different type from those initially used in the combining operation. It has been shown that the p-product can be applied to express various image processing transforms in computing form [66, 67, 68]. In this document, however, we consider only products between matrices having the same type of values. In the subsequent discussion, îðïNñ"òó2ôõ and the set of all ös÷ø matrices with entries from î will be denoted by îùcú2û . We will follow the usual convention of setting î ûü îhý ú1û and view î û as the set of all n-dimensional row vectors with entries from î . Similarly, the set of all m-dimensional column vectors with entries hü ý  from î is given by þ{î ù5
ÿ ²ü î ý ú1 ù  îùcú  Let öó øó and p be positive integers with p dividing both m and n. Define the following correspondences:     
 *,+.-0/-

þró

  û  

÷ ÿMü

þ

#21341

% '



û

ÿ&%

"!$#

')(

ró '

ó6587:9;#21< 1

ó

and =

>  ù  ?

<@  *,+.-E/ -

þBA ó0

#F1G 1

÷

ÿQü

 

 

ÿ

þBAC!D# '

'

ù

(

ró

ó657H9"#21AI1KJ '

Since = þ L0ó6M ÿ;N = þ L ó6M ÿ4O L N L GP / L ü L 57H9DM



 using the row scanning order as shown: ù  ÷ , ý

QR

T ý Uý V R R R R R R R R R R R R R R R

.. . ý

T^T

ù,T 



V ] ý V  ù, U ý V

T 6 ý ZU V Z  T U ZV W Z

XYXYX XYXYX

XYXYX

T

ù,



M

=

,

T ý6U V

XYXYX

U V W

XYXYX

ù,



bdc c c c c c c c c c c c c c c c c

T W U V W

XYXYX

[ T\[ U V

XYXYX

.. . T

linearizes the array

.. .

T_[] ý`V T\[ U V  Z Z

.. . ý

N

.. .

T\[ U V W

.. . S

© 2001 by CRC Press LLC

.. . ý V T_[ U ý V 

T\[^] R

ý T  U ý V W

T 6 ý WU V W  T U WV W W



.. . T U V Z

XYXYX

ù,T  ù,

a V ù U V

e

1. 8 The p-Product

43 fh, g ijlkmf j g

It follows that the row-scanning order on n`op6q.rIs
y j

is given by

n`op6q.rs$y j

n o^t p6qtvr

or, equivalently, by n`op6q.rIs
o~€o t‚„ƒH…

q s†q t‡

We define the one-to-one correspondence ˆ

j;‰  Ž f gŠ‹kŒfh? g ij kfj g f gŠ‹kŒfhg  n`’Cp “&pB”>r–• n ’p6y `n “.p6”r6r ‡ ‘ ˆ j ‰ j

The one-to-one correspondence allows us to re-index the entries of a matrix ™ š › ¡_¢› £0¤ ž Š_Ÿ h in terms of a triple index by using the convention ™ š › ¡¥¢u› £E¤ €¦™š › œŒw ©,ª&«  ­ « ¬

Example: Suppose

s†oCs¯®l°6±

´?µ·¶¸º¹{µ·»½¼¾¿†ÀÁµÂ¶

. Hence for

ÀxµÌ¶¸–¼Í¾¿ÌÇÉȘÎFÈ˹$ÃÄÀ³µÌÅ

Ý

y n`opBq&r§€‹¨p j ‚ƒ²… ¬ s†q s³± ‡

. Then

¹‹ÃÄÀÁµÂÅƸ)ÇÉÈËÊÁÈ

ÏеÌÑBÒ²Ó6Ô Õ6Öº×)ØFÙÚ²Û

ÜÞݏßzàHá0á

à²á Ù

à²á6â

à²á6ã

à²á6ä

à²á Û

à Ù á

à Ù0Ù

à Ù â

à Ù ã

à Ù ä

à Ù6Ûzå

, we have

ßzà á ÔYæ á Ô á6ç

à á ÔYæ á Ô Ù ç

à á ÔYævÙ8Ô á6ç

à á ÔYæ_Ù8Ô Ù ç

à á ÔYæ â Ô áBç

à á ÔYæ â Ô Ù ç

à Ù8ÔYæ á Ô á6ç

à Ù8ÔYæ á Ô Ù ç

à ÙèÔYævÙ8Ô á6ç

à Ù8ÔYæ_Ù8Ô Ù ç

à Ù8ÔYæ â Ô áBç

à ÙèÔYæ 8 â é Ù ç ålê

ëíï ì î

is decomposed in a similar

The factor ëCíì of the Cartesian product fashion. Here the row-scanning map is given by ñ ò ó

ë òì î

ö,÷HøEù ø³ñ ò²ú`ûü`ýþ ÿ  

ý



ì ô òõ ë í ú`û

ü 




ëìð

ë íì

 þ ú.þ xýü



This allows us to re-index the entries of a matrix triple index )+*-,/. 0+1. 2 by using the convention 

û





ÿÐú  þ

) *3,. 014. 265 ) 78. 2:9 + ;=< $&>?&@A' C 5 B ? DFE!GHIGKJL > N L MPORQSTJUL @ K L V#W+MYX

© 2001 by CRC Press LLC

€˜n ™š› œ r —

í!#"$&%('

in terms of a

44

CHAPTER 1.

Example: Suppose

Z[]\ ^`_a[cbedgf hCij[lk

. Hence for

iv[]k^wdxf h]qys{zNs|Z}moip[~k Ž

[

Á ¿

‚R‘I“

‚’+‘

‚’’

‚R’“

‚ “o‘

‚ “/’

“ “ ˜ ‚ +

‚/”o‘

‚/”/’

‚•”+“



‚‘’

Now let ŸT ]¡&¢g£-¤+¥4¦ §©¨(ª¬«®­ , A and B can be rewritten as  ä

–— —

 ‚‘+‘

Ž  [

and



IMAGE ALGEBRA

. Then

Znmoip[]k^rq:sutps

€[]‚ƒ„ …†`‡pˆn‰‹Š#Œ

, we have –— —

‚™ ‘ „ ‘š „ ‘

‚›™ ‘ „ ‘+š „ ’

‚›™ ‘ „ ‘š „ “

‚œ™ ‘ „ ’š „ ‘

‚›™ ‘ „ ’+š „ ’

‚›™ ‘ „ ’š „ “

‚ ™ ’ „ ‘š „ ‘

‚ ™ ’ „ ‘+š „ ’

˜ž ‚ ™ ’ „ ‘š „ “

‚ ™ ’ „ ’š „ ‘

‚ ™ ’ „ ’+š „ ’

‚ ™ ’ „ ’š „ “

¯±°e²³´-µ·¶ ¸¹º©¨¼»«#½

. Using the maps

and

¾À¿

°]Ã+Ä/Å ¶ Æ4Ç4¶ È+É4ÊœË «‹ÌÎÍÏFÐ!Ñ/ÒIÑNÓÔCÕ`Ô×Ö&ÍØÓÔPÙ&Ú ²ÜÛ ÍÝ ¹(°pÞß ÔCà:ͼáâã °

Ã+å ÆÈo¶ æ+É4¶èç ÊAé «¼êÍëÏFЮÑ+ÒIÑjÓÔ|ì Ú ² ÝÍ Þg¹(°|Û4ß Ô×íKáâ#ãKÓÔïînÔ|ðgñ

The p-product or generalized matrix product of A and B is denoted by the matrix õ ô °{òïó

ô ò×ó

¿ ¯

, and is

¿ ¯Tºa¨ ªèƬ»œö4¿É&«Æ·­÷ö4¿Aɬ½

defined by Á

ú Æ3ø¶ ù+É-Ƭ´4¶ ¸-É °

¿ Ã&þ ø¶ Æÿ´4¶ û É ³ Æ û ¶ ù+É4¶ ¸ Ê

û/ü¼ý

°




Ã&þ ø¶ Ƭ´4¶ ý É ³ Æ ý ¶ ù+É4¶ ¸ Ê ó

+ó

Ã&þ ø8¶ Ƭ´4¶ ¿oÉ ³ Æ ¿A¶ ù+ɶ ¸ Ê Í

  and  th column entry of C. Here we use the    th row       Thus, the matrix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

where Á Æ-ø8¶ ùÜÉ3Ƭ´4¶ ¸ É denotes the lexicographical order ² Í ¹ has the following form:

.. .

.. .

. . .

. . .

. ..

² Í e²

¹

²

ß ¹

ß Í

¹

Í

ß

Ò

Õ

°

Õ ß Í Þ

NÞ ß ñ

.. .

.. .

.. .

. . .

. . .

. . .

. . .

. ..

. ..

. ..

. ..

. ..

. ..

. ..

. ..

. ..

. ..

. ..

.. .

.. .

.. .

.. .

.. .

.. .

The entry

JLKMN OLPK!Q7N R%P

© 2001 by CRC Press LLC

in the (s,j)-row and (i,t)-column is underlined for emphasis.

1. 8 The p-Product

45

SUTEVXWZY[T]\W5^_TE`aWcbedafhgTji . Then lY kLmnT]iW5^okLmnT]VEbedaf]p qDrsqEV . Now let t Tvuwx*xBwXxyCwXxz{wXx7|}wx~CwXx w€yxBw‚y*yBw‚yz{w‚y|{w€y*~ƒw‚y1…„‡† ˆ yŠ‰…e‹ŒU and Ž •*– x*x x+y xz T‘’”“ yx “ y*y “ y+z — ŒU|e‰Xz™˜ “ zx “ zLy “ z1z † “ Ž |xelement “ |Ly “ |1z Then the (2,1)-row and (2,3)-column “ “ šL“ › y*œx+ › yœ z1 of the matrix ž ThŸ¡¢   y † Œ™£ ›,¤‚¥%¦ ‰ ›,§5¥%¦ ¨ T Œ |e‰© is given by « « L « ª ¬ y šL› yœx › yLœ z+ T w yœ ­+® › yœ ;¯ ® › œx7%œ z T w yLœ x ­+® › yœx”¯ ® › x*“Lœ° x%œ z   w yœ ­+® › yœ y+”¯ ® › yœx%œ z T w‚y+z ¯ xz   “Lw‚° y| ¯ z*z™˜ “° “ Thus, in order to compute š › elements of A are combined with yœx › yœ z1 “ , the two underlined

To provide an example, suppose that for p = 2, one obtains

the two underlined elements of B as illustrated:

•– *x x x+y xz y ‘’ ““ zyxx ““ zLy*yy ““ z*yzz — “ |x “ |Ly “ |*z w x*œ ­ “ ® ›› yœ y “ w x*œ ­ “ ® ›› zŠœx w x*œ ­ ® ›› zœ y ¢   y w yœ ­ ® yœ y w yœ ­ ® zŠœx w yœ ­ ® z ² y ±

u w x*x w x+y w xz w x| w x+~ w x+ ¢   w yx w y*y w yz w y+| w y*~ w y*…± T u w x*œ ­ ® ›› x1œx w x*œ ­ ® ›› x*œ y+ w x*œ ­ ® ›› yLœx w yœ ›­ ® x1œx w y›œ ­ ® x*œ y+ w y› œ ­ ® yLœx• –– ‘‘’ “L° ®® ›› x*x*œœy+x³³œœxx “L° ®® ›› x*x*œœy+x%%œœ yy “L° ®® ›› x*x*œœy+x%%œœ zz — “L° ® › yœx³œx “L° ® › yœx%œ y “L° ® › yœx%œ z “L° ® yœy+³œx “L° ® yœy+%œy “L° ® yœy+%œ z “L° “L° “L° š › x*œx › x*œx š › x*œx › x*œy ¯¯¯ š › x*œx7 › yœ z1 ¯¯Š¯ š › x*œx › zœ z1 • – T ‘’ ššL›› x*œy+ ›› x*œx šš›› x*œ y+ ›› x*œy ¯¯¯ šš›› x*œy ›› yœ z1 ¯¯Š¯ šLš ›› x*œy ›› zœ z1 — B¯¯¯ š› yœx7 › yœ z1 ¯¯Š¯ šL› yœx › zœ z1 šL› yyœœy+x › x*x*œœxx š› yyœœy+x › x*x*œœyyB š x*x š x+y ¯¯¯ š x ¯¯¯¯ ¯¯ š x© y•*œ– y yœ z1ƒ¯¯Š¯ yœ y zœ z1 T‘’ šš yx šš y*y ¯¯¯ šš y1 ¯¯¯ šš y1© — ˜ š |zxx š |LzLyy ¯¯¯¯¯¯ š |´z´ ¯¯¯¯¯¯ š z´|´©© In particular, V \ p • – \» pµ p¸p}pŠµ pŠµ}V¹¶ pCV µ}¶·` i ¢   ‘’ p{i}V — T ‘’ i pŠµ» pŠ¶ µ V¹µ u V} i}` pCµ}\ ± y V V}¶ pµ pL¼ p pŠ» µ}V¸\ º i µ·` p¹p \ pŠ\ pŠV pV © 2001 by CRC Press LLC

p` pi pV p ¼

i¸µ pŠº • – pŠV}V¸µ — pŠV}i¸µ ˜ µ Ve`

46

CHAPTER 1.

IMAGE ALGEBRA

½
¾À¿:Á  ¾ÊË̔ÍÎÏ Ñ*Ò Á ÁÅĐÆeÇaÈ]É ÓÕÔ Ã Ð Ï ¾ × ½ ½E× ¾ Î Î Á ¾ Î ÛÞÝ Ö Û É Ü Ø:Ù Üß Ö ØÚÙ ÉÛÜ Ö Í then Ô Ô Ô ÍÔÃ Ô ÔÃÐ This shows that the transpose property, which holds for the regular matrix product, is × for the × p-product. The reason is that the p-product is not a dual operation generally false Ü in the transpose property hold we define the dual Ø à ofdomain. Ø à byIn order½
to× make the transpose × ½ operation Ø àÜ ÉEáãâÉ Ü Ø à Ü,ä Ü ß ½E× ¾ × ½ It follows that Ø à É × âÉ Ü Ø àÜ Ü&ä Ü Ü ½E× ¾ × ½ Ø à â Ø É ä isÜ theÉ dual and the p-product operation of . In particular, we now have the transpose Ü ÜØ × Ü à à Ü. property ½å¾ ½ìthe ¾ operation Ø à is defined in termsÖæ€çè ofÛ matrix transposition, labeling of Since â æ ç*í îïðLñ ä Specifically, if matrix indices are reversed. ¾ convention ¾9isö an é™êÕë matrix, then A gets , using the reindexed as æ ç*í î,ï3ÁÞí ðLñ û æ û ç´í èlò ó à Öô ÁÞÛ û û Ô%õ Ô ÷5øùLú+ù õ ëcüLý ÆeÇaÈ ô ý ß ¾ É ÖþLçè Û þ î í ïLñ%í è Similarly, if is an ÿÕ
of B are relabeled as  , ê ¾matrix, then the entries ¾ using the convention þ î  í ïLñ%ÁÞí è û þ ç*û í è ò  à Ö ÁÞ û ô Û û  Ô Ô ÷5øXùLú+ù ô ý ÆeÇaÈ  ÿoüLý ß ½E× ¾ Ø àÜ É  is then defined by the equation The product ¾ à ¾ × × â â â æ ç´í î í ðLñ þ î í ñí è ä€ß ä ä ó î  í çñ³î,èí ðLñ ï   æ ç*í î,ïeí ðLñ þ î  í ïLñ%í è æ ç*í î *í ðLñ þ î  í ñ%í è à à If

  

Note that the dimension of C is

éà ê à  .

Ø àÜ , suppose that þ  þ  Ù Ñ*Ò þ Ù  þ Ù*Ù ÒÒ þ >þ Ù ÒÒ ß þ >þ  Ù þ !>þ  Ù Ó þ "!>þ " Ù

To provide a specific example of the dual operation

¾ ÌÊ æ  æ  Ù æ Ù Bæ Ù*Ù  æ  Bæ  Ù © 2001 by CRC Press LLC

ÊË

æ  æ  Ñ ÆeÇaÈ]É ¾ ËËËË Ì æ Ù Cæ Ù  Ó æCæ

×

1. 9 Exercises

47

#%$'&)(+*,$.-/(102$435(7698/:;<$4=

. Thus, for p = 2 and using the In this case we have scheme described above, the reindexed matrices have form

?XX Y F

F CCDD KECG G DDCC Y FJFJCCDD KECGGDD KK S[[ XX Y FMK!DCG DC Y FZK!DCGD K [[ ?@BA5CED FCDCGHAICD FJCD KGLAICD FMK!DCGNA5CD FOK!D KPG ST > $ A)K!D FJCDCGHAQK!D FJCD KEGLAQK!D FMK!DCGNA)K!D FOK!D KPG 6U8/:4VW$ X@ YY FMK!D KEG DC YY FZK!D KEGD K T][ \ AR D FJCDCG AR D FJCD KEG AR D FMK!DCG AR D FOK!D KPG Y F R DCG DC Y F R DCGD K Y F R D KEG DC Y F R D KEGD K >_`2^ aK VW$_b is a cedf- matrix given by According to the dual product definition, the matrix ?XX g K!CCC g KCEKK g KC RR g KEChh S[[ ?XXX g F F CCDD KECGMGMFFCCDDCCGG g FFCCDD KPCGMGMF F CCDD KEKEGG g FFCCDD KPCGMGMFMFMK!K!DDCCGG g F F CCDD KECGJGJFMFMK!K!DD KEKEGG S [[[ g g g XX g g g g [[ XX g [[ XX .. .. .. .. [[ XX .. .. .. .. [ .C .K .R .h [ .GMFCDCG .GMF CD KEG .GMFMK!DCG .GJFMK!D KEG [[ \ X R R R R M F ! K D J F K D J F K D M F ! K D X b$ @ g i g i g!i g!i T $ @ g g g g T .. .. .. .. .. .. .. .. g j. C g j. K g!j. R g!j. h g F R D R .GMFCDCG g F R D R .GMF CD KEG g F R D R .GMFMK!DCG g F R D R .GJFMK!D KEG R The underlined element g i is obtained by using the formula: K l l l k m Rg i $ g FMK!D R GMFJK DCG $ A R D F DCG Y FMK!D G D K $ A R D F CDCG Y FZK!DCGD K ^ A R D FZK!DCG Y FJK D KPG D K \ C R Thus, in order to compute g i , the two underlined elements of A are combined with the two underlined elements of B as illustrated:

?XX Y !KCCC ?@BA C C A CEK A C R A Ch ST XXX YY R C A K!C A KK A K R A KPh ` ^ aK @ Y h C A R C A R K A RR A R h YnC YiC

Y CEK Y KK YRK YhK YnK YiK

S[[

[[ T
[ \

As a final observation, note that the matrices A, B, and C in this example have the form of the transposes of the matrices B, A, and C, respectively, of the previous example.

1.9. 1.

Exercises a. b.

oqp]r s%()uWpvrwt s
pxrwt rBt

s
p7rt

and a point is a special case of Show that the addition of a scalar . the addition of two points is a special case of Hadamard Show that scalar multiplication of a point . multiplication of two points in

2. Show that the operations of point addition and scalar multiplication satisfy: a. b.

^
© 2001 by CRC Press LLC

, where , where

o|p}r and s%(Iu~p}rBt , o(  p}r and sp}rBt .

48

CHAPTER 1.

‚

IMAGE ALGEBRA

…ƒ „ † a point in ƒ…„ . The neighborhood (or sphere ˆŠ‰E‹ ŒU†BŽ and defined as ˆ ‰P‹ Œ †Ž…‘U’2“7ƒ „•” ‚5†%–P’Že—˜‡Q™›š Give a graphical representation of the neighborhood of the origin †7œ5–œ9Ž of radius 1 if a. ‚ is the Euclidean distance, b. ‚ is the chessboard distance, c. ‚ is the city block distance. 4. Let †%–P’W“_ƒB . For ž~ Ÿ , show that: a. ¡†%¡E¢£¤œ , b. ¡†%¡ ¢  œ if and only if †˜ œI–œ9Ž , c. ¡†¦¥]’§¡P¢}¨¡!’¦¥]†%¡P¢ , ª d. © «€¬ ­ † « ’ « © —®¡ †%¡ ¢ ¡ ’e¡ ¢ , ©© ©© e. ¡© †°¯2’§¡ ¢ © —¤¡†%¡ ¢ ¡’§¡ ¢ . 5. Let ± , ² , and ³ be any three sets in ƒB . Show that a. ±µ´¶²¸·|³%Ž¹  ±‘´q²]ŽB·2 ±‘´|³wŽ , b. ±µ·¶²¸´|³%Ž¹  ±4·º²]ŽB´2 ²µ·|³%Ž , c. ±4´º²7Ž< ±»· ² , d. ±4·º²7Ž< ±»´ ² , e. ±¤¼²'´|³wŽ¹½± ¼°²]ŽB´¶±,¼7³%Ž , f. ±¤¼²'·|³wŽ¹½± ¼°²]ŽB·¶±,¼7³%Ž . ­ ­ ­ ­ 6. Let ¾]“2ƒB¿ . Show that ¾°À ¾Á À ¾7W¾ and that ¾ÂÁ À€¾ºÀ€¾ÂÁ W¾Á . and 3. Let be a distance function on or ball) of radius of is denoted by

‡

7. Suppose

¾ –ÓvƒB¿

†

and

that Ħ» ¾Å¥2ÃBŽÇÆ
È . Show É%ÊË  ¾/Ž%Ä Á ­ À ÄIš

¾Ì“_ƒB¿ by ¾/Í , where ¾/Í< Î ¥˜¾Á ­ À¾ Ã'“WƒB¿ a. É%ÏIÐ ¾/Ž<ÒÑ É…Ê/Ð ¾/Ž Ó Í , b. ɧ¬ Ô Ð  ¾/Ž°ÒÑ É Ð ¾›Ž Ó Í , c. ¾ Í À)¾¨œ . 9. Suppose Õ ” ƒ
Ö¨ƒ denotes the absolute value function ՍE‡9ŽB‘× ‡I×– ‡•“]ƒ . Show that ÕÙØe¾°Ì¾ÚÆ|P¥Û¾Ž whenever ¾.“‘ƒB¿ . ß à are the following translation invariant templates: 10. Suppose Ü–ݹ“.ހƒ ߀àá 8. Define the Boolean complement of a real-valued image . If show that

1 s y=

1

2

1

t y=

3 -1

a. b.

âã%äå€æèç , âã1é å9æê , and âã%ëºåìæèç . í<î‘ã1ð ï å . Give a graphical representation of í ç .

Find Let

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.

1. 9 Exercises

49

ñ2ò óôBõö÷Ùø ù . The dual of ñ is denoted by ñìú ñìû›ú üýÇþŠÿ
ñ üþ ú . Show that   ñ ÿ ü ñ ú   ú þ ú ñ ò ó ô ùö ÷ ø õ . where  ¶

11. Suppose defined by

ó ô ùö÷ ø õ

and is the template

ñ9ú|ò

12. Suppose

3

sy=

2

ty =

1



are two translation invariant real-valued templates on where of a. b.

"  , and   ! .

1

1

1

1

2

1

1

1

1

 . Provide a graphical representation

is an element of $%&'%)(+* or ,%-. * . images 56 , 5 , 587 . Compute the image /102&37=?/4>A0 @ rectangular = 59:$ !; 6<( 5 :$5B6 ! 65  ( ! 65C7ED  14. Let 5A6 , 5 , 587 be as in Exercise 13. Find  a. #F
5A62G5 G5C7 , and  b. HI# . 15. Let J 6 and J  be two neighborhoods defined on A , 5 a rectangular KL&M realvalued image and NO ! . Show that 59  $J 6   J  (PQ$R5'  J 6 (   J  D 16. Let S /WVX0ZYZ[ \ UT VZ\][ /]0Z^P_a`bdc kml Z V ^ / e gfhi /W\ZV n D VZ \ZVZ 0j[Y Compute each S  ofe the following: a. $   S (+o , e  b. $ o   S o ( , e c. o  o  o .S S e Svu and e S arewup2&qK matrices 17. a. Suppose $S  (  rIg, S  rx.y, e ze u rx. and. r is a s&t matrix. Show that r are of dimension s{&vt . Show that b. Suppose u r .  and, e   u and   u $ e isr|(xan pP,&vK   S matrix . r . e c. Let the dimension of , , and r be sw&zt , }~&p , and K&zM , respectively. both p and K . Show that Suppose € divides both S  t u ande } ‚ , and  Sdivides u  e  ,  r .  ,  .  ‚ rƒD 13. A real- dual-valued image Construct three real dual-valued

#

This is the associative law for the generalized matrix product.

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50

1.10.

CHAPTER 1. IMAGE ALGEBRA

References

[1] G. Ritter, “Image algebra.” Unpublished manuscript, available via anonymous ftp from ftp://ftp.cise.ufl.edu/pub/src/ia/documents, 1994. [2] S. Unger, “A computer oriented toward spatial problems,” Proceedings of the IRE, vol. 46, pp. 1144–1750, 1958. [3] J. von Neumann, “The general logical theory of automata,” in Cerebral Mechanism in Behavior: The Hixon Symposium, New York, NY: John Wiley & Sons, 1951. [4] J. von Neumann, Theory of Self-Reproducing Automata. Urbana, IL: University of Illinois Press, 1966. [5] K. Batcher, “Design of a massively parallel processor,” IEEE Transactions on Computers, vol. 29, no. 9, pp. 836–840, 1980. [6] M. Duff, D. Watson, T. Fountain, and G. Shaw, “A cellular logic array for image processing,” Pattern Recognition, vol. 5, pp. 229–247, June 1973. [7] M. J. B. Duff, “Review of CLIP image processing system,” in Proceedings of the National Computing Conference, pp. 1055–1060, AFIPS, 1978. [8] M. Duff, “Clip4,” in Special Computer Architectures for Pattern Processing (K. Fu and T. Ichikawa, eds.), ch. 4, pp. 65–86, Boca Raton, FL: CRC Press, 1982. [9] T. Fountain, K. Matthews, and M. Duff, “The CLIP7A image processor,” IEEE Pattern Analysis and Machine Intelligence, vol. 10, no. 3, pp. 310–319, 1988. [10] L. Uhr, “Pyramid multi-computer structures, and augmented pyramids,” in Computing Structures for Image Processing (M. Duff, ed.), pp. 95–112, London: Academic Press, 1983. [11] L. Uhr, Algorithm-Structured Computer Arrays and Networks. New York, NY: Academic Press, 1984. [12] W. Hillis, The Connection Machine. Cambridge, MA: The MIT Press, 1985. [13] J. Klein and J. Serra, “The texture analyzer,” Journal of Microscopy, vol. 95, pp. 349– 356, 1972. [14] R. Lougheed and D. McCubbrey, “The cytocomputer: A practical pipelined image processor,” in Proceedings of the Seventh International Symposium on Computer Architecture, pp. 411–418, 1980. [15] S. Sternberg, “Biomedical image processing,” Computer, vol. 16, pp. 22–34, Jan. 1983. [16] R. Lougheed, “A high speed recirculating neighborhood processing architecture,” in Architectures and Algorithms for Digital Image Processing II, vol. 534 of Proceedings of SPIE, pp. 22–33, 1985. [17] E. Cloud and W. Holsztynski, “Higher efficiency for parallel processors,” in Proceedings IEEE Southcon 84, (Orlando, FL), pp. 416–422, Mar. 1984. [18] E. Cloud, “The geometric arithmetic parallel processor,” in Proceedings Frontiers of Massively Parallel Processing, George Mason University, Oct. 1988. [19] E. Cloud, “Geometric arithmetic parallel processor: Architecture and implementation,” in Parallel Architectures and Algorithms for Image Understanding (V. Prasanna, ed.), (Boston, MA.), Academic Press, Inc., 1991. [20] J. Wilson and E. Riedy, “Efficient SIMD evaluation of image processing programs,” in Parallel and Distributed Methods for Image Processing, vol. 3166 of Proceedings of SPIE, (San Diego, CA), pp. 199–211, July 1997. [21] H. Minkowski, “Volumen und oberflache,” Mathematische Annalen, vol. 57, pp. 447– 495, 1903.

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1.10 References

51

[22] H. Minkowski, Gesammelte Abhandlungen. Leipzig-Berlin: Teubner Verlag, 1911. ¨ [23] H. Hadwiger, Vorlesungen Uber Inhalt, Oberflæche und Isoperimetrie. Berlin: SpringerVerlag, 1957. [24] G. Matheron, Random Sets and Integral Geometry. New York: Wiley, 1975. [25] J. Serra, “Introduction a la morphologie mathematique,” booklet no. 3, Cahiers du Centre de Morphologie Mathematique, Fontainebleau, France, 1969. [26] J. Serra, “Morphologie pour les fonctions “a peu pres en tout ou rien”,” technical report, Cahiers du Centre de Morphologie Mathematique, Fontainebleau, France, 1975. [27] J. Serra, Image Analysis and Mathematical Morphology. London: Academic Press, 1982. [28] T. Crimmins and W. Brown, “Image algebra and automatic shape recognition,” IEEE Transactions on Aerospace and Electronic Systems, vol. AES-21, pp. 60–69, Jan. 1985. [29] R. Haralick, S. Sternberg, and X. Zhuang, “Image analysis using mathematical morphology: Part I,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 9, pp. 532–550, July 1987. [30] R. Haralick, L. Shapiro, and J. Lee, “Morphological edge detection,” IEEE Journal of Robotics and Automation, vol. RA-3, pp. 142–157, Apr. 1987. [31] P. Maragos and R. Schafer, “Morphological skeleton representation and coding of binary images,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-34, pp. 1228–1244, Oct. 1986. [32] P. Maragos and R. Schafer, “Morphological filters Part II : Their relations to median, order-statistic, and stack filters,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-35, pp. 1170–1184, Aug. 1987. [33] P. Maragos and R. Schafer, “Morphological filters Part I: Their set-theoretic analysis and relations to linear shift-invariant filters,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-35, pp. 1153–1169, Aug. 1987. [34] J. Davidson, “Simulated annealing and morphological neural networks,” in Image Algebra and Morphological Image Processing III, vol. 1769 of Proceedings of SPIE, (San Diego, CA), pp. 119–127, July 1992. [35] J. Davidson and A. Talukder, “Template identification using simulated annealing in morphology neural networks,” in Second Annual Midwest Electro-Technology Conference, (Ames, IA), pp. 64–67, IEEE Central Iowa Section, Apr. 1993. [36] J. Davidson and F. Hummer, “Morphology neural networks: An introduction with applications,” IEEE Systems Signal Processing, vol. 12, no. 2, pp. 177–210, 1993. [37] E. Dougherty, “Unification of nonlinear filtering in the context of binary logical calculus, part ii: Gray-scale filters,” Journal of Mathematical Imaging and Vision, vol. 2, pp. 185–192, Nov. 1992. [38] E. Dougherty, “Optimal binary morphological bandpass filters induced by granulometric spectral representation,” Journal of Mathematical Imaging and Vision, vol. 7, pp. 175–191, Mar. 1997. [39] D. Schonfeld and J. Goutsias, “Optimal morphological pattern restoration from noisy binary images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, pp. 14–29, Jan. 1991. [40] J. Goutsias, “On the morphological analysis of discrete random shapes,” Journal of Mathematical Imaging and Vision, vol. 2, pp. 193–216, Nov. 1992. [41] L. Koskinen and J. Astola, “Asymptotic behaviour of morphological filters,” Journal of Mathematical Imaging and Vision, vol. 2, pp. 117–136, Nov. 1992.

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52

CHAPTER 1. IMAGE ALGEBRA

[42] K. Sivakumar and J. Goutsias, “Morphologically constrained grfs: Applications to texture synthesis and analysis,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 21, pp. 99–131, Feb. 1999. [43] S. R. Sternberg, “Language and architecture for parallel image processing,” in Proceedings of the Conference on Pattern Recognition in Practice, (Amsterdam), May 1980. [44] S. Sternberg, “Overview of image algebra and related issues,” in Integrated Technology for Parallel Image Processing (S. Levialdi, ed.), London: Academic Press, 1985. [45] P. Maragos, A Unified Theory of Translation-Invariant Systems with Applications to Morphological Analysis and Coding of Images. Ph.D. dissertation, Georgia Institute of Technology, Atlanta, 1985. [46] J. Davidson, Lattice Structures in the Image Algebra and Applications to Image Processing. Ph.D. thesis, University of Florida, Gainesville, FL, 1989. [47] J. Davidson, “Foundation and applications of lattice transforms in image processing,” in Advances in Electronics and Electron Physics (P. Hawkes, ed.), vol. 84, pp. 61–130, New York, NY: Academic Press, 1992. [48] J. Davidson, “Classification of lattice transformations in image processing,” Computer Vision, Graphics, and Image Processing: Image Understanding, vol. 57, pp. 283–306, May 1993. [49] P. Miller, “Development of a mathematical structure for image processing,” optical division tech. rep., Perkin-Elmer, 1983. [50] J. Wilson, D. Wilson, G. Ritter, and D. Langhorne, “Image algebra FORTRAN language, version 3.0,” Tech. Rep. TR-89–03, University of Florida CIS Department, Gainesville, 1989. [51] J. Wilson, “An introduction to image algebra Ada,” in Image Algebra and Morphological Image Processing II, vol. 1568 of Proceedings of SPIE, (San Diego, CA), pp. 101–112, July 1991. [52] J. Wilson, G. Fischer, and G. Ritter, “Implementation and use of an image processing algebra for programming massively parallel computers,” in Frontiers ’88: The Second Symposium on the Frontiers of Massively Parallel Computation, (Fairfax, VA), pp. 587– 594, 1988. [53] G. Fischer and M. Rowlee, “Computation of disparity in stereo images on the Connection Machine,” in Image Algebra and Morphological Image Processing, vol. 1350 of Proceedings of SPIE, pp. 328–334, 1990. [54] D. Crookes, P. Morrow, and P. McParland, “An algebra-based language for image processing on transputers,” IEE Third Annual Conference on Image Processing and its Applications, vol. 307, pp. 457–461, July 1989. [55] D. Crookes, P. Morrow, and P. McParland, “An implementation of image algebra on transputers,” tech. rep., Department of Computer Science, Queen’s University of Belfast, Northern Ireland, 1990. [56] J. Wilson, “Supporting image algebra in the C++ language,” in Image Algebra and Morphological Image Processing IV, vol. 2030 of Proceedings of SPIE, (San Diego, CA), pp. 315–326, July 1993. [57] University of Florida Center for Computer Vision and Visualization, Software User Manual for the iac++ Class Library, Version 1.0, 1994. Available via anonymous ftp from ftp.cise.ufl.edu in /pub/ia/documents. [58] G. Birkhoff and J. Lipson, “Heterogeneous algebras,” Journal of Combinatorial Theory, vol. 8, pp. 115–133, 1970.

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53

[59] University of Florida Center for Computer Vision and Visualization, Software User Manual for the iac++ Class Library, Version 2.0, 1995. Available via anonymous ftp from ftp.cise.ufl.edu in /pub/ia/documents. [60] G. Ritter, J. Davidson, and J. Wilson, “Beyond mathematical morphology,” in Visual Communication and Image Processing II, vol. 845 of Proceedings of SPIE, (Cambridge, MA), pp. 260–269, Oct. 1987. [61] D. Li, Recursive Operations in Image Algebra and Their Applications to Image Processing. Ph.D. thesis, University of Florida, Gainesville, FL, 1990. [62] D. Li and G. Ritter, “Recursive operations in image algebra,” in Image Algebra and Morphological Image Processing, vol. 1350 of Proceedings of SPIE, (San Diego, CA), July 1990. [63] R. Gonzalez and P. Wintz, Digital Image Processing. Reading, MA: Addison-Wesley, second ed., 1987. [64] G. Ritter, “Heterogeneous matrix products,” in Image Algebra and Morphological Image Processing II, vol. 1568 of Proceedings of SPIE, (San Diego, CA), pp. 92–100, July 1991. [65] R. Cuninghame-Green, Minimax Algebra: Lecture Notes in Economics and Mathematical Systems 166. New York: Springer-Verlag, 1979. [66] G. Ritter and H. Zhu, “The generalized matrix product and its applications,” Journal of Mathematical Imaging and Vision, vol. 1(3), pp. 201–213, 1992. [67] H. Zhu and G. X. Ritter, “The generalized matrix product and the wavelet transform,” Journal of Mathematical Imaging and Vision, vol. 3, no. 1, pp. 95–104, 1993. [68] H. Zhu and G. X. Ritter, “The p-product and its applications in signal processing,” SIAM Journal of Matrix Analysis and Applications, vol. 16, no. 2, pp. 579–601, 1995.

© 2001 by CRC Press LLC

CHAPTER 2 IMAGE ENHANCEMENT TECHNIQUES

2.1.

Introduction

The purpose of image enhancement is to improve the visual appearance of an image, or to transform an image into a form that is better suited for human interpretation or machine analysis. Although there exists a multitude of image enhancement techniques, surprisingly, there does not exist a corresponding unifying theory of image enhancement. This is due to the absence of a general standard of image quality that could serve as a design criterion for image enhancement algorithms. In this chapter we consider several techniques that have proved useful in a wide variety of applications.

2.2.

Averaging of Multiple Images

The purpose of averaging multiple images is to obtain an enhanced image by averaging the intensities of several images of the same scene. A detailed discussion concerning rationale and methodology can be found in Gonzalez and Wintz [1].

Image Algebra Formulation

„†…ˆ‡ŠŽO‰‹1”‹4‹Œ‰m‘P ’ , ,letis ŽBgiven 3:‘“by’ be a family of images of the same scene. ›mŸ š  › “ œ ž ‡ Ž'•–… —™˜ Ž

For enhanced image,

For actual implementation the summation will probably involve the loop

Ž ¢8£• ¤¦…¥“¡ §4¨ Ÿ©Ÿ —|› ª £E£¬« ‡ ® ¨8Ž3¯ •–ª…£°Žˆ£¬«Ÿ ­Ž Ž • … —‡ Ž © 2001 by CRC Press LLC

The

Source Image

+η

1

+η

+η

2

k

Noisy Images

...

1 k

Σ

k=5

k = 20

Averaged Images

k = 100

Figure 2.2.1. Averaging of multiple images for different values of k. Additional explanations are given in the comments section.

Comments and Observations images is applicable when several noise degraded images, ±A²1³´±dµ³¶1¶4¶mAveraging ³·±d¸ , of themultiple same scene exist. Each ±8¹ is assumed to have pixel values of the form ±8¹·º»½¼)¾¿±dÀºRÁ¼AÂqù†ºRÁ¼4³ where ±dÀ is the true (uncorrupted by noise) image and ù·ºÁ¼ is a random variable representing the introduction of noise (see Figure 2.2.1). The averaging multiple images technique assumes that the noise is uncorrelated and has mean equal zero. Under these assumptions

© 2001 by CRC Press LLC

Ì ÅBÆÇÈ|ÉËÌÊ Î?Í Ï Å Î Æ ÇÈ approaches ÅdÐÑÆÇÈ . Thus, by averaging multiple images, it may be possible to assuageÊ degradation due

the law of large numbers guarantees that as

Ä

increases,

to noise. Clearly, it is necessary that the noisy images be registered so that corresponding pixels line up correctly.

2.3.

Local Averaging

Local averaging smooths an image by locally reducing the variation in intensities. This is done by replacing the intensity level at a point by the average of the intensities in a neighborhood of the point. a neighborhood of with Specifically, if denotes the source image and , then the enhanced image is given by

Õ·ÖŠ×´Ø Æ´ÒvÆRÓÔÈ·ÈÔÉ~ÙÛÚ9ÓwÜvÅ ÝÞ

ß ÒvÆRÓÔÈ ß ÆRÓÔÈPÉaÙ à â°ãEá ä åçæŠè ÅAƆÇÈé

Ó

Additional details about the effects of this simple technique can be found in Gonzalez and Wintz [1].

Image Algebra Formulation

Let Å{ÜvêPë be the source image, and ÒvÆRÓÔÈ-ìGÝAÞ a predefined neighborhood of the averaging function (see Section 1.7). The result Óimage wÜvÝAÞ . Let í, îderived êPëPï ðòñÛby ê localdenoteaveraging from Å{ÜvêPë is given by: ßÜzêPë ßgî–ÉaÅQô™ó õ é

Comments and Observations Local averaging traditionally imparts an artifact to the boundary of its result image. This is because the number of neighbors is smaller at the boundary of an image, so the average should be computed over fewer values. Simply dividing the sum of those neighbors by a fixed constant will not yield an accurate average. The image algebra specification does not yield such an artifact because the average of pixels is computed from the set of neighbors of each image pixel. No fixed divisor is specified.

2.4.

Variable Local Averaging

Variable local averaging smooths an image by reducing the variation in intensities locally. This is done by replacing the intensity level at a point by the average of the intensities in a neighborhood of the point. In contrast to local averaging, this technique allows the size of the neighborhood configuration to vary. This is desirable for images that exhibit higher noise degradation toward the edges of the image [2, 3]. The actual mathematical formulation of this method is as follows. Suppose denotes the source image and a neighborhood function. If denotes the number of points in , then the enhanced image is given by

ÅyÜòê ë

© 2001 by CRC Press LLC

ë ûÒ ÆÓÔÈüìýö Ò
îö÷ñùø ß ÆRÓÔÈ)É Ù à æ â°ã¬á ä åþæEè ÅAÆÇÈ1é

ß

úæ

Image Algebra Formulation

ÿ


    aÿ  

denote the source image and the specific neighborhood Let configuration function. The enhanced image is now given by

Comments and Observations Although this technique is computationally more intense than local averaging, it may be more desirable if variations in noise degradation in different image regions can is translation be determined beforehand by statistical or other methods. Note that if invariant, then the technique reduces to local averaging.



2.5.

Iterative Conditional Local Averaging

The goal of iterative conditional local averaging is to reduce additive noise in approximately piecewise constant images without blurring of edges. The method presented here is a simplified version of one of several methods proposed by Lev, Zucker, and , is replaced by Rosenfeld [4]. In this method, the value of the image at location , the average of the pixel values in a neighborhood of whose values are approximately the . The method is iterated (usually four to six times) until the image assumes same as the right visual fidelity as judged by a human observer. and for , let denote the For the precise formulation, let is a Moore neighborhood. Define desired neighborhood of . Usually,



ÿ

ÿ

 ÿ 

 !     ÿ
"$ #" % '&)(*+,-/. ÿ021ÿ0(3).547698;: where < denotes a user-defined threshold, and set =>?A@CBDFEGFHI0J . 

The conditional local averaging operation has the form

L M = NPORPQ SUT;V ÿKXW3YX(3Z:  ÿ. where ÿK5[ is the value at the \ th iteration and ÿ/]^¿

ÿK5?

Image Algebra Formulation

ÿ


`_ ab 

denote the source image and the desired neighborLet hood function. Select an appropriate threshold and define the following parameterized neighborhood function:

c

<

% ÿde0I&X(   _ . ÿ021ÿ(3X.54f<98 

The iterative conditional local averaging algorithm can now be written as

where

ÿ ] G ÿ .

© 2001 by CRC Press LLC

ÿhg _ aÿhgXW3Y+ i % †ÿ/j W3Ykl:

2.6.

Gaussian Smoothing

Gaussian smoothing is employed in numerous image processing and computer vision algorithms. As pointed out by Marr and Hildreth [5], there are two considerations in choosing a smoothing filter. First, it must reduce the range of scales over which intensity changes occur, thus it should be smooth and relatively band-limited in the frequency domain. Second, it should represent an averaging of nearby points, thus it should be smooth and localized in the spatial domain as well. Simple spatial domain smoothing filters such as local averaging address the second consideration but fail to satisfy the first. In fact, these are in mutual conflict. Leipnik [6] has shown that the filter that provides localization best in both the spatial and frequency domains is the Gaussian, namely

m$n[odpIq

r s2t 5u v^wx5z0{ ykz-z |

m$no ~p q F€ 5 ‚ y z0z { xhz ƒ z . |F} „ of z anw ximage …

and in two dimensions, The Gaussian smoothing Gaussian :

m

n„ o Ip q … n o p†^m$no ?p q ˆ ‡ |C} C| } |C} ‰ ‡

is achieved by convolving it with the

ˆ ‡ n Š pFmn[o$Ž Š Ž pJRŠ … |Œ‹ |C} ‹ ‹ ‰ ‡

Image Algebra Formulation

…’‘”“•

–

for some 2–dimensional point set be the source image. If we Let to have unbounded extent, then the Gaussian smoothing of were to imagine the set image would given by

…

where

™ ‘ ›“•^œ •

–

„

„ q …
˜š — ™

is defined by

™lUž;Ÿ  X¡ n[¢ ?p q r € |C£ u;vs w x¥¤ yZx¥¦[§¨z0z©{ xPz ¤ªƒJxP« §¬z  In practice, we must smooth bounded images with a bounded template. In such cases, truncating the extent of the template will lead to introduction of a bias by the smoothing even at those points where the template support is entirely contained within the image point set. This is due to the integral of the truncated Gaussian being less than one. To achieve an unbiased result, the template must be constructed as follows:

where

­

™XUž;Ÿ  Z¡ n¢ Ip q r € |C£ u;­hvs w xP¤yZxP¦F§ z0z { xPz ¤¨ƒJxR«;§ z is given by

­ q

© 2001 by CRC Press LLC

® Už5Ÿ  Z¡0¯;° n[±

¤²³ ²´§

r up 5vs € w xy z xPz ƒ z  0z {

Comments and Observations

µ$¶[·2¸F¹Pºf»¼µ¶·/ºJµ$¶[¹5º

. This The Gaussian is a separable function, that is, means that convolutions with the 2–dimensional Gaussian can be computed with iterated convolutions of 1–dimensional Gaussians, namely,

½ ¶F·3¸C¹;º¾^µ¶[·2¸k¹5ºI» ½ ¶F·2¸k¹5º¾^µ$¶[·dº¾^µ$¶[¹5ºX¿  Á ÃXÆ ¿ À »'¶ ½
šÁ ÃÄ º Å

This can be computed with two image-template linear product operations:

where

and

ÃÇFÈUÉ Ê Ë)Ì ¶[Í2¸k¹5ºI» Ï ÇCÐÎÑ Ò;Ó^Ô>ÕhÖ ×lÕhØFÙªÚ ¸ ÚÛ Ú Ã Æ È¬É;Ê ËZÌ ¶F·2¸FÜ;ºI» Ï Æ Ð ÎÑ Ò;Ó^Ô>ÕPÖUÝFÕPÞ[Ù Ú ¸ ÚÛ Ú ÏÇ» ÐÑ Î Ò5Ó Ô Õ× Ú Ú ¸ ȨÉ5Ê Ë)Ì0à;ß á ÚÛ ¶âCã º Ö¨äå ä0Ù ÏÆ » ÐÑ Î Ò5Ó^Ô ÕPÝÚ Ú ¿ ÈUÉ;Ê ËZÌ0à;ß á ÚÛ ¶â Öäå ä0Ù º Ã Ú Choice of the support of the template must be made wisely, balancing the

template’s size against the properties of the smoothing filter. Too small an extent will yield a filter that no longer provides good locality in both space and frequency. Too large an extent will require extra processing time for no measurable benefit. An appropriate choice of size should be driven by the standard deviation of the Gaussian, . More than 95% of the energy in the Gaussian is contained within a radius units from the origin and more than 99% of the area is contained within a radius of units. Either of these represent reasonable choices for template radius for a variety of of applications.

Ò;Ð æÐ

2.7.

Ð

Max-Min Sharpening Transform

The max-min sharpening transform is an image enhancement technique which sharpens fuzzy boundaries and brings fuzzy gray level objects into focus. It also smoothens isolated peaks or valleys. It is an iterative technique that compares maximum and minimum values with respect to the central pixel value in a small neighborhood. The central pixel value is replaced by whichever of the extrema in its neighborhood is closest to its value. The following specification of the max-min sharpening transform was formulated be the source image and denote a symmetric in Kramer and Bruchner [7]. Let . Define neighborhood of

ë ç”í

The sharpening transform



ïú>ïñòJòIó

½~çè3é

½GîðïñòIóõô÷öXøùXúïû3ò
ü9û,ý+þï0ñò©ÿ
ú 9ïñòó’ ô ùXúïû3ò
ü9û,ý þïìñò©ÿ 

 is defined as ú ,ïñò  Iú ,ïñò*úïñò ú ïñò!"#

The procedure can be iterated as

%$%&' © 2001 by CRC Press LLC

ê¶ìëº

ïú>ïñòJòIó  ï ($ ï0úïñòJòFò)

ú>ïñò ú
9ïñò

Image Algebra Formulation Let *,+.-/ be the source image, and 0214365 a desired neighborhood of The max-min sharpened image ; is given by the following algorithm:

37+98
: .

BDC *< = >?*A@ B C * EF=G>H*I J =G>K*LNMO*
P,QSR,TU* J J ;,=G>KVXWZY%1 5TU* L MOVX[
Y(1 5T)* P]\ The algorithm is usually iterated until ; stabilizes or objects in the image have assumed desirable fidelity (as judged by a human observer).

Comments and Observations Figure 2.7.1 is a blurred image of four Chinese characters. Figure 2.7.2 show the results, after convergence, of applying the max-min sharpening algorithm to the blurred C characters. Convergence required ^_R` iterations. The neighborhood used in this example is the von Neumann neighborhood.

Figure 2.7.1. Blurred characters.

© 2001 by CRC Press LLC

Figure 2.7.2. Result of applying max-min sharpening to blurred characters.

2.8.

Smoothing Binary Images by Association

The purpose of this smoothing method is to reduce the effects of noise in binary pictures. The basic idea is that the 1-elements due to noise are scattered uniformly while the 1-elements due to message information tend to be clustered together. The original image is partitioned into rectangular regions. If the number of 1’s in each region exceeds a given threshold, then the region is not changed; otherwise, the 1’s are set to zero. The regions are then treated as single cells, a cell being assigned a 1 if there is at least one 1 in the corresponding region and 0 otherwise. This new collection of cells can be viewed as a lower resolution image. The pixelwise minimum of the lower resolution image and the original image provides for the smoothed version of the original image. The smoothened version of the original image can again be partitioned by viewing the cells of the lower resolution image as pixels and partitioning these pixels into regions subject to the same threshold procedure. The precise specification of this algorithm is given by the image algebra formulation below.

Image Algebra Formulation Let T denote a given threshold and aNbdc(egf_h%i)j be the source image with kmlon
p . For a fixed integer qsrut , define a neighborhood function vxwyqz|{gkm}Kt j by

~ Here

vxwyqz4yw€z‚„ƒ
…xbNk†{ˆ‡

…

qX‰

‚Š‡

€

q‰
‹Œ

)Ž‘ means that if …ˆ‚’w”“f•gz , then )Ž –—‚Š˜)™gf–›šœ( . The smoothed image a
žbdc(efh)ij is computed by using the statement § aŸž {G‚]a¢¡s£X¤¥waN¦ v¨w”qz"z

© 2001 by CRC Press LLC

Œ

2. 8 Smoothing Binary Images by Association

63

If recursion is desired, define

©ª« ¬­©¢®°¯²±³´©ª4µ·¶¹­ ¸ º ´”»¢¸­¼
½7¾_¿"¿ for

¼ÁÀà, where ©‘Äd¬Å© .

The recursion algorithm may reintroduce pixels with values 1 that had been eliminated at a previous stage. The following alternative recursion formulation avoids this phenomenon:

© ª «G¬]© ª4µ¶ ®s¯ ±³ ´y© ª4µ¶ ¹7 ¸ º ´”»,¸7¼‘½Æ¾)¿¿Ç

Comments and Observations Figures 2.8.1 through 2.8.5 provide an example of this smoothing algorithm for

»x¬ÉÈ and Ê ¬ËÈ . Note that º ´»¿ partitions the point set X into disjoint subsets since Ì º ´”»¿”Íy´Î6¿¬ Ì º ´”»¿”́´4Ï¿xÐ Ñ)Ò Ó–ÔS¬ÅÑ_ÓgÕ Ô . Obviously, the larger Ì the k, the larger the ̺ º ´y»number ¿4Í4´Î¿ as pixels forming size of the cells Ì º ´y»¿4Í4´Î¿ . In the iteration, one views the cells ´”»,¸Æ¾¿4Íy´Î¿ . The effects of the two different iteration algorithms can the next partition be seen in Figures 2.8.4 and 2.8.5.

Figure 2.8.1. The binary source image a.

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CHAPTER 2.

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Ý7Þ

Ù4ÚÜÞ Û Figure 2.8.2. The lower-resolution image ÖX×
Ø(Ý­ left and the smoothened version ÚáNÖX×
ØفÚNÛ

Figure 2.8.3. The lower-resolution image

© 2001 by CRC Press LLC

ٔßàà is shown on the ٔßàà of a on the right.

éˆê â²ãZäå4æ·çè å”ë#ì"ì of the first iteration.

2. 9 Median Filter

65

Figure 2.8.4. The smoothened version

Figure 2.8.5. The image

÷7ø í
î,ïuíðsñXòZî%óíŸôöõ óyù#ú"ú of a.



û‘üýþû
ÿ
Züû
ÿ 

  .

As can be ascertained from Figs. 2.8.1 through 2.8.5, several problems can arise when using this smoothing method. The technique as stated will not fill in holes caused by noise. It could be modified so that it fills in the rectangular regions if the number of 1’s exceeds the threshold, but this would cause distortion of the objects in the scene. Objects that split across boundaries of adjacent regions may be eliminated by this algorithm. Also, if the image cannot be broken into rectangular regions of uniform size, other boundarysensitive techniques may need to be employed to avoid inconsistent results near the image boundary.  Additionally, the neighborhood  is a translation variant neighborhood function that needs to be computed at each pixel location y, resulting in possibly excessive computational overhead. For these reasons, morphological methods producing similar results may be preferable for image smoothing.

2.9.

Median Filter

The median filter is a smoothing technique that causes minimal edge blurring. However, it will remove isolated spikes and may destroy fine lines [1, 2, 8]. The technique

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CHAPTER 2.

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involves replacing the pixel value at each point in an image by the median of the pixel values in a neighborhood about the point. The choice of neighborhood and median selection method distinguish the various median filter algorithms. Neighborhood selection is dependent on the source image. The machine architecture will determine the best way to select the median from the neighborhood. A sampling of two median filter algorithms is presented in this section. The first is for an arbitrary neighborhood. It shows how an image-template operation can be defined that finds the median value by sorting lists. The second formulation shows how the familiar bubble sort can be used to select the median over a  neighborhood.

Image Algebra Formulation Let  be the source image,  !#"%$& a neighborhood function. Let ' (*) +,"- denote the median function described in Section 1.7. The median filtered image . is given by 4 56 . 0/132

Alternate Image Algebra Formulation

7

The alternate formulation uses a bubble sort to compute the median value over a neighborhood. Let 8( be the source image. The images 9 are obtained by

 9 0/:<; = 9?> @BADCADEF> where the functions

=9

are defined as follows:

=GIHJLK(/MJON HQP >R@ K =&STHJLK(/MJON HVU @>W@ K =RXHJLK(/MJON HVU @> PK =RYHJLK(/MJON HVU @> U @ K =&ZTHJLK(/MJON HQP > U @ K =&[THJLK(/MJON H @\> U @ K =&]THJLK(/MJON H @\> P&K 6 =R^HJLK(/MJON H @\>R@ K The median image .

is calculated with the following image algebra pseudocode:

`_ 0/: ab&c C(dIe P 6f6 gih bjbTk albcnm dReoC N @ 6p6 E h bjbk q 0/3 9sr t  t  /: 9`u  t q 9L0/ v ew h bjbk v ew h bxbk 6 .y /: Y © 2001 by CRC Press LLC

2. 9 Median Filter

67

Comments and Observations The figures below offer a comparison between the results of applying averaging filter and a median filters. Figure 2.9.1 is the source image of a noisy jet. Figures 2.9.2 and 2.9.3 show the results of applying averaging and median filter, respectively, over zn{7z neighborhoods of the noisy jet image.

Figure 2.9.1. Noisy jet image.

Figure 2.9.2. Noisy jet image smoothed with averaging filter over a

© 2001 by CRC Press LLC

|7}|

neighborhood.

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CHAPTER 2.

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Figure 2.9.3. Result of applying median filter over a ~€3~ neighborhood to the noisy jet image. Coffield [9] describes a stack comparator architecture which is particularly well suited for image processing tasks that involve order statistic filtering. His methodology is similar to the alternative image algebra formulation given above.

2.10.

Unsharp Masking

Unsharp masking involves blending an image’s high-frequency components and low-frequency components to produce an enhanced image [2, 10, 11, 12]. The blending may sharpen or blur the source image depending on the proportion of each component in the enhanced image. Enhancement takes place in the spatial domain. The precise formulation of this procedure is given in the image algebra formulation below.

Image Algebra Formulation Let ƒ‚…„(† be the source image and let ‡ be the image obtained from  by applying an averaging mask (Section 2.3). The image ‡ is the low-frequency component of the source image and the high-frequency component is €ˆ3‡ . The enhanced image ‰ produced by unsharp masking is given by

‰nŠ0‹3ŒOŽˆ‡‘“’”‡–• or, equivalently,

‰nŠ0‹3Œ€—˜’DŽš™–ˆ3Œ`‘›I‡
œ Œ”‚„ between ž and ™ results in a smoothing of the source image. A Œ greater than emphasizes the high-frequency components of the source image, which sharpens detail. Figure 2.10.1 shows the result of applying the unsharp masking technique to a mammogram for several values of Œ . A ~€8~ averaging neighborhood was used to produce the lowfrequency component image ‡ . A

™

© 2001 by CRC Press LLC

2. 10 Unsharp Masking

69

A more general formulation for unsharp masking is given by

Ÿn 0¡£¢¥¤&¦¨§˜©3ª(«L¬3­¤Iª
® where ¢–®­”¯° . Here ¢ is the weighting of the high-frequency component and weighting of the low-frequency component.

Source

γ = 0.0

γ = 0.5

γ=5

γ = 10

γ = 20

Figure 2.10.1. Unsharp masking at various values of ± . © 2001 by CRC Press LLC

­

is the

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CHAPTER 2.

IMAGE ENHANCEMENT TECHNIQUES

Comments and Observations Unsharp masking can be accomplished by simply convolving the source image with the appropriate template. For example, the unsharp masking can be done for the mammogram in Figure 2.10.1 using

²n³0´3µ·: ¶ ¸x¹

¸

where is the template defined in Figure 2.10.2. The values of À ¾x¿ Á¼ , respectively.

t=

v

v

v

v

Ãw

v

v

v

v

º

and

»

are

¼?½¿ ¾

and

Figure 2.10.2. The Moore configuration template for unsharp masking using a simple convolution.

2.11.

Local Area Contrast Enhancement

In this section we present two methods of local contrast enhancement from Harris [13] and Narendra and Fitch [14]. Each is a form of unsharp masking (Section 2.10) in which the weighting of the high-frequency component is a function of local standard deviation.

Image Algebra Formulation Let ÄÅÆÇ , and N be a neighborhood selected for local averaging (Section 2.3). The von Neumann or Moore neighborhoods are the most commonly used neighborhoods for this unsharp masking technique. The local mean image of Ä with respect to this neighborhood function is given by

ÏÑ ÈÊÉ0Ë ÄÍÎÐ Ì

The local standard deviation of

Ä

is given by the image

ΠπÜnÞÝ Ñ Û Ò É0Ë3ÓOÔMÕ&Ö ÄØ× È£ÙÚ Ì Î-ÏÜnÞÝ

Ӑßáà The image ÕRÖ ÄØ× È£ÙÚ Ì Ò actually represents the local standard deviation while is a lower bound applied to in order to avoid problems with division by zero in the next step of this technique. The enhancement technique of [13] is a high-frequency emphasis scheme which uses local standard deviation to control gain. The enhanced image â is given by Ñ Ò ã å Ö ÄØ× È£Ù â 0É Ëä

As seen in Section 2.10, Ò Ä× È represents a highpass filtering of Ä in the spatial domain. The local gain factor ælç is inversely proportional to the local standard deviation. Thus, © 2001 by CRC Press LLC

2. 12 Histogram Equalization

71

a larger gain will be applied to regions with low contrast. This enhancement technique is useful for images whose information of interest lies in the high-frequency component and whose contrast levels vary widely from region to region. The local contrast enhancement technique of Narendra and Fitch [14] is similar to the one above, except for a slightly different gain factor and the addition of a lowfrequency component. In this technique, let

èêé ë í—îjïWðFì ñòsó›ô denote the global mean of ò . The enhanced image

ò õ

is given by

è õ é ë÷öù ø ø ñò7ú3û£ó“ü3ûþý where ÿ ö ì The addition of the low-frequency component û is used to restore the average intensity level within local regions. We need to remark that in both techniques above it may be necessary to put limits on the gain factor to prevent excessive gain variations.




2.12.



Histogram Equalization

Histogram equalization is a technique which rescales the range of an image’s pixel values to produce an enhanced image whose pixel values are more uniformly distributed [15, 1, 16]. The enhanced image tends to have higher contrast. denote the source image, The mathematical formulation is as follows. Let ò Oë Fñ ó , and be the number of times the gray level occurs in the image ò . Recall that ë ÿ ý ì ý øRøRø ý ú ì . The enhanced image õ is given by

    

  

 



#%$'& +  õ ñ Lóë !"ô (*) ,  

Image Algebra Formulation

- 

be the source image and let Let ò histogram of ò as defined in Section 10.11. The enhanced image õ is given by

.

õ é0ë ñFú ì ó ø ñ .0/ ò`ó

denote the normalized cumulative



Comments and Observations Figure 2.12.1 is an illustration of the histogram equalization process. The original image and its histograms are on the left side of the figure. The original image appears dark (or underexposed). This darkness manifests itself in a bias toward the lower end of the gray scale in the original image’s histogram. On the right side of Figure 2.12.1 is the equalized image and its histograms. The equalized histogram is distributed more uniformly. This more uniform distribution of pixel values has resulted in an image with better contrast.

© 2001 by CRC Press LLC

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CHAPTER 2.

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Original Image

Equalized Image

6000

6000

5000

5000

4000

4000

3000

3000

2000

2000

1000

1000

0 0

50

100

150

200

0

250

0

Histogram of Original Image

50

100

150

200

250

Histogram of Equalized Image

300000

300000

250000

250000

200000

200000

150000

150000

100000

100000

50000

50000

0

0

0

50

100

150

200

250

0

Cumulative Histogram of Original Image

50

100

150

200

250

Cumulative Histogram of Equalized Image

Figure 2.12.1. Left: Original image and histograms. Right: Equalized image and histograms.

2.13.

Histogram Modification

The histogram equalization technique of Section 2.12 is a specific example of image enhancement using histogram modification. The goal of the histogram modification technique is to adjust the distribution of an image’s pixel values so as to produce an enhanced image in which certain gray level ranges are highlighted. In the case of histogram equalization, the goal is to enhance the contrast of an image by producing an enhanced image in which the pixel values are evenly distributed, i.e., the histogram of the enhanced image is flat. Histogram modification is accomplished via a transfer function

132546879;:

© 2001 by CRC Press LLC

2. 14 Lowpass Filtering

73

F = F ? @ F @ FHGAI < =><?A@B<
@C<+DE=

represents a gray level value of the source image and where the variable is the variable that represents a gray level value of the enhanced image. The method of deriving a transfer function to produce an enhanced image with a prescribed distribution of pixel values can be found in either Gonzalez and Wintz [1] or Pratt [2]. Table 2.13.1 lists the transfer functions for several of the histogram modification is the cumulative probability distribution of examples featured in Pratt [2]. In the table the input image and is an empirically derived constant. The image algebra formulations for histogram modification based on the transfer functions in Table 2.13.1 are presented next.

JLKNMPO

Q

Table 2.13.1 Histogram Transfer Functions Name of modification Uniform modification: Exponential modification:

Transfer function T(z)

RTSVUWRXZY[R;\]^_`'ab[R;\ RTS[R \ Y c[\ dfe U%gHY^_`'ah]

Rayleigh modification:

\z R3SiR \ b,jWk'lnm d%epo \qsrE\ tvu+wxy m

Hyperbolic cube root modification:

| R3S o j{R X h\ zh| } Y R \ \zN| y UW^_`'a8]~b5R \ \zN| x

Hyperbolic logarithmic modification:

r*t{u+w

R3SiR \ j ‚ €  y

Image Algebra Formulation

ƒ…„-† ‡ ˆ

‰

denote the source image and the normalized cumulative histogram Let and denote the grey value bounds for the of as defined in Section 10.11. Let enhanced image. Table 2.13.2 below describes the image algebra formulation for obtaining the enhanced image using the histogram transform functions defined in Table 2.13.1.

Š

‹Œ

Ž



2.14.

Lowpass Filtering

Lowpass filtering is an image enhancement process used to attenuate the highfrequency components of an image’s Fourier transform [1]. Since high-frequency components are associated with sharp edges, lowpass filtering has the effect of smoothing the image. where . Let denote the Fourier transform of , Suppose and denote the lowpass filter transfer function. The enhanced image is given by

Ÿž

Š…‘-’”“

•—–™˜ši›œ˜ 

Šž

 E¡¢*£+¤¥>–™¦¨§ HŒ © EŠ ž ¡ª£«'¥­¬ Ÿ ž ¡ª£;«¥;®n£ Ÿ Œ where ¯ is the filter which attenuates high frequencies, and ¦ §

 

Š

denotes the inverse Fourier transform. Sections 8.2 and 8.4 present image algebra implementations of the Fourier transform and its inverse.

© 2001 by CRC Press LLC

74

CHAPTER 2.

IMAGE ENHANCEMENT TECHNIQUES

Table 2.13.2 Image Algebra Formulation of Histogram Transfer Functions Name of modification

Image algebra formulation

Uniform modification:

°²±´³¶µf·¸º¹}·»¼ ½ ¾A¿”ÀÂÁÃ5·;»

Exponential modification:

°Ä±´³Å· » ¹ Æ» %Ç È µfɔ¹}½ ¾0¿”ÀÁ ¼

Rayleigh modification:

» Û °Ä± ³Ê·»ËÃÍÌ%Î'Ï*Ð Ç%ÈœÑ »ÒsÓ »ÔÕNÖØׂÙÂÚ Ð

Hyperbolic cube root modification:

Ý °Ü±´³ Ñ Ì · ¸ » ÛNÝ º ¹ · » »ÛNÝ Ú ½ ¾0¿”ÀÂÁ*ÃB· » »ÛÝ Ù

Hyperbolic logarithmic modification:

ã â ½ä*åæ'Á

Let plane; that is,

°²± ³Ê·» Ì ÞNáÞ ß à Ú

be the distance from the point

½ä*åæ'Á

Ó ÔÕÖH×

to the origin of the frequency

ã â ½ä*åæ'Á­³çä Ð Ã}æ Ðáè h» Û HÐ é

The transfer function of the ideal lowpass filter is given by

É ïñð ã ã â ½+ëåì'Á”òîó ê â ½ëå;ìÁ>³îí ô ² Wï ðöõ ½ëå;ìÁ”÷øó å

where

ù

is a specified nonnegative quantity, which represents the cutoff frequency. The transfer function of the Butterworth lowpass filter of order is given by

ê â ½ëåìÁ˳ where

ÿ

É å ã É>Ãüû*Ì â ½ëåìTÁNýÂó Ú Ð þ Typical values for ÿ are É and ÎA¹ É .

ú

is a scaling constant. The transfer function of the exponential lowpass filter is given by



ê â ½ëå;ìÁ>³
  ¹  Ñ ã â ½+ëå;ìÁýÂó þ é Ù

É



ç Îè

Typical values for are and . The images that follow illustrate some of the properties of lowpass filtering. When filtering in the frequency domain, the origin of the image is assumed to be at the center of the display. The Fourier transform image has been shifted so that its origin appears at the center of the display (see Section 8.2).

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2. 14 Lowpass Filtering

75

Figure 2.14.1 is the original image of a noisy angiogram. The noise is in the form of a sinusoidal wave. Figure 2.14.2 represents the power spectrum image of the noisy angiogram. The noise component of the original image shows up as isolated spikes above and below the center of the frequency image. The noise spikes in the frequency domain are easily filtered out by an ideal lowpass filter whose cutoff frequency falls within the distance from the center of the image to the spikes. Figure 2.14.3 shows the result of applying an ideal lowpass filter to 2.14.2 and then mapping back to spatial coordinates via the inverse Fourier transform. Note how the washboard effect of the sinusoidal noise has been removed. One artifact of lowpass filtering is the “ringing” which can be seen in Figure 2.14.2. Ringing is caused by the ideal filter’s sharp cutoff between the low frequencies it lets pass and the high frequencies it suppresses. The Butterworth filter offers a smooth discrimination between frequencies, which results in less severe ringing.



Figure 2.14.1. Noisy angiogram. The lowpass filtering used above was successful in removing the noise from the angiogram; however, the filtering blurred the true image. The image of the angiogram before noise was added is seen in Figure 2.14.4. The blurring introduced by filtering is seen by comparing this image with Figure 2.14.3. Lowpass filtering blurs an image because edges and other sharp transitions are associated with the high frequency content of an image’s Fourier spectrum. The degree of blurring is related to the proportion of the spectrum’s signal power that remains after of is defined by filtering. The signal power

  !"$# 0/13254$6 0 /1798;:=< &%('*),+.+ @A The percentage of power that remains after filtering > using an ideal filter ? frequency B is DC >F? EG@ ? AH ELK,MNMO< I,>GJ  © 2001 by CRC Press LLC

with cutoff

76

CHAPTER 2.

IMAGE ENHANCEMENT TECHNIQUES

Figure 2.14.2. Fourier transform of noisy angiogram.

Figure 2.14.3. Filtered angiogram using ideal filter.

P

RQ S

P

As increases, more and higher frequencies are encompassed by the circular region that makes up the support of . Thus, as increases, the signal power of the filtered image increases and blurring decreases. The top two images of Figure 2.14.5 are those of an original image (peppers) and its power spectrum. The lower four images show the blurring that results from filtering with ideal lowpass filters whose cutoff frequencies preserve and of the original image’s signal power. The blurring caused by lowpass filtering is not always undesirable. In fact, lowpass filtering may be used as a smoothing technique.

T=UWVXT=YWVXT=ZWV

TNT=[

Image Algebra Formulation The image algebra formulation of the lowpass filter is roughly that of the

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2. 14 Lowpass Filtering

77

Figure 2.14.4. Angiogram before noise was introduced. mathematical formulation presented earlier. However, there are a few subtleties that a programmer needs to be aware of when implementing lowpass filters. The transfer function

\ ] ^`_3a7b$cedgfixhkjml5n _p_poeoeq&q&bLbLoo rLr s7s7tt ovovuzww jyl n ^ x a x c . On the other hand, represents a disk of radius d of unit height centered at the origin ‚ d p  … ƒ † „ p  ‡ , results in a complexthe Fourier transform of an image {}|5~e , where € \ ƒ a ‡ ther location of the origin with respect to { ˆ is not ˆ |DŠ . In other words, valued image {‰ n the center of X, which is the point o o , but the upper left-hand corner of { ˆ . Therefore, ˆ ‹ ] is undefined. Simply shifting the image { ˆ so that the midpoint of X the product {Œ moves to the origin, or shifting the disk image so that the disk’s center will be located at the midpoint of X, will result in properly aligned images that can be multiplied, but will not result in a lowpass filter since the high frequencies, which are located at the corners of , would be eliminated. There are various options for the correct implementation of the lowpass filter. One such options is to center the spectrum of the Fourier transform at the midpoint of X (Section 8.3) and then multiplying the centered transform by the shifted disk image. The result of this process needs to be uncentered prior to applying the inverse Fourier transform. The exact specification of the ideal lowpass filter is given by the following algorithm. and , where m and n are even integers. Define Suppose the point set

{ˆ



and set

© 2001 by CRC Press LLC

{ˆ |†Š

€ d ƒ „Ž ‡

d…fv^`3a7‘$c‰’”“=•, –˜— ™›š o q‚•,‘œ–˜™eš ož sŸt o uw*  ]5ˆ ’d¡^7¢1£ ¤vc,£ ¥G¦

78

CHAPTER 2.

Original

IMAGE ENHANCEMENT TECHNIQUES

Power spectrum

Filtered at 90%

Filtered at 93%

Filtered at 97%

Filtered at 99%

Figure 2.14.5. Lowpass filtering with ideal filter at various power levels.

§ § ¨ © « ¨¬®­¯;°W§±X¯=²´§³`µ‰¶ ª · ©«¹¸º¼»1½¾ ¨ ¿ À1Á ¬®­¯;°W±X¯,²N³9µÂ¶ŸÃOÄ

The algorithm now becomes

In this algorithm, 1 and 0 denote the complex-valued unit and zero image, respectively.

«ÇÆ1³7ȼÉËÊL¶‰©‰Ì9ÈOÍÎ&ÊNÍÏ »7Ð ÍvÑÓÒ®Ô instead, then it becomes Å À À necessary to define Õ as Õ ©«×ÖÙØÛÚ Ü”Î Ì,Ý Í É=Þ Í Ï,ß Ú à , where the extension to 0 is on the array À X, prior to multiplying ¨ Õ by Õ . Figure 2.14.6 provides a graphical interpretation of this algorithm. Another method, which avoids centering the Fourier transform, is to specify the set « ` ³ 3 È Ë É ÊL¶© Ì ÈpÍeÎáÊNÍ Ï »7Ð ÍvÑÒ Ô and translate the image Ø1Ú Ü directly to the corners of Æ Å If U is defined as

© 2001 by CRC Press LLC

2. 14 Lowpass Filtering

a

1 v




79 U



d

v

d

u u

center

shift v

v

multiply

u

u

v

u



center

v

u

inverse F.T.

Figure 2.14.6. Illustration of the basic steps involved in the ideal lowpass filter. X in order to obtain the desired multiplication result. Specifically, we obtain the following form of the lowpass filter algorithm:

   

      !  "#$%!'&( )*  +  -,/.1032457 6 1 8:9

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80

CHAPTER 2.

;=< >?

IMAGE ENHANCEMENT TECHNIQUES

means restricting the image d to the point set X and then extending d on X Here to the zero image wherever it is not defined on X. The basic steps of this algorithm are shown in Figure 2.14.7.

a

1 U

v

v

A

@

u

u multiply

shift and extend v

u

B

inverse F.T.

Figure 2.14.7. Illustration of the basic steps as specified by the alternate version of the lowpass filter.

C

As for the ideal lowpass filter, there are several way for implementing the thorder Butterworth lowpass filter. The transfer function for this filter is defined over the point set , and is defined by

DFEHGIKJ LNMOGIQPL

[

S:R TUV%WX E

LY cedfk Lg'hji L lnm%o V Y:Z\[^]_%`ba S:R T#UV%WX pR

SR

where denotes the scaling constant and the value needs to be converted to a complex value. For correct multiplication with the image , the transfer function needs to be shifted to the corners of X. The exact specification of the remainder of the algorithm is given by the following two lines of pseudocode:

S R q E S R S R TtuV(vnXw S R Tv VyXw S T#t$V%yX#w l3} ~  Zzr|{ Z  q E-€/_ 1‚Zsr p R r ] S RZ w:ƒ Zxr Z S/R TU1V(W„X E†…ˆ‡„‰7‹%Š ŒŽ=‘ U m ’ Z W m “ oˆ”• V

The transfer function for exponential lowpass filtering is given by

© 2001 by CRC Press LLC

¥ ¤ –¦—1˜%™nš

2. 15 Highpass Filtering

–#—˜™›š/œ"žQŸ  ¢¡$žQ£ 

81

where and needs to be a complex valued. The remainder of the algorithm is identical to the Butterworth filter algorithm.

2.15.

Highpass Filtering

Highpass filtering enhances the edge elements of an image based on the fact that edges and abrupt changes in gray levels are associated with the high-frequency components of an image’s Fourier transform (Sections 8.2 and 8.4). As before, suppose denotes the Fourier transform of the source image . If denotes a transfer function which attenuates low frequencies and lets high frequencies pass, then the filtered enhancement of is the inverse Fourier transform of the product of and . That is, the enhanced image is given by

§¤

§ ¥¤

§¤ ¥ ¤ ¨©–#ª˜f«›š:¬­¯®°n±³§ ² –(´˜ˆµš/¶ ¥ ² –%´˜fµ„šf·©˜

­ ®¹¸

where

§

¨

denotes the inverse Fourier transform. The formulation of the highpass transfer function is basically the complement of the lowpass transfer function. Specifically, the transfer function of the ideal highpass filter is given by

¿ ² –—1˜(™„šÈÇ ¬ÊÉ—eË:Ì͙ËÏÎ °Ð Ë

where

¥ ² –%´˜fµ›š:¬»ºNà ¼F½ ¾ ¿ ¿ ² –À´˜ˆµš‘Á»Â ½Ä¾ ¤ –%´˜fµ„š‘ÅÆÂe˜

is a specified nonnegative quantity which represents the cutoff frequency and . The transfer function of the Butterworth highpass filter of order is given by

ð

where

¥ ² –Ò´˜ˆµš:¬ º °Ó/Ô%Õ×ÖÀ؄ڛ٠° Û'ÜÝ ÞÒß×àâáäãæåÄç=è¦éê(ë„ìQîî í è#ï4ê%ïnì ï åÄç=è¦é1ê%ënì è#ï4ê%ïnìÀê ð ñ ò óõôöñ ø ÷ è%ùêfúì î»ü ýÿû þ ô ÷ è%ùêfú›ì åÄç¯è#éê%ëìKî í è#ï4ê%ïnì ï åÄç¯è#éê%ëì î è#ï4ê%ïnì ñ òó

Ñ

is a scaling constant. Typical values for are and . The transfer function of the exponential highpass filter is given by




 





Typical values for  are

and 

 .

Image Algebra Formulation set .

î

 #"$  Let !

denote the source image, where % by

> ?BADC /&1032 46587:9<;4 , and define @ =

î

'&)(+*,&)- . Specify the point

W ^a` H)WXYKNWZN[]\ _ > E FIH1JLKNMGO'PRQTSU'V G W da`fe H)WXYKNW Z []\ _ b ScU V

Once the transfer function is specified, the remainder of the algorithm is analogous to the lowpass filter algorithm. Thus, one specification would be > E gO h {

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E Xlk >j E 6 i >j X FImnJ

E e g O}|~ [  k q € E  >1o

Q

E a MpoqX'k >j X F Q

J]rsMpotXuk >jXYFmvJLrsMoswyx z =

82

CHAPTER 2.

IMAGE ENHANCEMENT TECHNIQUES

This pseudocode specification can also be used for the Butterworth and exponential highpass filter transfer functions which are described next. The Butterworth highpass filter transfer function of order ‚ with scaling constant ƒ is given by „ …†‡:ˆ‰ ŠG‹Œ

where

†I§1ˆ¨©Šq­@®¯‹±°1²3³

†¡§1ˆ]¨©Š3@ ª‹ †«ˆL«™Š Ž « ŽIL‘“’”L•—–™˜L)š›˜]œIŸž ˜¡  ˜c¢¤£¦¥ †¡§1ˆL¨™ŠG‹@†«ˆL«™Šˆ £¬¥

°s²µ

. ˜6´ for ˜ exponential highpass filtering is given by The transfer function „ ÆLÉ:Ê¶Ë ¸ ¹<º
»¼¾½¡¿tÀÂÁ à ÄsÅ §yÆÇ#¨NÆ¶È …†‡sˆ¶‰ Š‹· Ž]” «

†I§1ˆL¨NŠ3Ì ª‹ †]«fˆ«¶Š £¬¥ †I§1ˆL¨NŠG‹Ì†]«fˆ«¶Š¶Í £¬¥

2.16.

Exercises

1. Construct a synthetic image and add random perturbations to each pixel value. In your construction, choose a representation that clarifies the effects of local averaging when using the neighborhoods described below. a. b.

Implement the local averaging algorithm using the Moore neighborhood. Repeat 1.a using the skew neighborhood

N =

2.

Construct a synthetic image consisting of vertical stripes of different widths. a. b.

Implement the local averaging algorithms using the Moore neighborhood. Repeat 2.a using the template

ty =

c.

1

2

1

2

4

2

1

2

1

and multiply the resulting image by 1/16. Explain the different effects of these two smoothing operations.

3. Consider the following algorithm. At each pixel location, calculate the difference Î between the pixel values of the two vertical neighbors above and below the pixel. Calculate the difference Ï between the pixel values of the two horizontal neighbors to the left and right of the pixel. If Î exceeds Ï , then the value of the pixel is replaced by the weighted average of the pixel and its two horizontal neighbors. Otherwise, it is replaced by the average of the pixel and its two vertical neighbors. a. b. c.

Write an image algebra formulation of this algorithm. Implement the algorithm. The algorithm can be implemented in terms of templates with weights ÐÒÑ and ÓÔÑ . Investigate the effects of this algorithm for different combinations of weights and also for repeated application.

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2. 16 Exercises

83

4. Prove that the Max-Min Sharpening Transform stabilizes, i.e., show that there exists a positive integer Õ such that Ö ×sØÚÙyÛDÜ'֛ÝÞØcÙyÛ for all integers ßnà@Õ . 5.

æ}ç

ØcÙâáäãGÛjå Show that while æ#ç æYç æ#ç

ØÚÙ<áëãÛ3ì

ܱÙ,ì

áíã6ì

æ}ç ÜèÙéå

æ¯ç áÌãêå

, it is not true in general that . This shows that local averaging is a linear operation

while median filtering is not. 6. A generalization of unsharp masking that is often used for object enhancement and background suppressions is given by the operation ãØIîpïÚð ÛÜêñ

ÙsØIõïLßÛ1ö ñ

ÙsØIõ¡ïßÛñ ñ

ñ

ñ ñ

çÒùñò óô

ñ

ò óø÷

ñ

where çâ is an ÕäúûÕ rectangular neighborhood centered at Ù denotes the input image, ü Øî]ï¡ðÛ , ú ý rectangular neighborhood centered at ØIîpï¡ð Û , and Õ,þBý . is an ý±v a. b. c. d.

Provide an image algebra formulation of this algorithm. Implement this algorithm on a synthetic image where the object intensity is greater ç ù . than the background and the object size can be covered by Repeat 6.b except that the object intensity is less thanç the background. çü ù but less than . Repeat 6.b except that the object size is greater than

7. Implement the Gaussian smoothing algorithm on at least two different images using different template support. Analyze your results. 8.* Implement the five histogram modifications described in Table 2.13.2 and describe their effects and differences when applied to an image. 9.* Develop a program to display histogram plots of digital images similar to those shown in Figure 2.12.1. Test your program on suitable images and provide an image algebra formulation of your program. ç

denotes an ÕnúvÕ rectangular neighborhoodæ} and 10.* It follows from Section 1.7 that if ç æ ÿ ç ÿ ÿ ü Ù á a template with Ø
ÛGÜDØ Û for each , then the image is equivalent to Ù@å ÿ ÿ as long as one ignores boundary effects and Ø:Û Ü NÕ whenever  DØ Û . a.

Showù ù

×

b.

×  Ý

ù that ×  



the

ù

×

  

ü"!

#%$

Fourier Ý&('p×*)



transform ü+!#-,

of

ÿ 

is

given

by



  Ø 1ï Û

Ü

Ý&('p× . 

÷

Using 10.a, establish a relationship between smoothing by averaging and lowpass filtering.

11.* Implement the three highpass filters described in Section 2.15. æ

ÿ

ÿ

12.* Consider the unsharp masking algorithm given by ùã/.Ü_Ù á , where denotes an ÕÔú3Õ ÿ  ÿ  ÿ  ù rectangular template with Ø
ÛGÜ and Ø
:Û܍ö whenever 21Ü34 DØ Û . a. b.

ÿ



×0

Find the Fourier transform of . Using 12.a, establish a relationship between unsharp masking and highpass filtering.

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84

2.17.

CHAPTER 2. IMAGE ENHANCEMENT TECHNIQUES

References

[1] R. Gonzalez and P. Wintz, Digital Image Processing. Reading, MA: Addison-Wesley, second ed., 1987. [2] W. Pratt, Digital Image Processing. New York: John Wiley, 1978. [3] G. Ritter, J. Wilson, and J. Davidson, “Image algebra: An overview,” Computer Vision, Graphics, and Image Processing, vol. 49, pp. 297–331, Mar. 1990. [4] A. Lev, S. Zucker, and A. Rosenfeld, “Iterative enhancement of noisy images,” IEEE Transactions on Systems, Man, and Cybernetics, vol. SMC-7, pp. 435–442, June 1977. [5] D. Marr and E. Hildreth, “Theory of edge detection,” in Proceedings of the Royal Society of London B, vol. 207, pp. 187–217, 1980. [6] R. Leipnik, “The extended entropy uncertainty principle,” Information and Control, vol. 3, pp. 18–25, 1960. [7] H. Kramer and J. Bruchner, “Iterations of a non-linear transformation for the enhancement of digital images,” Pattern Recognition, vol. 7, pp. 53–58, 1975. [8] R. Haralick and L. Shapiro, Computer and Robot Vision, vol. I. New York: AddisonWesley, 1992. [9] P. Coffield, “An architecture for processing image algebra operations,” in Image Algebra and Morphological Image Processing III, vol. 1769 of Proceedings of SPIE, (San Diego, CA), pp. 178–189, July 1992. [10] R. Gonzalez and R. Woods, Digital Image Processing. Reading, MA: Addison-Wesley, 1992. [11] L. Loo, K. Doi, and C. Metz, “Investigation of basic imaging properties in digital radiography 4. Effect of unsharp masking on the detectability of simple patterns,” Medical Physics, vol. 12