Computer Vision Algorithms in Image Algebra

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H A N D B O O K O F

s e c o n d e d i t i o n

Computer Vision Algorithms in

Image Algebra

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Boca Raton London New York Washington, D.C.

CRC Press

Gerhard X. Ritter Joseph N. Wilson

H A N D B O O K O F

s e c o n d e d i t i o n

Computer Vision Algorithms in

Image Algebra

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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.

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

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

disclaimer Page 1 Monday, August 21, 2000 2:37 PM

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Preface

The present edition differs from the first in several significant aspects. Typo- graphical 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 presentiac++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 Theiac++library supports the use of image algebra in the C++ programming language and is available via anonymous ftp fromftp://ftp.cise.ufl.edu/pub/src/ia/.

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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 theiac++library. We also wish to thank Mr. Jeffrey Palm for preparing the fractal and iterated function system images.

Wewish to express our gratitude to those at Wright Laboratory for their encour- agement 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.

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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 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.

%+*

The set

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

.

%'&

* The set

%'&

*

,

"!#!!#.+$

.

%)2 *

The set

%'2 * 3 /4.6578#!"!"!/98 #8#!!#!:.;/<=$

.

> >?& >@( >BADC

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

E

The set of complex numbers.

F

An arbitrary set of values.

F?G

The set

F

unioned with

H $

.

F ( G

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

F?2

G

The set

F

unioned with

/ H H $

.

I

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

=J The power set of (the set of all subsets of ).

K "is an element of."

L

K "is not an element of."

M "is a subset of."

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Symbol Explanation

NPORQ

Union

NTSUQPV3WX6YX6Z[N]\^X6Z[Q0_

.

O

`8a-b N `

Let ^WN ^` ^_ ^`8a-b be a family of sets indexed by an indexing set

c

. ^O

`8a-b N `

V3Wd[Y8d[Z[N

`fe

\^?g-hjilk"g-mhB\nk4opZ c _

.

q

O

rts)u

N r q

O

rvs)u N r

VwN

u

SUN0x?Sy#y"yS0N

q .

z

O

rts)u

N r zO

rvs)u N r

V3W#d{YdUZ[N

r+e

\^|m\}0k4~|Z{'@_

.

NPRQ

Intersection

NTUQPV3WX6YX6Z[Ng-n0XZ{Q0_

.

`8a-b N `

Let ^WN ^` ^_ ^`8a-b be a family of sets indexed by an indexing set

c

.

`8a-b N `

V3Wd[Y8d[Z[N

` e

\^g=iliopZ c _

.

q

rts)u

N r q

rvs)u N r

VwN

u

UN

x

y#y"y0N

q .

z

rts)u

N r z

rvs)u N r

V3W#d{YdUZ[N

r e

\^?g=ili~Z'?_

.

N[Q

Cartesian product

N[QPVW8d):8YdUZ{N{6Z[Q0_

.

q

rts)u

N r q

rvs)u N r

V3W-d

u

dx=#y"y#y:d

q

Yd

r

ZN

r_

.

z

rts)u

N r z

rvs)u N r

V3W-d

u

dx=:dy"y"y@Yd

r

Z{N

r _

.

q

The Cartesian product of copies of , i.e., ^qV

q

rts)u

.

N@Q

Set difference

Let ^N and^Q be subsets of some universal set ,

NT@QPV3WdUZ[NYdZ[Q0_

.

N{

Complement

N'V

@N

, where is the universal set that contains^N .

¡¢-£

¤N{

The cardinality of the set^N .

¥§¦

~

#¨

¤N{

A function that randomly selects an element from the set ^N .

Point and Point Set Operations

Symbol Explanation

©«ªR¬

If ^©

¬ Zpq

, then^©0ª®¬

V,d

u ª u

)y#y"y§d

q ª q

.

©[¯¬ If ^© ^¬ ^Zpq , then^©U¯¬ ^V,d ^u ^¯ ^u )y#y"y§d

q ¯ q

.

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Symbol Explanation

°{±² If ^{°?³²´pµ¶} , then°[±²{·3¸¹+º¼»8º-³'½"½#½§³¹

¶ »

¶¾.

°À¿=²

If ^{°?³²´pµ¶} , then°À¿=²{·,¸¹'º¿»8º#³)½#½"½§³¹

¶

¿8»

¶¾ .

°«ÁU²

If ^{°?³²´pµ¶} , then°0ÁU²·¸¹+º+Á[»8º-³'½#½"½'³¹

¶

ÁU»

¶¾.

°«ÂU²

If ^{°?³²´pµ¶} , then^{°0ÂU²·¸¹} ^º ^Â[» ^º ³'½#½"½'³¹

¶

ÂU»

¶ ¾.

°'Ã² In general, if ^{°?³²´µ|¶} , and Ã{Äµ®Å{µÆÇµ

, then

°'Ã²È·3¸¹ º Ã» º ³'½"½"½³§¹

¶

Ã»

¶ ¾.

É

Ãj° If ^É ^´µ ,^°´{µ|¶ , and Ã{Ä8µÊÅ[µÆÇµ

, then

É

Ã°·3¸

É

Ã§¹ º ³'½#½"½§³

É

Ã¹

¶ ¾.

°ÀË² If ^{°?³²´pµ¶} , then°ÀË²{·¹'º»8ºÍÌÊ¹§Î"»-Î?Ì±"±#±'ÌÊ¹

¶ » ¶ .

°ÅU² If ^{°?³²´pµ¶} , then

°Å«²·3¸¹ Î »"ÏÐÊ¹jÏ#» Î ³¹jÏ-» º ÐÈ¹ º »"Ï³¹ º » Î Ð¹ Î » º ¾.

°Ñ²

If ^°´{µ|¶ and^²´pµÒ , then

°:Ñ²·,¸¹'º"³À½"½#½§³¹

¶

³»8º³'½"½#½'³»

ÒÓ¾.

Ð° If ^°´{µ|¶ , then Ð°p·Ô¸:Ð4¹ º ³À½"½"½³§Ð4¹

¶ ¾ .

Õ

°Ö If ^°´{µ|¶ , then

Õ

°ÖÓ·3¸

Õ

¹'º¤Öj³À½"½#½§³ Õ¹ ¶ Ö ¾ .

×

°Ø If ^°´{µ|¶ , then

×

°ØÓ·3¸

×¹ ºØj³À½"½#½§³

×¹ ¶ Ø ¾ .

Ù

°§Ú If ^°´{µ|¶ , then

Ù

°§ÚD·3¸

Ù¹ º Ú³'½#½"½§³ Ù¹ ¶ Ú¾.

ÛjÜ ¸Ý°

¾ If ^°{·¸¹ ^º ^³¹ ^Î ³"½"½#½³:¹

¶ ¾

´pµ¶

, then^ÛjÜ ^¸Ý° ^¾ ^·1¹ ^Ü.

Þ °

If ^°´{µ|¶ , then ^Þ °·¹'º|ÌÊ¹§Î?Ì±"±#±'ÌÊ¹

¶ .

ß °

If ^°´{µ|¶ , then ^ß °[·7¹+º¼¹§Î6±#±"±¹

¶ .

Á|° If ^°´{µ|¶ , then Á|°·¹+º+Á[¹§Î?Áà±#±"±Á¹

¶ .

Â|° If ^°´{µ|¶ , then ^Â|°·¹ ^º ^Â[¹ ^Î Âà±#±"±Â¹

¶ .

á ° á Î

If ^°´{µ|¶ , then

á ° á Î

·àâ ¹ Îº

Ìã±#±"±Ìä¹ Î¶ .

á ° á º

·æå¹'ºåÌ7å¹§ÎåÌ±"±"±DÌãå¹ ¶ å

.

á ° á#ç

·æå¹+º=åÁå¹Î8åÁà±#±"±Áæå¹ ¶ å

.

èéÝê

¸°

¾ If ^°´{µ|¶ , then

èé¤ê

¸¤°

¾

·ë .

ì

Ìí If ^ì ^{³íïîðµ|¶} , then ^ì Ìíñ·3ò#°0Ì®²óÄô°õ´

ìö

ë è

²´í÷

.

ì

Ðí If ^ì ^{³íïîðµ|¶} , then ^ì Ð{íñ·3ò#°UÐ²óÄô°õ´

ìö

ë è

²´í÷

.

ì

Ìø If ^ì ^î1µ|¶ and^Û ^´{µ|¶ , then ^ì Ìø·3ò#°0Ì®øÄô°{´

ì ÷

.

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Symbol Explanation

ùûúü

If ^ù]ý1þ|ÿ and ^þ|ÿ , then ^ùæúü^[úü ^ù .

ù

If ^ù^ïýðþ|ÿ , then^ù ^ù .

ù!

If

ùïýðþ|ÿ

, then

ù! "#$

ù"%'&)(*

.

ù!+,

If ^ù^ïýðþ|ÿ , then

ù-+.#$/

ù.%'&)(*

ù0

.

ù21,

If

ùïýðþ|ÿ

, then

ù1.#4356798!

ù%:&)(;7

.

úù

If ^ù]ý1þ|ÿ , then ^úù^"^8ú^<= ^ù> .

?

ù

If

ù]ý1þ|ÿ

, then

?

ù"#

þA@,%'&)(!*

ù

.

BC 3Ýù-8

If ^ù]ý1þ|ÿ , then ^B$C ^3Ýù-86 the supremum of^ù . If

ù"ED$F9G4HHHIF

ÿ

4

then ^B$C 3Ýù-86JEDLK9G6KH$HHMK,

ÿ .

N ù

For a point set

ù

with total order ^O ,

LP< N ùQR

O

9PSUTV

ù 9P

.

W5&)X)3ù!8

If ^ù]ý1þ|ÿ , then WY&)XZ3ù-86

the infimum of ^ù . If

ù"

D

F

G

H$HHF

ÿ

, then WY&)XZ3ù-86J DL[

G\[

H$HH [ ÿ .

] ù

For a point set^ù with total order ^O ,

P ]

ùQR)^

O_

UTV

ù P

.

`a W

`$b 3¤ù!8

If

ù]ý1þ|ÿ

, then

`a W

`b 3ù-8

ùc3d%'&)(:efhg

`a

Bb

& b fb e b

&ZiI8

.

`

%'(j3¤ù!8

If ^ù]ý1þ|ÿ , then ^` ^%'(j3^¤ù-8Eki ^aVb` %'('WY&)%'fhWliIgmXUù

. In particular, if ^ù

D

nnn

ÿ

, then ^` ^%'(j3¤ù!8Eo&

.

Morphology

In the following table, ^p

q,Mr

and ^s denote subsets of

þÿ

.

Symbol Explanation

pt The reflection of^p across the origin^u #3wvjIvj$HHHMv8

þÿ

.

pyx The complement of^p ; i.e., ^px ^þ|ÿ^{! /*}^p .

pyz

${}|~ {

p

.

p

1-q

Minkowski addition is defined as

p

1q#${m|~ {

p F~

qy

(Section 7.2).

p

q

Minkowski subtraction is defined as^p, ^q3^pyx ^1q ^t ⁸ ^x (Section 7.2).

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Symbol Explanation

<

The opening of by is denoted ^< and is defined by

<d,4-

(Section 7.3).

<

The closing of by is denoted^< and is defined by

<dS

(Section 7.3).

/

Let

#w

be an ordered pair of structuring elements.

The hit-and-miss transform of the set

is given by

J#/4!V ,¡5¢

(Section 7.5).

Functions and Scalar Operations

Symbol Explanation

£

¤¥§¦

£

is a function from^¤ into^¦<¨

©'ª«,¬'d® £

The domain of the function

£ ¤¥§¦

is the set

¤¨

¯

¬'®±°:²

£

The range of the function ^£ ^¤¥§¦ is the set

£ U³±:³´!¤µ¢

.

£A¶A·

The inverse of the function ^£ .

¦¸

The set of all functions from

¤

into

¦

, i.e., if

£ ´-¦¸

, then

£ ¹¤.¥§¦¨

£6º»

Given a function ^£ ^S¤.¥§¦ and a subset^¼ ^¤ , the restriction of ^£ to^¼ ,^£Lº^» ^¼ ^¥§¦ , is defined by

£Lº» ¬ E £ ¬

for ^¬ ^´ ^¼ .

£Eº½

Given ^£ ^¼ ^¥§¦ and

°

4¾¥§¦

, the extension of^£ to

°

is defined by ^£Eº^½ ^³±E=¿

£ ³VÁÀÃÂ\³´

¼

° w³±ÄÀÃÂ\³´¾/Å

¼ ¨

° £

Given two functions

£

¤Æ¥§¦

and

°

4¦¥RÇ

, the composition

° £

¤¥RÇ

is defined by

° £

Èw³±E

° £

³VF

, for every^³´¤ .

£É °

Let

£

and

°

be real or complex-valued functions, then

£É ° M³±L £ U³± É ° ³V

.

£Ê °

Let ^£ and^° be real or complex-valued functions, then

£Ê ° Md³VE £ ³± Ê ° d³V

.

Ë Ê$£

Let

£

be a real or complex-valued function, and ^Ì be a real or complex number, then

£

´ÍÏÎ

,

Ë Ê£

Ðd³VE

Ë Ê £

³±

.

º£Eº º£Eºw³±6 º£ w³± º

, where^£ is a real (or complex)-valued function, and ^º^£ ^w³± ^º denotes the absolute value (or magnitude) of ^£ ^³V.

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Symbol Explanation

ÑÐÒ

The identity function ^Ñ ^{ÒÓÔÆÕÖÔ} is given by ^Ñ ^{Òm×ØÚÙ6ÛØ} .

ÜSÝ Ó!Þß

àháAâ Ô à

Õ§Ô

Ý

The projection function^Ü4Ý onto the^ã th coordinate is defined by^Ü4Ý

×dØ

âäåå$åMä Ø Ý

äå$ååMä Ø Þ

ÙLÛØ

Ý .

æç'èé

×YÔÙ

The cardinality of the set

Ô

.

æê±ëì5æ$í

×YÔÙ

A function which randomly selects an element from the set

Ô

.

Ømî!ï

For ^Ø ^ä ^ïmðñ ,^Øî!ï is the maximum of ^Ø and^ï .

Ømò!ï

For

Ø ä ïmðñ

,

Øò!ï

is the minimun of

Ø

and

ï

.

óØô

For

Øðñ

the ceiling function

óØSô

returns the smallest integer that is greater than or equal to

Ø

.

õ

Øö

For ^Øðñ the floor function

õ

ØSö

returns the largest integer that is less than or equal to ^Ø .

÷

Øùø

For

Øðñ

the round function returns the nearest integer to

Ø

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

Øú

ëé

ï

For ^Ø ^ä ^ïmðû , ^Øú ^ëé ^ïmÛ ^è if there exists^ü ^ä ^è ^ðû with

èmý

ï

such that ^ØÛï ^üÿþ ^è .

×dØVÙ

The characteristic function is defined by

×UØ±ÙEÛ Ñ

Øð

Images and Image Operations

Symbol Explanation

ä<ä

Bold, lowercase characters are used to represent images.

Image variables will usually be chosen from the beginning of the alphabet.

ð"!#

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

% ð!#

Let

!

be a set with unit ^Ñ . Then ^% denotes an image, all of whose pixel values are ^Ñ .

&

ð!# Let

!

be a set with zero

. Then ^& denotes an image, all of whose pixel values are

.

'(

The domain restriction of ^ð"!# to a subset ⁾ of^$ is defined by ^'⁽ ^Û ^+* ^× ^)-, ^! ^Ù. Thus,^'⁽ ^ð.!

(

.

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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

<LKM 4IBHN/O@M F 4

9QP , then^/10R254 ^9S .

/TUWVYX2[Z

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

J

<LK` 4 \

N/a@` F 4

9bP,^/T^UcVdX^2Z ⁴ ^98S .

/Te Let ^B and^f be subsets of the same topological space. The extension of ^/54.67 to^g ^4.61h is defined by

/TeO@M F

<ji /O@M

Fkml M 4IB

g @M Fnkml

M 4

fpo B.q

@r/T

gsFut

@v/OwxT/y[TuzRz3zT/{

F Row concatenation of images ^/ and^g , respectively the row concatenation of images ^/ ^w ^t^/ ^y ^tu|R|3|t ^/ ^{ .

}~ /g

Column concatenation of images ^/ and^g .

@C/

F If ^/A4"67 and

Nx6

, then the image

@C/

F

4.7

is given by

/ , i.e.,

@C/

F

<LK

@M

t

@M

FF

N @M F <

@C/O@

M

FFut

M

4IB

P .

/

If ^N ^f ^B and^/A4"67 , the induced image^/ ^4"6h is defined by ^/ ^<K ^@v ^t ^/O@ ^@C ^{FF F} ^N.4 ^fP .

/

g If is a binary operation on ⁶ , then an induced operation on

67 can be defined. Let ^/ ^{t g} ^4.68 ; the induced operation is given by ^/ ^g ^<LK ^@^M ^t ^@^M ^FF ^N ^@ ^F ^< ^/a@^M ^F ^g ^@^M ^Ft ^M ^4IB ^P .

/ Let

4"6

,^/A467 , and be a binary operation on ⁶ . An

induced scalar operation on images is defined by

/ <K @M

t

@M

FF

N @M F <

/O@ M

F3t M 4IB

P .

/e

Let ^/ ^{t g} ^4"7 ; ^/e ^< ^@^M ^t ^@^M ^{F F} ^N ^@^M ^F ^< ^/O@^M ^F ^e

U

Z t M 4AB.

.

¡W¢£

e /

Let ^/ ^{t g} ^{4@ ¤} ^F⁷ ;

¡W¢3£

e / <j¥ @M t

@M

FF

N @M F <

¡c¢3£

e

U

Z

/O@

Ft M 4AB§¦

.

/©¨ If ^/A4"67 and⁶ has a conjugation operation *, then the

pointwise conjugate of image ^/ ,^/d¨@^M ^F ^< ^@r/O@^M ^{F F} ^¨ .

ª / ª /

denotes reduction by a generic reduce operation

ª

Nx67-6

(Section 1.4).

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 .

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Symbol Explanation

«¬ «E¬¯®°¬O±v²³´Oµ^¬O±v²1¶3´Oµ·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

Dot product,

¬+ÀbÁ.®ÃÂ±r¬=·ÁQ´®Ä«

ÅÆÇ

±v¬a±È²1´·uÁb±v²1´ ´

.

É¬

Complementation of a set-valued image

¬

.

¬©Ê

Complementation of a Boolean image

¬

.

¬Ë

Transpose of image

¬

.

Templates and Template Operations

Symbol Explanation

ÌÎÍ ÏÍCÐ

Bold, lowercase characters are used to represent templates.

Usually characters from the middle of the alphabet are used as template variables.

ÏÒÑÔÓCÕ ÇbÖR×

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 .

Ï3Ü

Let^ÏÒÑ ^Ó^Õ ^Ç ^Ö ^× . For each^Ý ^Ñ ^Ø ,^Ï ^Ü

® Ï ±Ý ´

. The image

Ï3Ü¯Ñ"Õ

Ç

is given by^ÏRÜ

®ÞÎ±v²

ÍÏRÜ

±v²1´´

Ú ² Ñ

Ùß .

à ±

ÏRÜ

´

If ^Õ;Ñ

Þá

Í â

ß and^ÏÒÑ ^Ó ^Õ

Ç Ö ×

, then the support of ^Ï is denoted by

à ±Ï Ü ´

and is defined by

à ±Ï Ü

´Q®LÞ²

Ñ Ù ÚÏ Ü

±r²1´Òã®Ãä

ß .

à8å

±Ï Ü ´

If^ÏÒÑ ^Ó

á Çå Ö ×

, then ^àOå

±Ï Ü

´Q®LÞ²

Ñ Ù ÚÏ

Ü

±r²1´Òã®Ãæ

ß .

àsç å ±Ï Ü ´

If^ÏÒÑ ^Ó

á

Çç

å Ö ×

, then

à ç å ±

ÏRÜ

´Q®LÞ²

Ñ Ù ÚÏRÜ

±v²1´èã®éæ

ß .

à8ê å ±

Ï3Ü

´

If^ÏÒÑ-Ó

á Çê å

ÖR×

, then ^à ^ê ^å

±Ï Ü

´Q®LÞ²

Ñ Ù ÚÏ Ü

±v²1´èã®ÃëÒæ

ß .

(14)

Symbol Explanation

ìÎíïîdð

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

ì3ü

Let^ìÒý ^÷^ñsù ^ú ^û . The transpose^ì3üOý ^÷ ^ñ û ú ù is defined as

ì3üþdívÿð°ì

í1ð

.

Image-Template Operations

In the table below, ^ó is a finite subset of .

Symbol Explanation

ì

Let ^í^ñ

ð

be a semiring and ^ý ^ñù ,^ì+ý ^{÷ÈñsùúRû} , then the generic right convolution product of with^ì is defined as

ìsívÿ

írÿððõÿý

ò

írÿð

í1ð

ì

í1ð

ì

With the conditions above, except that now ^ìÒý ^÷ ^ñ

û ú ù , the generic left convolution product of with^ì is defined as

ì

!írÿ

"

íWÿð ðbõÿ;ý

ò írÿð#

$

í%1ð ì þ

ívÿð

'&

ì

Let^ò)( ^* , ^ý ^ñù , and^ìÒý ^÷^ñsù ^ú ^û , where ^ñ ^ý,+.- ^0/ . The right linear convolution product is defined as

'&

ì21èívÿ

3

íWÿð ðbõÿý

ò

ívÿð465

þ87

ù:98;=<?>A@CB

í%1ðED ì í%1ðFG

ì &

With the conditions above, except that^ìÒý ^÷Cñ

û ú ù , the left linear convolution product is defined as

ì &

)H IJ

írÿ

"

íWÿð ðõÿ.ý

ò$

íWÿð 5

þ87

ù:98;

í

>K

@ ð

í%1ðED ì þ írÿðEL M

N

PO QR

For^STVUWXY and^R ^{T[ZUWXY]\^} , the right morphological

max convolution product is defined by

S_

`

RbadcfeAgh"i:egkjlj:mng

Tpo

hiqe%gj4a r

s8t

W:u8v8w8xy?zA{.|

S

e%}kj~RCe%}kjC

R _

` S

For^STVUWXY and^R ^T ^Z^U ^

XY \ W , the left morphological max convolution product is defined by

R _` S a

e%gqh3iqeg4j"jmCg

Tpo

hi:egja r

s8t

W:u8v

wnx

e

z

{ j

S

eA}kj~GR

s

eg4j=

SP

` R

For^S'TVUWXY and^R T[ZUWXkY]\$^ , the right morphological min

convolution product is defined by

S

` Rba c eAg:h"iqeg4j"j:mng

To

hiqegja

s8t

Wqu x y?z { |

S

eA}kj~fR eA}kj

(15)

Symbol Explanation

For^V and ^[]

, the left morphological min convolution product is defined by

¡¢£

¬

¨$©:¦§ª

¢

®8¯

°8±=²

¦%³3´

µ ª

¦¶kªk·f¸

®

¦§ª=¹º

»G¼

'½

¾

For ^0¿ÁÀ

and ^#Â ^q¿À

Ã

, the right multiplicative max convolution product is defined by

½

¾ ¢dÄ

f¬

¨l©q¦§4ª

¢ÆÅ

®8¯

q°±8Ç

³

µCÈ

¦%¶kªÊÉ

$Ë

¦%¶kª$Ì

¼

½

¾

For'ÍAq¿ÁÀ

and ^Â ^Aq¿ÀÎ

Ã

, the left multiplicative max convolution product is defined by

½

¾

¢£

f¬

¨©q¦%§4ª

¢ Å

®8¯

:°8±

¦³ ´

µ ª

¦%¶kªÉ

®

¦§4ªE¹º

»Ï¼

¾

For ^q¿À

and ^#Â ^q¿À

Ã

, the right multiplicative min convolution product is defined by

¾ ¢dÄ

p¬

¨l©q¦%§ª

¢

®8¯

:°8± ² Ç³

µCÈ

¦%¶kªqÉ¸

Ë

¦A¶kªÌ

¼

¾

For^' ^q¿ÁÀ

and ^Â ^q¿À

Ã

, the left multiplicative min convolution product is defined by

¾

=¢£

p¬

¨©q¦%§ª

¢

®8¯

:°8± ²

¦%³

´µ ª

¦A¶kªqÉ4¸

®

¦§4ªE¹º

»Ï¼

Neighborhoods and Neighborhood Operations

Symbol Explanation

Ð

¨lÑ

Italic uppercase characters are used to denote neighborhoods.

Ñ

[Òn

A neighborhood is an image whose pixel values are sets of points. In particular, a neighborhood from^¬ to^Ó is a function

ÑÔ«

¬ÖÕ×ÒC

.

Ñ,¦?ØÙª

A parameterized neighborhood from ^¬ to^Ó with parameters in^Ú is a function of the form

Ñ)«

Ú Õ ÒC

.

Ñ ¸

Let

Ñ ÒC

, the transpose

Ñ'¸

ÒC

is defined as

Ñ'¸¦¶kª

¢Û

§

¬

«C¶

ÑÜ¦%§4ªÞÝ

, that is,

¶

Ñ,¦%§ªßáà'§

ãâ

¸¦%¶kª

.

Ñ¡äå·

¾

Ñdæ

The dilation of

Ñ]ä

by

Ñdæ

is defined by

Ñ,¦§4ª

¢ ç

è

¯=éê3ë?ì=íEî3ï]ð$î%ñòó[îAô'õñ4ò3ò

.