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Page 1: T g Ju - University of Winnipegion.uwinnipeg.ca/~sliao/pdf/thesis.pdf · aract er recognit ion are an alyze d as w ell v. Ac kno wle dgem en t s Man ypeo p le h a v epro vid e d advice

Image Analysis by Moments

by

Simon Xinmeng Liao

A Thesis

Submitted to the Faculty of Graduate Studies

in Partial Ful�llment of the Requirements

for the Degree of Doctor of Philosophy

The Department of

Electrical and Computer Engineering

The University of Manitoba

Winnipeg� Manitoba� Canada

c� Simon Xinmeng Liao ����

i

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��������������������

To the students who were massacred in Beijing� June �th� �����

and could not �nish their education�

ii

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I hereby declare that I am the sole author of this thesis�

I authorize the University of Manitoba to lend this thesis to other institutions or

individuals for the purpose of scholarly research�

Simon Xinmeng Liao

I further authorize the University of Manitoba to reproduce this thesis by photo�

copying or by other means� in total or in part� at the request of other institutions or

individuals for the purpose of scholarly research�

Simon Xinmeng Liao

iii

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The University of Manitoba requires the signatures of all persons using or photo�

copying this thesis� Please sign below� and give address and date�

iv

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Abstract

To select a set of appropriate numerical attributes of features from the interested

objects for the purpose of classi�cation has been among the fundamental problems

in the design of an imagery pattern recognition system� One of the solutions� the

utilization of moments for object characterization has received considerable atten�

tions in recent years� In this research� the new techniques derived to increase the

accuracy and the eciency in moment computing are addressed� Based on these

developments� the signi�cant improvement on image reconstructions via Legendre

moments and Zernike moments has been achieved� The eect of image noise on

image reconstruction� the automatic selection of the optimal order of moments for

image reconstruction from noisy image� and the usage of moments as image features

for character recognition are analyzed as well�

v

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Acknowledgements

Many people have provided advice� support� and encouragement to the author� during

the research which led to this thesis� I would like to express my heartfelt appreciation

to�

My supervisor� Prof� Dr� Miroslaw Pawlak� for his generous support and intellec�

tual guidance throughout my years as a graduate student� his insightful advice� clear

vision� many suggestions� and endless eorts to be available for many educational

discussions� were invaluable�

Prof� Dr� David Erbach� whose friendship and encouragement were invaluable

and kept me thinking that there really was a light at the end of the tunnel� and who

provided many valuable comments on drafts of this thesis�

my committee members� Prof� Dr� Richard Gordon and Prof� Dr� Waldemar

Lehn� for valuable insights and suggestions which have signi�cantly improved this

thesis in both structure and contents�

my External Examiner� Prof� Dr� Adam Krzyzak� for his critical comments and

constructive suggestions on this thesis�

my wife� Dr� Ming Yang� who shared with the pains and happiness during the

course of this work� her endless support� sacri�ce� and understanding kept me going

through it all�

and �nally� my parents� Li Bofan and Liao Cuichuan� who �rst taught me the

importance of education�

vi

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Contents

Abstract v

Acknowledgements vi

List of Figures xiv

List of Tables xiv

List of Symbols xv

� Introduction �

� Theory of Moments �

�� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� Geometric Moments in Image Processing � � � � � � � � � � � � � � � � �

� �� Preliminaries � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � Properties of Geometric Moments � � � � � � � � � � � � � � � � �

� �� Moment Invariants � � � � � � � � � � � � � � � � � � � � � � � � �

�� Complex Moments � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Orthogonal Moments � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

���� Legendre Moments � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Zernike Moments � � � � � � � � � � � � � � � � � � � � � � � � � ��

���� Pseudo�Zernike Moments � � � � � � � � � � � � � � � � � � � � �

vii

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� Accuracy and E�ciency of Moment Computing ��

��� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

�� Geometric Moments Computing � � � � � � � � � � � � � � � � � � � � � �

��� Legendre Moments Computing � � � � � � � � � � � � � � � � � � � � � � �

����� Approximation Error � � � � � � � � � � � � � � � � � � � � � � � �

���� Eciency � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��� Zernike Moments � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

���� Geometric Error � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� The Lattice Points of a Circle Problem � � � � � � � � � � � � � ��

����� Approximation Error � � � � � � � � � � � � � � � � � � � � � � � ��

����� A New Proposed Solution to Reduce Approximation Error � � ��

��� Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

����� Legendre Moment Computing � � � � � � � � � � � � � � � � � � ��

���� Zernike Moment Computing � � � � � � � � � � � � � � � � � � � ��

� Image Reconstruction from Moments ��

��� Inverse Moment Problem � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Method of Legendre Moments � � � � � � � � � � � � � � � � � � � � � � ��

�� �� Theory of Image Reconstruction from Legendre Moments � � � ��

�� � Reconstruction Error Analysis � � � � � � � � � � � � � � � � � � ��

�� �� Experimental Results � � � � � � � � � � � � � � � � � � � � � � � ��

��� Method of Zernike Moments � � � � � � � � � � � � � � � � � � � � � � � ��

����� Theory of Image Reconstruction from Zernike Moments � � � � ��

���� Reconstruction Error Analysis � � � � � � � � � � � � � � � � � � ��

����� Experimental Results � � � � � � � � � � � � � � � � � � � � � � � ��

��� Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

viii

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����� Image Reconstruction via Legendre Moments � � � � � � � � � � �

���� Image Reconstruction via Zernike Moments � � � � � � � � � � � ��

� Reconstruction of Noisy Images via Moments ��

��� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Legendre Moments � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� The Reconstruction Error � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Data�Driven Selection of the Optimal Number � � � � � � � � � � � � � �

� Character Recognition via Moments �

��� Introduction � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Character Recognition via Central Moments � � � � � � � � � � � � � � ��

��� Character Recognition with Legendre Moments � � � � � � � � � � � � � ��

��� Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

� Conclusions and Recommendations ��

��� Conclusions � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ���

�� Recommendations � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Bibliography ��

A ���

ix

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List of Figures

�� Moments projections onto x and y axes� � � � � � � � � � � � � � � � � ��

� The plots of some two�dimensionalPm�x�Pn�y�Legendre polynomials�

�a� P��x�P��y�� �b� P��x�P��y�� �c� P��x�P��y�� and �d� P��x�P��y�� � ��

�� The plots of some two�dimensionalPm�x�Pn�y�Legendre polynomials�

�a� P��x�P��y�� �b� P��x�P��y�� �c� P��x�P��y�� and �d� P��x�P��y�� � ��

�� The plots of the magnitudes of some Vnm�x� y� polynomials� �a� jV���x� y�j��b� jV���x� y�j� �c� jV���x� y�j� and �d� jV�����x� y�j� � � � � � � � � � � � ��

�� The plots of the magnitudes of some Vnm�x� y� polynomials� �a� jV���x� y�j��b� jV���x� y�j� �c� jV���x� y�j� and �d� jV���x� y�j� � � � � � � � � � � � �

��� Normalized E��s obtained by applying �ve dierent numerical integ�

ration rules to a constant image� � � � � � � � � � � � � � � � � � � � � � ��

�� Dierent areas covered by a disk and all pixels whose centres fall

inside the disk� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� ��dimensional formula I� � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� ��dimensional formula II� � � � � � � � � � � � � � � � � � � � � � � � � � �

��� ���dimensional formula � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Normalized approximation errors obtained by applying �ve dierent

types of multi�dimensional cubature formulas on a constant image� � � ��

x

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��� Normalized EA�s obtained by applying �ve dierent types of multi�

dimensional cubature formulas on a constant image with the new pro�

posed technique� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Five original Chinese characters used in image reconstruction via Le�

gendre moments� From left to right are C�� C�� C�� C�� and C�� � � � ��

�� Five Chinese characters and their reconstructed patterns viaLegendre

moments� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Normalized reconstruction errors for the �ve reconstructed Chinese

characters� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Five original Chinese characters used in image reconstruction via

Zernike moments� From left to right are C�� C�� C�� C�� and C�� � � ��

��� The Chinese character C� and its reconstructed patterns via Zernike

moments� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� The normalized mean square errors from appling �ve dierent formu�

las to character C�� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� The Chinese character C� and its reconstructed patterns via the mod�

i�ed Zernike moments� � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Normalized reconstruction errors from the reconstructed �ve Chinese

characters via the new proposed Zernike moment technique� � � � � � ��

��� The �ve Chinese characters and their reconstructed patterns via the

modi�ed Zernike moments with ��dimensional formula II� � � � � � � �

��� Square error Error�egMmax �� �� � ���� � � � � � � � � � � � � � � � � � � ��

�� Noisy version of C�� with �� � ���� and its reconstructed versions� � � ��

��� Optimal moments numbers� � � � � � � � � � � � � � � � � � � � � � � � ��

��� Five original Chinese characters used for testing� � � � � � � � � � � � � ��

xi

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�� Representations of the �ve Chinese characters in the central moment

feature space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Ninety Chinese characters� � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Representations of the ninety Chinese characters in the central mo�

ment feature space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Representations of the �ve Chinese characters in the Legendre mo�

ment feature space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Representations of the ninety Chinese characters in the Legendre

moment feature space� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

xii

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List of Tables

��� N is the number of points which are equally spaced apart by constant

h inside a single interval� � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Range of geometric errors for several commonly used image sizes� � � ��

��� Values of the normalized approximation errors from appling �ve dif�

ferent types of multi�dimensional cubature formulas on a constant

image� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Values of the normalized EA�s from appling �ve dierent types of

multi�dimensional cubature formulas on a constant image with the

new proposed technique� � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� The values of normalized reconstruction errors for the �ve reconstruc�

ted Chinese characters� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Values of the normalized mean square errors from appling �ve dier�

ent formulas to character C�� � � � � � � � � � � � � � � � � � � � � � � � ��

��� Values of the normalized reconstruction errors from the reconstruc�

ted �ve Chinese characters with the new proposed Zernike moment

technique� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Values of the normalized reconstruction errors from the reconstruc�

ted �ve Chinese characters via the new proposed Zernike moment

technique� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

xiii

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��� Square reconstruction error Error�egMmax � with �� � ���� � � � � � � � ��

��� Values of the �ve Chinese characters in the central moment feature

space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Values of the ninety Chinese characters in the central moment feature

space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

��� Values of the �ve Chinese characters in the Legendremoment feature

space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Values of the ninety Chinese characters in the Legendre moment

feature space� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��

��� Values of the ninety Chinese characters in the Legendre moment

three�dimensional feature space� � � � � � � � � � � � � � � � � � � � � � ��

xiv

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List of Symbols

Some of the most frequently occurring abbreviations and symbols used in the text

are tabulated here� Other symbols are explained where used�

Anm � Zernike moments of order n with repetition m

bAnm � Digital version of Anm

Cnf � cubature formula

Cpq � complex moments

EA �PP j bAnmj� m � n �� �

E� �PMmax

m�

Pmn�

b��mn m � n �� �bf�x� y� � reconstructed image from f�x� y�

F �u� v� � characteristic function of the image function f�x� y�

g�x� y� � noisy degraded version of f�x� y�

Mpq � geometric moments of order �p�q�

N � the number of points which are spaced apart by a constant step

h inside a single interval

Pn�x� � Legendre polynomials

Rnm � Radial polynomials

Vnm � Zernike polynomials

�mn � Kronecker symbol

�mn � Legendre moments

b�mn � Digital version of �mn

xv

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e�mn � Legendre moments from g�x� y�

�pq � central moments

xvi

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Chapter �

Introduction

One of the basic problems in the design of an imagery pattern recognition system

relates to the selection of a set of appropriate numerical attributes of features to be

extracted from the object of interest for the purpose of classi�cation� The recognition

of objects from imagery may be achieved with many methods by identifying an

unknown object as a member of a set of known objects� Ecient object recognition

techniques abstracting characterizations uniquely from objects for representation and

comparison are crucially important for a given pattern recognition system�

Research on the utilization of moments for object characterization in both in�

variant and noninvariant tasks has received considerable attention in recent years�

The principal techniques explored includeMoment Invariants� Geometric Moments�

Rotational Moments� Orthogonal Moments� and Complex Moments� Various forms

of moment descriptors have been extensively employed as pattern features in scene

recognition� registration� object matching as well as data compression�

The mathematical concept of moments has been around for many years and has

been used in many diverse �elds ranging from mechanics and statistics to pattern

recognition and image understanding� Describing images with moments instead of

other more commonly used image features means that global properties of the image

are used rather than local properties�

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Historically� Hu�������� published the �rst signi�cant paper on the utilization of

moment invariants for image analysis and object representation in ����� Hu�s ap�

proach was based on the work of the nineteenth century mathematicians Boole� Cay�

ley� and Sylvester� on the theory of algebraic forms� Hu�s Uniqueness Theorem states

that if f�x� y� is piecewise continuous and has nonzero values only in the �nite part

of the �x� y� plane� then geometric moments of all orders exist� It can then be shown

that the moment set fmpqg is uniquely determined by f�x� y� and conversely� f�x� y�is uniquely determined by fmpqg� Since an image segment has �nite area and� inthe worst case� is piecewise continuous� a moment set can be computed and used to

uniquely describe the information contained in the image segment� Using nonlinear

combinations of geometric moments� Hu derived a set of invariant moments which

has the desirable properties of being invariant under image translation� scaling� and

rotation� However� the reconstruction of the image from these moments is deemed

to be quite dicult�

The Rotational moment is an alternative to the regular geometric moment� The

Rotational moments are based on a polar coordinate representation of the image and

can be used to extend the de�nition of moment invariants to arbitrary order in a

manner which ensures that their magnitudes do not diminish signi�cantly with in�

creasing order� Smith and Wright���� used a simpli�edRotationalmoment technique

to derive invariant features from noisy low resolution images of ships� Boyce and

Hossack���� derived the Rotational moments of arbitrary order that are invariant to

rotation� radial scaling� and intensity change�

In ����� Teague���� presented two inverse moment transform techniques to de�

termine how well an image could be reconstructed from a set of moments� The �rst

method� called moment matching� derives a continuous function

g�x� y� � g�� � g��x� g��y � g��x� � g��xy � g��y

� �

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g��x� � g��x

�y � g��xy� � g��y

� � ����

whose moments exactly match the geometric moments fmpqg of f�x� y� through or�der n� However� this technique is impractical for calculation as it requires one to

solve an increasing number of coupled equations when higher order moments are

considered� Then� Teague suggested the notion of orthogonal moments to recover

the image from moments based on the theory of orthogonal polynomials� Teague

introduced the rotationally invariant Zernike moment� which employs the complex

Zernike polynomials as the moment basis set� and the Legendre moment� using

Legendre polynomials as its basis set� Signi�cant eorts have been made in vari�

ous experimental image reconstruction tasks performed by Teague� then Boyce and

Hossack����� Teh and Chin����� Taylor and Reeves�� �� and more recently� Khotan�

zad and Hong�������� with both Zernike and Legendre methods� However� no

high quality multi�graylevel image has ever been successfully reconstructed from its

original version�

Later� the notion of Complexmomentswas introduced by Abu�Mostafa and Psaltis���

as a simple and straightforward way to derive a set of invariant moments� Abu�

Mostafa and Psaltis used Complex moments to investigate the informational proper�

ties of moment invariants� However� comparing Complex moments with the Zernike

moments� they concluded that the Complex moment invariants are not good image

features� In other work� Abu�Mostafa and Psaltis� � examined the utilization of mo�

ments in a generalized image normalization scheme for invariant pattern recognition�

They rede�ned the classic image normalizations of size� position� rotation� and con�

trast� in terms of Complex moments� Moment invariants were shown to be derivable

from Complex moments of the normalized image as well�

Teh and Chin���� performed an extensive analysis and comparison of the most

common moment de�nitions� They examined the noise sensitivity and information

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redundancy of Legendre moments along with �ve other types of moments� Teh and

Chin concluded that higher order moments are more sensitive to noise� Among the

explored techniques� Complex moments are least sensitive to noise while Legendre

moments are most severely aected by noise� In terms of information redundancy�

Legendre� Zernike� and pseudo�Zernike moments are uncorrelated and have the

least redundancy� In terms of overall performance� Zernike and pseudo�Zernike

moments are the best� In general� orthogonal moments are better than other types

of moments in terms of information redundancy and image representation�

More recently� Prokop and Reeves�� � reviewed the basic geometric moment the�

ory and its application to object recognition and image analysis� The geometric

properties of low�order moments were discussed along with the de�nition of several

moment�space linear geometric transforms� Prokop and Reeves also presented an

extensive review summarizingmost of research advancements related to the moment�

based object representation and recognition techniques over the past �� years�

The speed of computing image moments is extraordinarily important when higher

order moments are involved� Several schemes of hardware architectures have been

performed to speed up the computation of image moments� Reeves���� proposed

a parallel� mesh�connected SIMD computer architecture for rapidly manipulating

moment sets� The architecture oered a reasonable speeding up over a single pro�

cessor for high speed image analysis operations and was expected to be implemented

in VLSI technology� Andersson��� developed a VLSI moment�generating chip and

presented a real�time system by implementing the processor� Hatamian���� proposed

an algorithm and single chip VLSI implementation to generate raw moments� It is

claimed that �� geometricmoments�mpq�p � �� �� � �� q � �� �� � ��� of a �� ��� ��bit image can be computed at �� frames�sec� The moment algorithm is based on

using the one�dimensional discrete moment�generating function as a digital �lter�

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The organization of this thesis is as follows� Chapter will review the general

characteristics of various types of moments and their properties� In Chapter ��

the new techniques derived to increase the accuracy and the eciency in moment

computing� will be addressed� Chapter � will discuss the reconstruction algorithms

of the Legendre moments and the Zernike moments� and provide signi�cantly

improved reconstructed images from these orthogonal moments� Then� the eect

of image noise on image reconstruction and the automatic selection of the optimal

order of moments for image reconstruction will be analyzed in Chapter �� Several

speci�c recognition aspects of proposed moment techniques for character recognition

are studied in Chapter �� Finally� Chapter � will summarize the important results

and conclusions of the entire study�

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Chapter �

Theory of Moments

��� Introduction

Numerous problems in mechanics� physics� and engineering lead to the problem of

characterization of a function in terms of some functionals� In particular� moment

functionals have attracted great attention���� due to their mathematical simplicity

and numerous physical interpretations�

A complete characterization of moment functionals over a class of univariate func�

tions was given by Hausdor� �� in �� ��

Let f�ng be a real sequence of numbers and let us de�ne

�m�n �mXi�

����i�mi

��ni� � ���

Note that �m�n can be viewed as the mth order derivative of �n�

By Hausdor�s theorem� a necessary and sucient condition that there exists a

monotonic function F �x� satisfying the system

�n �Z �

�xndF �x�� n � �� �� � ��� � � �

is that the system of linear inequalities

�k�n � � k � �� �� � ��� � ���

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should be satis�ed� I�e�� if f�x� is a positive function �which is the case in image

processing�� then the set of functionals

Z �

�xnf�x�dx� n � �� �� ���

completely characterizes the function�

A necessary and sucient condition that there exists a function F �x� of bounded

variation satisfying � � � is that the sequence

pXm�

�pm

�j�p�m�mj p � �� �� � ���

should be bounded�

These results were extended to the two�dimensional case by Hildebrandt and

Schoenberg���� in �����

Since then� moments and functions of moments have been utilized in a num�

ber of applications to achieve both invariant and noninvariant recognitions of two�

dimensional and three�dimensional image patterns�� ��

In this chapter� the various types of moments are de�ned and their properties

are summarized� It is assumed that an image can be represented by a real valued

measurable function f�x� y��

��� Geometric Moments in Image Processing

����� Preliminaries

The two�dimensional geometric moment of order �p � q� of a function f�x� y� is

de�ned as

Mpq �Z a�

a�

Z b�

b�xp yq f�x� y� dxdy� � ���

where p� q � �� �� � ������ Note that the monomial product xpyq is the basis functionfor this moment de�nition�

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A set of n moments consists of all Mpq�s for p � q � n� i�e�� the set contains

���n� ���n� � elements�

The use of moments for image analysis and pattern recognition was inspired by

Hu���� and Alt���� Hu stated that if f�x� y� is piecewise continuous and has nonzero

values only in a �nite region of the �x� y� plane� then the moment sequence fMpqgis uniquely determined by f�x� y�� and conversely� f�x� y� is uniquely determined by

fMpqg� Considering the fact that an image segment has �nite area� or in the worstcase is piecewise continuous� moments of all orders exist and a complete moment

set can be computed and used uniquely to describe the information contained in the

image� However� to obtain all of the information contained in an image requires an

in�nite number of moment values� Therefore� to select a meaningful subset of the

moment values that contain sucient information to characterize the image uniquely

for a speci�c application becomes very important�

����� Properties of Geometric Moments

The lower order moments represent some well known fundamental geometric prop�

erties of the underlying image functions�

Central Moments

The central moments of f�x� y� are de�ned as

�pq �Z a�

a�

Z b�

b��x� �x�p �y � �y�q f�x� y� dxdy� � ���

where �x and �y are de�ned in � �����

The central moments �pq de�ned in Eq� � ��� are invariant under the translation

of coordinates�����

x� � x� ��

y� � y � �� � ���

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where � and � are constants�

Mass and Area

The de�nition of the zeroth order moment� fM��g� of the function f�x� y�

M�� �Z a�

a�

Z b�

b�f�x� y� dxdy � ���

represents the total mass of the given function or image f�x� y�� When computed for

a binary image� the zeroth moment � ��� represents the total area of the image�

Centre of Mass

The two �rst order moments�

M�� �Z a�

a�

Z b�

b�x f�x� y� dxdy � ���

and

M�� �Z a�

a�

Z b�

b�y f�x� y� dxdy � ���

represent the centre of mass of the image f�x� y�� The centre of mass is the point

where all the mass of the image could be concentrated without changing the �rst

moment of the image about any axis� In the two�dimensional case� in terms of

moment values� the coordinates of the centre of mass are

�x �M��

M��� �y �

M��

M��� � ����

As a usual practice� the centre of mass is chosen to represent the position of an

image in the �eld of view� The equations in � ���� de�ne a unique location of the

image f�x� y� that can be used as a reference point to describe the position of the

image�

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Orientations

The second order moments� fM���M���M��g� known as the moments of intertia�may be used to determine an important image feature� orientation� In general� the

orientation of an image describes how the image lies in the �eld of view� or the

directions of the principal axes�

In terms of moments� the orientations of the principal axes� � are given by����

��

tan���

������ � ���

�� � ����

In � ����� is the angle of the principal axis nearest to the x axis and is in the range

��� � � ���

Projections

An alternative means of describing image properties represented by moments is

to consider the relationship between the moments of an image and those of the

projections of that image� The moments in the sets fMp�g and fM�pg are equivalentto the moments of the image projection onto the x axis and y axis respectively�

Consider the horizontal projection� h�y�� of an image f�x� y� onto the y axis given

by

h�y� �Z a�

a�f�x� y� dx� � �� �

Then� the one�dimensional moments�Mq� of h�y� are obtained by

Mq �Z b�

b�yq h�y� dy� � ����

Substituting � �� � into � ���� gives

Mq �Z a�

a�

Z b�

b�yq f�x� y� dxdy �M�q� � ����

Figure �� illustrates the projections of an object onto the x axis and y axis and

the moment subsets corresponding to the projections�

��

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v(x)

h(y)

M40

M31 M30

M22 M21 M20

M13 M12 M11 M10

M04 M03 M02 M01 M00

Figure ��� Moments projections onto x and y axes�

��

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����� Moment Invariants

The earliest signi�cat work employing moments for image processing and pattern

recognition was performed by Hu���� and Alt���� Based on the theory of algebraic

invariants� Hu�������� derived relative and absolute combinations of moments that

are invariant with respect to scale� position� and orientation�

The method of moment invariants is derived from algebraic invariants applied to

the moment generating function under a rotation transformation� The set of absolute

moment invariants consists of a set of nonlinear combinations of central moments that

remain invariant under rotation� Hu de�nes the following seven functions� computed

from central moments through order three� that are invariant with respect to object

scale� translation and rotation�

�� � ��� � ��� � ����

�� � ���� � ����� � ����� � ����

�� � ���� � ������ � ����� � ����� � ����

�� � ���� � ����� � ���� � ����

� � ����

�� � ���� � ��������� � ��������� � ����� � ����� � ����

��

������ � �������� � ���������� � ����� � ���� � ����

�� � ����

�� � ���� � ��������� � ����� � ���� � ����

��

��������� � �������� � ���� � � ��

�� � ����� � �������� � ��������� � ����� � ����� � ����

��

����� � ��������� � ���������� � ����� � ���� � ����

��� � � ��

The functions �� through �� are invariant with respect to rotation and re�ection

while �� changes sign under re�ection�

Hu�s original result has been slightly modi�ed by Reiss���� in ����� Reiss revised

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the fundamental theorem of moment invariants with four new invariants� The cor�

rection presented by Reiss aects neither similitude �scale� nor rotation invariants

derived using the original theorem� but it does aect features invariant to general

linear transformations�

The de�nition of the geometric moments � ��� has the form of the projection of

the image function f�x� y� onto the monomial xpyq� However� with the Weierstrass

approximation theorem����� the basis set fxpyqg� while complete� is not orthogonal�

��� Complex Moments

The notion of complex moments was introduced in ��� as a simple and straightfor�

ward technique to derive a set of invariant moments� The two�dimensional complex

moments of order �p� q� for the image function f�x� y� are de�ned by�

Cpq �Z a�

a�

Z b�

b��x� jy�p �x� jy�q f�x� y� dxdy� � � �

where p and q are nonnegative integers and j �p���

The complex moments of order �p� q� are a linear combination with complex coef�

�cients of all of the geometric moments fMnmg satisfying p � q � n�m�

In polar coordinates� the complex moments of order �p � q� can be written as

Cpq �Z ��

Z �

� pq ej�p�q � f� cos� sin� d d� � � ��

If the complex moment of the original image and that of the rotated image in the

same polar coordinates are denoted by Cpq and Crpq� the relationship between them

is

Crpq � Cpqe

�j�p�q �� � � ��

where is the angle that the original image has been rotated�

The complex moment invariants can be written in the form of

CrsCktu � CsrC

kut� � � ��

��

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where

�r � s� � k�t� u� � �� � � ��

This combination of complex moments cancels both the imaginary moment and the

rotational phase factor� and thus provides real�valued rotation invariants�

However� these complex moment invariants are not� in general� good features����

They suer from information loss� suppression� and redundancy which limit their

discrimination power�

��� Orthogonal Moments

����� Legendre Moments

Legendre Polynomials

The nth � order Legendre polynomial is de�ned by

Pn�x� �nX

j�

anj xj �

nn�

dn

dxn�x� � ��n� � � ��

The Legendre polynomials have the generating function

�p�� rx � r�

��Xs�

rs Ps�x�� r � �� � � ��

From the generating function� the recurrent formula of the Legendre polynomials

can be acquired straightforwardly�

d

dr�

�p�� rx � r�

� �d

dr��Xs�

rs Ps�x��

x� r

��� rx � r������

�Xs�

s rs�� Ps�x�

�x� r��Xs�

rs Ps�x� � ��� rx� r���Xs�

s rs�� Ps�x��

Then we have

xPk�x�� Pk���x� � �k � ��Pk��x�� xkPk�x� � �k � ��Pk���x��

��

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or� the recurrent formula of the Legendre polynomials�

Pn��x� � n � �

n � �xPn�x�� n

n� �Pn���x�� � � ��

The Legendre polynomials fPm�x�g ���� are a complete orthogonal basis set onthe interval ���� ��� Z �

��Pm�x�Pn�x�dx �

m � ��mn� � ����

where �mn is the Kronecker symbol�

Figure � and Figure �� show some of the two�dimensional Legendre polyno�

mials in the image space�

Legendre Moments

The �m�n�th order of Legendre moment of f�x� y� de�ned on the square ���� ������� �� is

�mn � � m� ��� n � ��

Z �

��

Z �

��Pm�x�Pn�y� f�x� y� dxdy� � ����

where m�n � �� �� � ����

Using � ���� � � ��� and � ����� we have

�mn �� m� ��� n � ��

Z �

��

Z �

��Pm�x�Pn�y� f�x� y� dxdy

�� m� ��� n � ��

Z �

��

Z �

��

mXj�

amj xj

nXk�

ank yk f�x� y� dxdy

�� m� ��� n � ��

mXj�

nXk�

amj ank

Z �

��

Z �

��xj yk f�x� y� dxdy�

Therefore� the Legendre moments and geometric moments are related by

�mn �� m� ��� n � ��

mXj�

nXk�

amj ankMjk� � �� �

The above relationship indicates that a given Legendre moment depends only on

geometric moments of the same order and lower� and conversely�

��

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( a )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

( b )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

( c )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

( d )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

Figure � � The plots of some two�dimensional Pm�x�Pn�y� Legendre polynomials��a� P��x�P��y�� �b� P��x�P��y�� �c� P��x�P��y�� and �d� P��x�P��y��

��

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( a )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

( b )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

( c )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

( d )

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

X

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

Y

F

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

Figure ��� The plots of some two�dimensional Pm�x�Pn�y� Legendre polynomials��a� P��x�P��y�� �b� P��x�P��y�� �c� P��x�P��y�� and �d� P��x�P��y��

��

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����� Zernike Moments

The usage of Zernike polynomials in optics dates back to the early �th century�

while the applications of orthogonal moments based on Zernike polynomials for

image processing were pioneered by Teague���� in �����

Zernike Polynomials

A set of orthogonal functions with simple rotation properties which forms a complete

orthogonal set over the interior of the unit circle was introduced by Zernike����� The

form of these polynomials is

Vnm�x� y� � Vnm� sin� cos� � Rnm� � exp�jm� � ����

where n is either a positive integer or zero� andm takes positive and negative integers

subject to constraints n � jmj � even� jmj � n� is the length of the vector from

the origin to the pixel at �x� y�� and is the angle between vector and the x axis

in the counterclockwise direction�

The Radial polynomial Rnm� � is de�ned as

Rnm� � ��n�jmj ��X

s�

����s �n� s��

s� �njmj� � s�� �n�jmj� � s�� n��s� � ����

with Rn��m� � � Rn�m� ��

Figure �� and Figure �� show some of the Vnm�x� y� polynomials�

The Zernike polynomials � ���� are a complete set of complex�valued functions

orthogonal on the unit disk x� � y� � ��

Z Zx�y���

�Vnm�x� y���Vpq�x� y� dxdy �

n � ��np �mq� � ����

or� in polar coordinates

Z ��

Z �

��Vnm�r� ��

� Vpq�r� � r drd �

n � ��np �mq� � ����

��

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( a )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

( b )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

( c )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

( d )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

Figure ��� The plots of the magnitudes of some Vnm�x� y� polynomials��a� jV���x� y�j� �b� jV���x� y�j� �c� jV���x� y�j� and �d� jV�����x� y�j�

��

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( a )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

( b )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

( c )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

( d )

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 90 . 9 8

X

- 0 . 9 8

- 0 . 4 9

0 . 0 0

0 . 4 9

0 . 9 8

Y

F

0 . 0 0

0 . 2 5

0 . 5 0

0 . 7 5

1 . 0 0

Figure ��� The plots of the magnitudes of some Vnm�x� y� polynomials��a� jV���x� y�j� �b� jV���x� y�j� �c� jV���x� y�j� and �d� jV���x� y�j�

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where the asterisk denotes the complex conjugate�

As is seen from � ���� and � ����� the real�valued radial polynomials fRnm�r�gsatisfy the relation Z �

�Rnl�r�Rml�r� r dr �

�n� ���mn� � ����

The radial polynomials Rnm� � have the generating function

�� � t�q�� t��� �� � t� �m

� t �mq�� t�� � �� � t�

��Xs�

tsRm�s�m� �� � ����

When m � �� it is interesting to see that the equation � ���� reduces to

�q�� t��� �� � t�

��Xs�

tsPs��� ��� � ����

and becomes the generating function for the Legendre polynomials of argument

� � �� so thatR�n��� � � Pn�

� � ��� � ����

Zernike Moments

The complex Zernike moments of order n with repetition m for an image function

f�x� y� are de�ned as

Anm �n� �

Z Zx�y���

f�x� y�V �nm� � � dxdy� � ����

or� in polar coordinates

Anm �n� �

Z ��

Z �

�f� � �Rnm� � exp��jm� d d� � �� �

where the real�valued radial polynomial Rnm� � is de�ned in � �����

Due to the conditions n � jmj � even and jmj � n for the Zernike polynomials

� ����� the set of Zernike polynomials contains ���n����n� � linearly independent

polynomials if the given maximum degree is n�

Since A�nm � An��m� then jAnmj � jAn��mj� therefore� one only needs to consider

jAnmj with m � ��

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Rotational Properties of Zernike Moments

Under a rotation transformation� the angle of rotation of the Zernike moments is

simply a phase factor� Therefore� the Zernike moments are invariant under image

rotation�

If the original image and the rotated image in the same polar coordinates are

denoted by f� � � and f r� � � respectively� the relationship between them is

f r� � � � f� � � ��� � ����

where � is the angle that the original image has been rotated� Using � �� �� the

Zernike moment of the rotated image is

Arnm �

n� �

Z ��

Z �

�f� � � ��Rnm� � exp��jm� d d

�n� �

Z ��

Z �

�f� � � ��Rnm� � exp��jm� � �� ��� d d

�n� �

Z ��

Z �

�f� � � ��Rnm� � exp��jm� � ��� exp��jm�� d d�

therefore� the relationship between Arnm and Anm is

Arnm � Anm exp��jm��� � ����

Equation � ���� indicates that the Zernikemoments have simple rotational trans�

formation properties� The magnitudes of the Zernike moments of a rotated image

function remain identical to the original image function� Thus the magnitude of

the Zernike moment� jAnmj� can be employed as a rotation invariant feature of thefundamental image function�

����� PseudoZernike Moments

If we eliminate the condition n � jmj � even from the the Zernike polynomials

de�ned in � ����� fVnmg becomes the set of pseudo�Zernike polynomials� The set of

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pseudo�Zernike polynomials was derived by Bhatia and Wolf�� � and has properties

analogous to those of Zernike polynomials�

For the pseudo�Zernike polynomials� the real�valued radial polynomial Rnm� �

is de�ned as

Rnm� � �n�jmjXs�

����s � n� � � s��

s� �n� jmj � s�� �n� jmj� �� s�� n�s� � ����

where n � �� �� � ����� and m takes on positive and negative integers subject to

jmj � n only� Unlike the set of Zernike polynomials� this set of pseudo�Zernike

polynomials contains �n � ��� linearly independent polynomials instead of

���n� ���n� � if the given maximum order is n�

The de�nition of the pseudo�Zernike moments is the same as that of the Zernike

moments in � ���� and � �� � except that the radial polynomials fRnm� �g in � ����are used�

Since the set of pseudo�Zernike orthogonal polynomials is analogous to that of

Zernike polynomials� most of the previous discussion for the Zernike moments can

be adapted to the case of pseudo�Zernike moments�

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Chapter �

Accuracy and E�ciency of Moment

Computing

��� Introduction

An essential issue in the �eld of pattern analysis is the recognition of patterns and

characters regardless of their positions� sizes� and orientations� As discussed in

the previous chapter� moments and functions of moments can be employed as the

invariant global features of an image in pattern recognition� image classi�cation�

target identi�cation� and scene analysis�

Generally� these features are invariant under image translation� scale change� and

rotation only when they are computed from the original two�dimensional images�

In practice� one observes the digitized� quantized� and often noisy version of the

image and all these properties are satis�ed only approximately� The problem of the

discretization error for moment computing has been barely investigated� though some

initial studies into this direction for the case of geometric moments were performed

by Teh and Chin�����

In this chapter� the detailed analysis of the discretization error for moment com�

puting is addressed� Several new techniques developed to increase the accuracy in

moment computing are provided�

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��� Geometric Moments Computing

Geometric moments are the most popular type of moments� The de�nition of geo�

metric moments � ��� is rewritten here for convenience�

Mpq �Z �

��

Z �

��xp yq f�x� y� dxdy� �����

If an analog original image function f�x� y� is digitized into its discrete version

f�xi� yj� with an M � N array of pixels� the double integration of ����� must be

approximated by double summations� In fact� in digital image processing� one

can observe f�x� y� only at discrete pixels� i�e�� instead of ff�x� y�� �x� y� � Rg�ff�xi� yj�� � � i � M� � � j � Ng is used� It has been a common prescription toreplace Mpq in ����� with its digital version

cMpq �MXi�

NXj�

xpi yqj f�xi� yj��x�y� ��� �

where �x and �y are sampling intervals in the x and y directions� However� when

the moment order increases� ��� � cannot produce accurate results�

By ������ one obtains

Mpq �Z �

��

Z �

��xp yq f�x� y� dxdy

�Z Z

Axp yq f�x� y�dxdy

�Z a�

a�

Z b�

b�xp yq f�x� y�dxdy� �����

where

A � �a�� a��� �b�� b��

is the area covered by the image�

Then� it is clear that

Mpq �MXi�

NXj�

Z xi�x�

xi��x�

Z yj�y

yj��y

xp yq f�x� y� dxdy� �����

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where �x � xi � xi�� and �y � yj � yj�� are the sampling intervals in the x and y

directions� and

xM ��x

� a�� �����

x� � �x

� a�� �����

yN ��y

� b�� �����

y� � �y

� b�� �����

By the second mean value theorem for integration� if f and g are integrable func�

tions on the set A� and if f is also continuous� thenZAf�z�g�z�dz � f���

ZAg�z�dz �����

for some � � A�

Applying this result to ����� yields

Mpq �MXi�

NXj�

f��i� �j�Z xi

�x�

xi��x�

Z yj�y

yj��y

xp yq dxdy� ������

where ��i� �j� belongs to the �i� j� pixel�

Let us assume without loss of generality that each pixel is quantized to one value�

it is normal to replace f��i� �j� by f�xi� yj�� This gives the following approximation

of Mpq� cMpq �MXi�

NXj�

hpq�xi� yj� f�xi� yj�� ������

where

hpq�xi� yj� �Z xi

�x�

xi��x�

Z yj�y

yj��y

xp yq dxdy ���� �

represents the double integration of xpyq over the pixel �xi � �x�� xi �

�x���

�yj � �y� � yj �

�y� ��

Note that

hpq�xi� yj� ���xi �

�x� �

p� � ��xi � �x� �

p��

�p � ��

��yj ��y� �

q� � ��yj � �y� �

q��

�q � ��� ������

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Then the question is how to acquire the double integration ���� �� The simplest

method to carry out the computation of hpq�xi� yj� is to use the following formula�

h�pq�xi� yj� � xpi yqj �x�y ������

to replace ���� �� However� the above approximation will result in a substantial error

when the order p � q increases�

Since the double integration in ���� � can be separated as

hpq�xi� yj� �Z xi

�x�

xi��x�

xp dxZ yj

�y�

yj��y�

yq dy� ������

for simplicity� we consider the single integration

hp�xi� �Z xi

�x�

xi��x�

xp dx� ������

and replace hp�xi� with

h�p�xi� � xpi �x� ������

When p � �� h�p�xi� holds�

When the order p increases to � from ������� we have

h��xi� �Z xi

�x�

xi��x�

x� dx

��

���xi �

�x

�� � �xi � �x

���

� x�i �x���x��

h��xi� � h���xi� ���x��

� �

The approximation error for the single integration is �x�

�� �

In the case of p � ��

h��xi� �Z xi

�x�

xi��x�

x� dx

��

���xi �

�x

�� � �xi � �x

���

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� x�i �x�xi ��x��

h��xi� � h���xi� �xi ��x��

��

The error is increased�

The approximation error will quickly get out of control when the order p increases�

Obviously� when the higher order moments are involved� the problem of numerical

approximation error in the moment computing must be solved before any implement�

ation�

By the well�known techniques of numerical integration����� the integration of ������

can be approximated with various accuracies� For example� applying Simpson�s rule

in the case of moment order p � �� we getZ xi�x�

xi��x�

x� dx ��x

��

��xi � �x

�� �

�x�i �

��xi �

�x

���

� x�i �x�xi ��x��

��

This is the same result as that of the integration�

Evidently� when the order p goes higher� more accurate rules are required to limit

the approximate error to a tolerable level�

As the solution� the alternative extended Simpson�s ruleZ xN

x�f�x�dx �

h

�����f� � ��f� � ��f� � ��f� � f� � f� � ���

�fN�� � ��fN�� � ��fN�� � ��fN�� � ��fN �

�O��

N�� ������

is employed in this research to compute the moments numerically����� In ������� N

is the number of points which are equally spaced apart by constant h inside a single

interval�

The above discussion about the approximation errors in geometric moment calcu�

lations certainly can be extended to the acquisition of the Legendre moments�

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��� Legendre Moments Computing

����� Approximation Error

The �m� n�th order Legendre moment is de�ned in � ���� as

�mn �� m� ��� n � ��

Z �

��

Z �

��Pm�x�Pn�y� f�x� y�dxdy�

where the mth order Legendre polynomial is

Pm�x� ��

mm�

dm

dxm�x� � ��m�

Similar to the case of geometric moments� we can approximate �mn by

b�mn �� m� ��� n � ��

mXi�

nXj�

h�mn�xi� yj� f�xi� yj�� ������

where

h�mn�xi� yj� �Z xi

�x�

xi��x�

Z yj�y�

yj��y�

Pm�x�Pn�y� dxdy� ��� ��

Since the Legendre polynomials Pm�x� and Pn�y� are independent� the double

integration in ��� �� can be written as

Z xi�x�

xi��x�

Z yj�y

yj��y

Pm�x�Pn�y� dxdy �Z xi

�x�

xi��x�

Pm�x�dxZ yj

�y

yj��y

Pn�y�dy� ��� ��

Therefore� similar to the case of geometricmoments� the alternative extended Simpson�s

rule can be applied in Legendremoment calculations to limit the approximate error

to a certain level�

By using the alternative extended Simpson�s rule� the approximation errors are

reduced dramatically� It makes further use of the Legendre moments possible as

well�

To show the improvement of accuracy in Legendre moment computing by adopt�

ing the alternative extended Simpson�s rule� an experiment was designed�

If we assume that the image function f�x� y� is a constant image with graylevel

a� i�e�� f�x� y� � a� it is easily seen that all Legendre moments should equal zero

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except ��� � a� We use the sum of all Legendre moment squares except for the

case of m � n � � as the measure to evaluate the approximation error� which has

the form of

E� �MmaxXm�

mXn�

b��mn m � n �� �� ��� �

Clearly� the smaller the E� value in ��� �� the better the performance of the

approximation� Five dierent numerical integration rules� N � �� N � �� N �

��� N � ��� and N � � are employed and all normalized E��s are illustrated in

Figure ���� The highest Legendre moment order used in this experiment is ���

10 15 20 25 30 35 40 45 50 55 60Moment Order

0

0.2

0.4

0.6

0.8

1

Squ

ared

App

roxi

mat

ionn

Err

or

N=3N=8N=13N=18N=23

Figure ���� NormalizedE��s obtained by applying �ve dierent numerical integrationrules to a constant image�

Only the E��s which are less than ��� are presented in Figure ���� Each E�

increases very sharply after the moment order is over a certain number� As expected�

the higher accuracy approximation rules perform better than the lower ones do�

��

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Order N � � N � � N � �� N � �� N � �� ��������� ��������� ��������� ������� � ������� ������ �� ���� ������� � ������� ������� ������� � ������� ����� ������ � ������� ������� ��������� ����� � ����� � ������ � ��� �� ������� ������� ��������� ����� ������� ������ ������ �� ��� � � ������ ����� � ��������� �� � �� ������� ������� ��������� ��� ��� ������� ������� �������� ������� ���� �� ����� � ���� ���� ������� ������� ���� �� ��������� ���� �� ����� � ��������� ������� ������� ��������� �� ��� ��� ��� �������� ��� � � ��� ��� ��������� ������� ������� ��������� ������� ������� �������

Table ���� N is the number of points which are equally spaced apart by constant hinside a single interval�

��

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����� Eciency

With the appearance of more powerful computers� it becomes practical to compute

and use the higher order moments� However� the computation of moments� speci�c�

ally� if the higher order moments are involved� is still a time consuming procedure�

Since most of computing work in this thesis was achieved with a �MHz ��� personal

micro�computer� reducing the computing time became even more critical�

From the discussion in the previous section� the Legendre moments of an image

function f�x� y� can be obtained numerically by the formula

b�mn �� m� ��� n � ��

mXi�

nXj�

h�mn�xi� yj� f�xi� yj� ��� ��

where

h�mn�xi� yj� �Z xi

�x�

xi��x�

Z yj�y

yj��y�

Pm�x�Pn�y� dxdy� ��� ��

As we have discussed� when the higher order Legendre polynomials Pm�x� and

Pn�y� are involved� to keep the approximation error under a certain level� the multi�

interval step alternative extended Simpson�s rule can be employed� However� if the

well accepted recurrent formula � � �� of the Legendre polynomials is applied to

compute the Legendre polynomials Pm�x� and Pn�y�� under the situation that N

takes a moderate value ��� even when the image consists of a small number of pixels�

for example� � by �� the computing time could be too long to be tolerated�

To speed up the computation� the most important measure is to avoid using the

recurrent formula � � �� of theLegendre polynomials� The fastest� the most ecient

measure� of course� is to use the Legendre polynomials themselves� Based on this

requirement� the Legendre polynomials up to order �� are worked out�

Some of the higher order Legendre polynomials are included in Appendix A�

To speed up the computation of Legendre polynomials further� the well known

Horner�s Rule has been applied� For instance� a real polynomial f�x� of degree n

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or less is given by

f�x� � anxn � an��x

n�� � ���� a�x� � a�x� a� ��� ��

with the coecients a�� a�� a�� ���� an��� and an representing real numbers� In

programming practice� assuming that all coecients are nonzero� a straightforward

naive approach to compute this polynomial will cost n�n� �

multiplications and n

addition operations� However� with Horner�s Rule� the polynomial f�x� can be

expressed by writing

f�x� � �������anx� an���x� an���x� ����x� a��x� a�� ��� ��

With this new formula� it requires only n multiplications and n additions to compute

the polynomial� Since the operation of multiplication takes much longer than that

of addition� in terms of the computation time� the new formula is about n�� times

faster than the straightforward naive approach�

Adopting the high order Legendre polynomials listed in Appendix A and Hor�

ner�s Rule has dramatically reduced the computing time required in Legendre

moments computation� and more importantly� made this research possible�

��� Zernike Moments

����� Introduction

As mentioned in the previous chapter� the complex Zernike moments of order n

with repetition m for a continuous image function f�x� y� are de�ned as

Anm �n� �

Z Zx�y���

f�x� y�V �nm� � �dxdy ��� ��

in the xy image plane� and

Anm �n� �

Z ��

Z �

�f� � �Rnm� � exp��jm� d d ��� ��

��

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in polar coordinates� The real�valued radial polynomial Rnm� � is de�ned as

Rnm� � �

n�jmj�X

s�

����s �n� s��

s� �njmj�� s�� �n�jmj

�� s��

n��s� ��� ��

where n� jmj � even and jmj � n�

The feature of invariance under image rotation makes the Zernike function one of

the most important moments� However� the nature of Zernike moment computing�

using the summation of square pixels to achieve the computation de�ned on a unit

disk� makes it more dicult to solve the accuracy problems�

For a digitized image function f�x� y�� as discussed in the previous chapter� the

double integration of � ���� can be approximated by double summation�

bAnm �n � �

Xxi

Xyj

hAnm�xi� yj� f�xi� yj�� x�i � y�j � �� ������

where

hAnm�xi� yj� �Z xi

�x�

xi��x�

Z yj�y�

yj��y

V �nm� � �dxdy� ������

From the de�nitions of bAnm and hAnm�xi� yj�� we can �nd that there are two kinds

of major errors in the computation of the Zernike moments bAnm� geometric and

approximate�

����� Geometric Error

When computing the Zernike moments� if the centre of a pixel falls inside the

border of unit disk x� � y� � �� this pixel will be used in the computation� if the

centre of the pixel falls outside the unit disk� the pixel will be discarded� Therefore�

the area covered by the moment computation is not exactly the area of the unit

disk�

Figure �� shows the dierent areas covered by a unit disk and all pixels whose

centres fall inside the unit disk�

��

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Figure �� � Dierent areas covered by a disk and all pixels whose centres fall insidethe disk�

��

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In the case of Zernike moment� the unit disk is located in a units � units

square which is composed of n� n pixels� Therefore� the area of the unit disk is �

If A�n� represents the number of pixels whose centres fall inside the unit disk� the

summation of the areas of all these pixels is

Apixels � A�n��

n�� ���� �

Now� the geometric error between the unit disk and the summation of all the pixels

used in the Zernike moment computation is

R�n� � A�n��

n�� � ������

For the Zernikemoment computing� it is crucial to know� when n tends to in�nity�

i�e�� if the number of pixels is increasing� how fast the geometric error R�n� converges

to zero�

In fact� this issue is closely related to a famous problem in analytic number theory�

due originally to Gauss and referred as The Lattice Points of a Circle Problem �����

����� The Lattice Points of a Circle Problem

Let A�x� be the number of lattice points �u� v� inside or on the circle u�� v� � x� so

that A�x� x as x tends to in�nity� Let

R�x� � A�x�� x� ������

and let � be the lower bound on the number such that

R�x� � O�x��� ������

We list some signi�cant steps in the history of the estimation of R�x� here�

Gauss ������� � �� � �������������

Sierpinski ������� � �� � ����������������

��

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Wal�sz ��� ��� � ������

� ��� �����������

Titchmarsh���� ������� � ����� ��� �����������

Hua���� ���� �� � ����� ��� ��������

and more recently

Iwaniec and Mozzochi���� ������� � ���� �������������� �

In the other direction� it has long been known that � � �� and this result also hasbeen improved by Hardy����� Landau����� and Ingham���� to�

limx��

R�x�

x�� �logx�

��

� �� ������

and

limx��

R�x�

x�

� ��� ������

This result shows that the smallest possible cannot reach � �� � This still remains

an open problem in the number theory�

Comparing our problem with The Lattice Points of a Circle Problem� we �nd that

the x in ������ is equivalent to n� in ������ when both x and n tend to in�nity�

On the other hand� the number of lattice points in The Lattice Points of a Circle

Problem and the number of pixels within the unit disk in our problem are identical

when the area of each lattice is �� and the area of each pixel is �n�� Then� it follows

that ������ can be

R�n� � A�n��

n��

� A�x��

x�

� R�x��

x

R�n� � O�x����� ������

Therefore� we obtain

R�n� � O�n����� �� ������

��

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With the latest result from Iwaniec and Mozzochi� the geometric error in the

Zernike moment computing is

R�n� � O�n���

�� �� ������

n� � n��� n��� n�� �n�� ��������� ��� ����� ������ �� ������� �

n���

�� ��������� ��������� ��������� ��������

n��

� ��������� ��������� ��������� ���������

Table �� � Range of geometric errors for several commonly used image sizes�

Several commonly used image sizes are employed here to show the range of geo�

metric errors in cases of n��� n���

�� � and n��

� � respectively� The results are displayed

in Table �� �

Like the case in our experiment� when n is �� with the best result from Iwaniec

and Mozzochi����� the geometric error is at the range of

n���

�� � �������������

Obviously� this is not a very encouraging result� Since the higher order Zernike

moments are the accumulations of the lower order computed Zernike moments� if

the geometric error is around O�n���

�� �� when the order of Zernike moments goes

higher� the accumulated geometric errors would quickly get out of control and the

use of higher order Zernike moments would be severely handicaped�

����� Approximation Error

As discussed previously� in the xy image plane with a digitized image function f�x� y��

the Zernike moments of order n with m repetitions are

bAnm �n � �

Xxi

Xyj

hAnm�xi� yj� f�xi� yj�� x�i � y�j � �� ������

��

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where

hAnm�xi� yj� �Z xi

�x�

xi��x�

Z yj�y

yj��y

V �nm� � �dxdy� ���� �

The Zernike polynomials V �nm� � � are de�ned as

Vnm� � � � Rnm� � � exp�jm�� ������

where the Radial polynomial Rnm� � is

Rnm� � ��n�jmj ��X

s�

����s �n� s��

s� �njmj�

� s�� �n�jmj�

� s�� n��s� ������

Unlike the Legendre polynomials Pm�x� and Pn�y�� which are independent� the

Zernike polynomials Vnm� � � are two�dimensional functions of and � Therefore�

to reduce the approximation error in the Zernike moment computation is more

complex than that of Legendre moments�

Under this particular situation� naturally� the way to reduce the approximation

error is to compute the double integrations and hAnm�xi� yj� by using some well

known cubature formulas� ���

Suppose we have a two�dimensional domain and wish to approximateR� f�x�d �

Let f � � and aT � �a� b� � � We have the Taylor expansion of the integrandfunction f�x��

f�x� � f�x� y�

� f�a� � �x� a�fx�x� � �y � b�fy�a�

��

��x� a��fxx�a� � �x� a��y � b�fxy�a� � �y � b��fyy�a��

������

�n� ��� �n��Xi�

�n� �i

��x� a�n�i���y � b�i

�n��f�a�

�xn�i���yi�

�error� ������

Let

I�f �Z�f�x�d �

��

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then it follows� �� that�

I�f � j jf�a� � fx�a�I��x� a� � fy�a�I��y � b� � ���

��

�n� ���n��Xi�

�n� �i

��n��f�a�

�xn�i���yiI���x� a�n�i���y � b�i�

�error� ������

and

Cnf � f�a�nXi�

Ai � fx�a�nXi�

Ai�xi � a� � fy�a�nXi�

Ai�yi � b� � ���

��

�n� ��� �n��Xj�

�n� �j

��n��f�a�

�xn�j���yj

nXi�

Ai�xi � a�n�j���y � b�j�� ������

Taking a � �� and comparing I�f with Cnf � we obtain the following equations

which can determine the weights of a cubature formula�

nXi�

Aixk�ji yji � I�x

k�jyji j � �� �� ���� k � n� ������

where n is the number of nodes inside � and

I� �Z�f�x� y�d � ������

This is a linear system of equations for the weights Ai� For example� taking j � �

we obtain the equations

nXi�

Ai � j jnXi�

Aixi � I�x�nXi�

Aiyi � I�y

nXi�

Aix�i � I�x

��nXi�

Aixiyi � I�xy�nXi�

Aiy�i � I�y

��

To achieve sucient accuracy� traditionally� we can increase the number of nodes

in each pixel� Solving the linear system equations obtained from the ������� we can

�nd the weights for all nodes inside each pixel�

��

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s s s

s

s

s

Figure ���� ��dimensional formula I�

One simple formula which can be adopted to increase the approximation accuracy

is shown in Figure ���� Using the unit height� we can determine the weights of the

cubature formula

C�f � A�f��� �� �A�f��� �� �A�f��� �� �A�f������ �A�f���� ��� ������

Employing ������� we obtain �ve linear system equations��������������������������������������������

�Xi�

� j j�X

i�

Aixi � I�x

�Xi�

Aiyi � I�y

�Xi�

Aix�i � I�x

�Xi�

Aiy�i � I�y

������

where

I� �Z�f�x� y�d �

From ������� straightforwardly� we can obtain the ��dimensional cubature formula

by solving the following �ve linear system equations����������������

A� �A� �A� �A� �A� � ��A� �A� � �

�A� �A� � ��A� �A� � ��

�A� �A� � ���

���� �

��

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The solutions of the above �ve linear system equations lead to

C�f ��

�f�f��� �� � f��� �� � f��� �� � f������ � f���� ��g� ������

where f�x�� x�� is a two dimensional function�

We can use another type of the ��dimensional formula� which is shown in Fig�

ure ���� as well�

s s s

s

s

s

Figure ���� ��dimensional formula II�

With the same �ve equations in ������ but dierent xi and yi values� we have a

set of �ve linear system equations���������������

A� �A� �A� �A� �A� � �����A� ����A� � �

����A� ����A� � ���� �A� ��� �A� � ��

��� �A� ��� �A� � ���

������

which produce the ��dimensional cubature formula�

C�f ��

�f�f��� �� � f��� �� � f��� �� � f������ � f���� ��g� ������

The number of nodes in each pixel can be increased further to achieve higher accur�

acy� An example is to use the ���dimensional cubature formula� which is illustrated

in Figure ���� to reduce the approximation error�

We can determine the weights of the cubature formula

C��f � A�f��� �� �A�f����� ���� �A�f���������� �A�f�����������

�A�f������ ���� �A�f��� �� �A�f��� �� �A�f��� �� �A�f������

�A��f������ �A��f������� �A��f���� �� �A��f���� �� ������

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s s s

s s

s s s

s s

s s s

Figure ���� ���dimensional formula�

the same way we did for the ��dimensional formula by employing ������ and solving

thirteen linear system equations� The equations are listed in the Appendix of this

chapter� and the solutions of these equations lead to

C��f ��

��f� �f��� ��

����f����� ���� � f���������� � f����������� � f������ �����

���f��� �� � f��� �� � f������ � f���� ���

���f��� �� � f������ � f������� � f���� ���g ������

It is possible to impose additions on the weights Ai to reduce the number of linear

system equations and make the determination of the Ai easier� For example� we can

assume that the thirteen dimensional cubature formula is

C��f � A���� ��

� A��f����� ���� � f���������� � f����������� � f������ �����

� A��f��� �� � f��� �� � f������ � f���� ���

� A��f��� �� � f������ � f������� � f���� ���� ������

��

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which leads to ���������������������������������

��Xi�

� j j��Xi�

Aix�i � I�x

��Xi�

Aix�i y

�i � I�x

�y�

��Xi�

Aix�i � I�x

��

������

and ���������A� ��A� ��A� ��A� � �

�A� � A� ��A� � ���� �A� ��A� � ����� �A� � A� ��A� � ���

������

Therefore� the ���dimensional cubature formula ������ can be written as�

C��f ��

��f���f��� ��

� ���f����� ���� � f���������� � f����������� � f������ �����

� �f��� �� � f��� �� � f������ � f���� ���

� ��f��� �� � f������ � f������� � f���� ���g� ������

If we ignore the geometric error and let the image function f�x� y� be a constant

image with graylevel a� like the case of Legendre moments� all Zernike moments

should equate to zero except A�� � a� Therefore� we can use the following measure

to evaluate the approximation errors of the Zernike moments

EA �XX j bAnmj� m � n �� �� ���� �

The two dierent types of ��dimensional cubature formulas� the ���dimensional

formula with dierent sets of weights� and the simplest ��dimensional formula are

employed to evaluate the approximation errors in the computation of the Zernike

moment� All normalized EA�s which are less that ��� are illustrated in Figure ����

and their values are listed in Table ����

��

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0 5 10 15 20 25 30Moment Order

0

0.2

0.4

0.6

0.8

1S

quar

ed A

ppro

xim

atio

n E

rror

1-D5-D(I)5-D(II)13-D(I)13-D(II)

Figure ���� Normalized approximation errors obtained by applying �ve dierenttypes of multi�dimensional cubature formulas on a constant image�

Order ��D ��D�I� ��D�II� ���D�I� ���D�II� ������ ������ ������ ������ ������� ������ ������ ������ ������ ������� ������ ������ ������ ������ ������� ��� �� ���� ������ ���� � ���� ��� ����� ������ ������ ������ ������� ������ ������ ������ ������ �������� ������ ������ ������ ������ �������� �� ��� ������ ������ ������ �������� ������ ������ ������ ����� �� � � ������ ����� �� ��� ����� � ������ � ������

Table ���� Values of the normalized approximation errors from appling �ve dierenttypes of multi�dimensional cubature formulas on a constant image�

��

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Table ��� and Figure ��� show that the multi�dimensional formulas could not pro�

duce better results than the simplest ��point formula did� In other words� the tradi�

tional method to reduce the approximation errors could not improve the accuracy in

this particular situation�

The reason that these multi�dimensional cubature formulas do not work is that

the Zernike moments are de�ned within the unit disk x� � y� � �� Since we use

all pixels whose centres fall into the unit disk for the Zernike moment computing�

the one�dimensional formula will not produce extra errors because all f�xi� yj� used

in the computing are covered by the de�nition� However� when multi�dimensional

cubature formulas are adopted� on the boundary of the unit disk� some f�xi� yj�

used to compute the pixels on the boundary will not �t the condition x� � y� � ��

For example� with the condition x� � y� � �� there will be respectively ��� ���

and ��� nodes used in the Zernike moment computation which fall outside the

unit disk for ��dimensional formula I� ��dimensional formula II� and ���dimensional

formulas� This certainly brings extra errors to the Zernike moments and makes the

approximation errors go up quickly�

����� A New Proposed Solution to Reduce Approximation

Error

We rede�ne the digitized version Zernike moments as

bAnm �n� �

Xxi

Xyj

hAnm�xi� yj� f�xi� yj�� x�i � y�j � �� �� ������

where

hAnm�xi� yj� �Z xi

�x�

xi��x�

Z yj�y

yj��y

V �nm� � � dxdy� ������

and � is an adjustable factor� For example� in our case� we let

� ��x��y

�� �� ������

��

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where � is an arbitrary small number� Then� with this new condition

x�i � y�j � �� �x��y

�� ��

the number of nodes that fall outside the unit disk will be reduced to ��� �� and

�� for the ��dimensional formula I� ��dimensional formula II� and ���dimensional

formulas� respectively� Obviously� under this condition� the geometric errors will be

higher�

Employing ������ as the � value� we re�evaluate all �ve dierent formulas discussed

above� Table ��� and Figure ��� show the results�

0 5 10 15 20 25 30 35Moment Order

0

0.2

0.4

0.6

0.8

1

Squ

ared

App

roxi

mat

ion

Err

or

1-D5-D(I)5-D(II)13-D(I)13-D(II)

Figure ���� Normalized EA�s obtained by applying �ve dierent types of multi�dimensional cubature formulas on a constant image with the new proposed tech�nique�

Compared with Figure ���� Figure ��� shows that the error EA goes up quickly

to the level of ��! for all �ve dierent formulas� then the ratios of increase slow

��

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Order ��D ��D�I� ��D�II� ���D�I� ���D�II� ������ ������ ������ ������ ������� ������ ������ ������ ������ ������� ����� �� � � �� �� �� ��� �� ���� ������ ��� �� ������ ��� �� ��� ���� ����� ������ ������ ������ ��� ��� ���� � ������ ���� � ������ �������� ������ ������ ���� � ������ �������� ������ ������ ������ ������ �������� ������ ���� � ������ ������ ���� � � ������ ������ ������ ������ ������ ������ ��� � ������ ������ ������ � ������ ������ ������ ������ ������ � ���� � ������ ������ ������ ������ � ������ ����� ������ ������ �������� ������ ������ ��� ��� ������ ������ ���� ��� ������ ��� �� ���� ��� ������

Table ���� Values of the normalized EA�s from appling �ve dierent types of multi�dimensional cubature formulas on a constant image with the new proposed technique�

��

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down� As expected� all four multi�dimensional formulas produce better results than

the simplest one�dimensional formula does�

The results shown in Figure ��� and Table ��� are better than those of Figure ���

and Table ���� However� one is reluctant to call the digitized version of Zernike

moments under the new condition a �awless solution to the approximation error

problem of the Zernike moment computing� Though it indeed controls the increase

ratio of the error to a lower level under certain circumstances� we would like to use

it as an alternative rather than call it a complete solution to compute the Zernike

moments�

��� Conclusions

In this chapter� the problems of accuracy and eciency in moment computing are

discussed�

����� Legendre Moment Computing

It has been shown that most problems concerning accuracy and eciency in the

Legendre moment computing have been solved� Therefore� we are able to use the

higher order of the Legendre moments in further research con�dently�

����� Zernike Moment Computing

Because of the nature of the Zernike moment calculations� the two major problems

in the Zernike moment computing� geometric and approximation errors� are more

dicult�

Geometric Error

We adopted the latest results from a classical problem in Number Theory� The

Lattice Points of a Circle� in our study on the geometric error of Zernike moment

��

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computing� It shows that the geometric error is

R�n� � O�n����� �� ������

For example� in the case of n � �� n���

�� � �������������

We have to admit that the errors in the range of O�n���

�� � are too large to be

ignored� More seriously� since the higher order Zernike moments are the accumu�

lations of the lower order computed Zernike moments� when the order of Zernike

moments goes higher� the accumulated geometric errors will be quickly out of control�

Increasing the size of an image� or n� will indeed make the geometric errors R�n�

for the individual moments smaller� However� in many cases� to increase n will

result in higher order moments being required to provide the needed image features�

Therefore� to increase the size of an image in order to reduce the geometric errors is

not recommended�

Certainly� the existence of the geometric errors severely handicaps the usage of the

Zernike moments�

Approximation Error

The approximation error in the Zernike moment computing is discussed in this

chapter as well� To reduce the approximation error� some well known cubature for�

mulas are applied� However� to implement the multi�dimensional cubature formulas

cannot improve the accuracy signi�cantly�

The digitized Zernikemoments are achieved from the summation of square pixels�

whose centres fall inside the unit disk� However� on the unit disk boundary� a pixel

whose centre falls inside the boundary does not mean that the entire pixel falls

into the unit disk� Therefore� the multi�dimensional cubature formulas� which use a

number of nodes inside a pixel to achieve sucient accuracy� will no longer be valid�

��

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To make the the multi�dimensional cubature formulas valid in the Zernikemoment

computation� we proposed a new condition in the Zernike moment computing�

The new condition

x�i � y�j � � � ��

where

� ��x��y

�� �

in our case� was employed as an alternative solution� The results show that� for all

four multi�dimensional formulas� the approximation errors go up quickly in the early

stage� then the ratios slow down� From the approximation error point of view� the

multi�dimensional formulas under the new condition provide better results than the

simplest one�dimensional formula does�

Though it is premature to say that changing the condition is a perfect solution

to reduce approximation errors in the Zernike moment computing� careful selection

of the multi�dimensional formula and modi�cation on the condition on the choice

of better points� i�e� x�i � y�j � � � �� will indeed improve the performance of the

Zernike moment computation signi�cantly�

��

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Appendix

Following is a set of thirteen linear system equations�����������������������������������������������������������������������������������������������������������������������

��Xi��

j�j

��Xi��

Aixi I�x

��Xi��

Aiyi I�y

��Xi��

Aix�i I�x

��Xi��

Aixiyi I�xy

��Xi��

Aiy�i I�y

��Xi��

Aix�i I�x

��Xi��

Aix�i yi I�x

�y

��Xi��

Aixiy�i I�xy

��Xi��

Aiy�i I�y

��Xi��

Aix�i I�x

��Xi��

Aix�i y I�x

�y

��Xi��

Aix�i y

� I�x�y��

������

which leads to�������������������������������

A� A� A� A� A� A A A� A� A� A�� A�� A�� ���A� ��A� ���A� ���A� A A� A� �A�� �A�� �A�� ���A� ���A� ���A� ��A� A A �A� �A� �A�� A�� ����A� ���A� ���A� ���A� A A� A� A�� A�� A�� ������A� ����A� ���A� ����A� A �A� A�� �A�� ����A� ���A� ���A� ���A� A A A� A� A�� A�� �������A� ����A� �����A� �����A� A A� A� �A�� �A�� �A�� �����A� �����A� �����A� ����A� A �A� �A�� A�� �����A� ����A� �����A� �����A� A A� �A�� �A�� �����A� �����A� �����A� ����A� A A �A� �A� �A�� A�� ������A� �����A� �����A� �����A� A A� A� A�� A�� A�� ��������A� ������A� �����A� ������A� A �A� A�� �A�� ������A� �����A� �����A� �����A� A A� A�� A�� ����

������

Solving these equations gives us the formula in ������ used in Chapter ��

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Chapter �

Image Reconstruction from

Moments

In Chapter �� the problems of accuracy and eciency in moment computing have

been studied� In this chapter� we want to verify how much information is contained

in moments� This issue can be addressed by analyzing the reconstruction power of

the moments�

A problem which is raised here can be stated as follows� if only a �nite set of

moments of an image are given� how well can we reconstruct the image� We start

the investigation by discussing the inverse moment problem�

��� Inverse Moment Problem

Consider the characteristic function���� for the image function f�x� y��

F �u� v� �Z �

��

Z �

��f�x� y� ej�uxvy dxdy� �����

Provided that f�x� y� is piecewise continuous and the integration limits are �nite�

F �u� v� is a continuous function and may be expanded as a power series in u and v�

Therefore�

F �u� v� �Z �

��

Z �

��f�x� y�

�Xk�

�Xl�

�jux�k

k�

�jvy�l

l�dxdy

��

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�Z �

��

Z �

��

�Xk�

�ju�k

k�

�Xl�

�jv�l

l�xk yl f�x� y� dxdy

��Xk�

�Xl�

jkl

k� l�uk vl

Z �

��

Z �

��f�x� y�xk yl dxdy

F �u� v� ��Xk�

�Xl�

jkl

k� l�uk vlMkl� ��� �

where the interchange of order of summation and integration is permissible� and the

momentMkl is the geometric moment of order �k � l� of the image function f�x� y�

Mkl �Z �

��

Z �

��f�x� y�xk yl dxdy�

We see from ��� � that the moment Mkl is the expansion coecient to the ukvl

term in the power series expansion of the characteristic function of the image function

f�x� y��

Then� we consider the inverse form of the characteristic function F �u� v�� From

��� � and the two�dimensional inversion formula for Fourier transforms� it follows

that

f�x� y� ��

��

Z �

��

Z �

��F �u� v� e�j�uxvy dudv

f�x� y� ��

��

Z �

��

Z �

��

�Xk�

�Xl�

jkl

k� l�uk vlMkl e

�j�uxvy dudv� �����

However� the order of summation and the integration in ����� cannot be inter�

changed� Thus we conclude that the power series expansion for F �u� v� cannot be

integrated term by term� Particularly� if only a �nite set of moments is given� we

cannot use a truncated series in ����� to learn about the original image function

f�x� y��

The diculty encountered in ����� could have been solved if the basis set fukvlgwere orthogonal� Unfortunately� with the Weierstrass approximation theorem�����

the basis set fukvlg� while complete� is not orthogonal�

��

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To solve this problem� we need a set of basis functions which are orthogonal over

a �nite interval� Based on this requirement� the Legendre polynomials would be

the appropriate set�

��� Method of Legendre Moments

����� Theory of Image Reconstruction from Legendre Mo

ments

As mentioned in Chapter � the Legendre polynomials fPm�x�g���� are a completeorthogonal basis set on the interval ���� ���

Z �

��Pm�x�Pn�x� dx �

m� ��mn� �����

By the orthogonality principle� and considering that f�x� y� is piecewise continuous

over the image plane� we can write the image function f�x� y� as an in�nite series

expansion�

f�x� y� ��X

m�

mXn�

�m�n�n Pm�n�x�Pn�y�� �����

where the Legendre moment of f�x� y� with order �m� n� is de�ned by

�mn �� m� ��� n � ��

Z �

��

Z �

��Pm�x�Pn�y� f�x� y� dxdy� �����

However� in practice� one has to truncate in�nite series in ������ If only Legendre

moments of order �Mmax are given� the function f�x� y� can be approximated by a

truncated series�

f�x� y� fMmax�x� y� �MmaxXm�

mXn�

�m�n�n Pm�n�x�Pn�y�� �����

Furthermore� �m�n�n�s must be replaced by their approximations given by �������

yielding the following reconstruction scheme

bfMmax�x� y� �MmaxXm�

mXn�

b�m�n�n Pm�n�x�Pn�y�� �����

��

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This is actually the basic equation used in the image reconstruction via the Legendre

moments� It is important to note that when the given order Mmax is increased� the

previously determined b�m�n�n�s do not change������ Reconstruction Error Analysis

To measure the error between the original image and its reconstructed version� the

following formula is employed

Error� bfMmax � �Z �

��

Z �

��� bfMmax�x� y�� f�x� y���dxdy� �����

where Mmax is the highest moment order involved in reconstruction� and bf�x� y�represents the reconstructed image from f�x� y��

Since

bfMmax�x� y� �MmaxXm�

mXn�

Pm�n�n�x�Pn�y� b�m�n�nand

f�x� y� ��X

m�

mXn�

Pm�n�n�x�Pn�y��m�n�n�

therefore

bfMmax�x� y�� f�x� y� �MmaxXm�

mXn�

Pm�n�n�x�Pn�y� b�m�n�n�

�Xm�

mXn�

Pm�n�n�x�Pn�y��m�n�n

�MmaxXm�

mXn�

Pm�n�n�x�Pn�y� �b�m�n�n � �m�n�n�

��X

mMmax�

mXn�

Pm�n�n�x�Pn�y��m�n�n� ������

Then� we have

Error� bfMmax � �Z �

��

Z �

��� bfMmax�x� y�� f�x� y���dxdy

�Z �

��

Z �

���MmaxXm�

mXn�

Pm�n�n�x�Pn�y� �b�m�n�n � �m�n�n��� dxdy

��

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� Z �

��

Z �

���MmaxXm�

mXn�

Pm�n�n�x�Pn�y� �b�m�n�n � �m�n�n��

��X

mMmax�

mXn�

Pm�n�n�x�Pn�y��m�n�n� dxdy

�Z �

��

Z �

���

�XmMmax�

mXn�

Pm�n�n�x�Pn�y��m�n�n�� dxdy� ������

Since the second term in ������ is zero and Legendre polynomials Pm�x� and

Pn�y� are orthogonal� applying � ���� to ������� we have

Error� bfMmax � � �MmaxXm�

mXn�

�m� n� � �

n � ��b�m�n�n � �m�n�n�

���X

mMmax�

mXn�

�m� n� � �

n� ���m�n�n� ���� �

As shown in ���� �� the reconstruction error Error� bfMmax� consists of two parts�

The �rst part comes from the discrete approximation of the true moment ��m�n�

while the second part is a result of using a �nite number of moments�

With the new techniques introduced in the previous chapter� we can reduce the

discrete approximation error to a tolerable low level� Based on these new techniques�

the experimental results of image reconstruction via Legendre moments� which will

be presented in the following section� indicate that when the maximum given order

Mmax reaches a certain value� bfMmax�x� y� can be very close to the original image

function f�x� y��

����� Experimental Results

The proposed approach was implemented in the C language and tested on a �MHz

��� computer� In the experiments� a set of �ve Chinese characters� shown in Fig�

ure ���� is used as the test images� Each image consists of � � � pixels and therange of graylevels for each pixel is � � All characters have the gray level �� and

the background has the value ��

��

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Figure ���� Five original Chinese characters used in image reconstruction via Le�gendre moments� From left to right are C�� C�� C�� C�� and C��

The reason we use these �ve Chinese characters is that they are very similar to

each other� Actually� among more than ������ Chinese characters� one cannot �nd

another set of �ve�or even set of three or four� in which the individual characters

are so similar to each other� Therefore� it seems that if these �ve characters can be

recognized successfully� the method can be applied to all the Chinese characters with

con�dence�

The normalized mean square error between the original image f�x� y� and the

reconstructed image bfMmax�x� y� is de�ned by

e�Mmax�

Error� bfMmax �R R�f�x� y��� dxdy

R R� bfMmax�x� y�� f�x� y��� dxdyR R

�f�x� y��� dxdy� �� � x� y � ��

������

which is considered as a measure of the image reconstruction ability of the moments

and adopted here�

The alternative extended Simpson�s rule with order N � � is applied to compute

the Legendre moments in this experiment� Table ��� and Figure ��� show the

e�Mmaxvalues from the reconstructed Chinese characters from order up to order ���

It should be noted that the e�Mmaxdecreases monotonically in the cases of all �ve

characters�

Figure �� shows the �ve original Chinese characters and their reconstructed pat�

terns� The �rst column illustrates �ve original characters� The second column to the

ninth column display the reconstructed patterns of all characters in the �rst column

with order �� � � ��� ��� ��� ��� � and ��� respectively�

��

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Order C� C� C� C� C�

�������� ������ �������� �������� ��������� ������ � �������� �������� ������ � ��������� ������ � �������� ������ � �������� ���� ��� ���� ��� �������� �������� �������� ������� �� ������� �������� ������ �������� ������� � �������� �������� ����� �� �������� ���������� ���� ��� ������� �������� �������� ���������� ������ � ���� ��� �������� �������� ���� �� �� ��� ���� ��� ���� ��� ���� ��� ���� ��� �� � � ��� ���� ��� ���� ��� ���� ��� ���� ��� ���� ��� ���� ��� � �� ��� �� ��� ���� ��� �� � � �������� �������� �������� �������� ��� �� � � �������� �������� �������� �������� ����� �� � ���� ��� ���� ��� �������� ����� � ���������� �������� �������� ���� � ���� ��� ���� ���� �������� �������� �������� �������� ������ ��� �������� �������� �������� �������� ���������� �������� ������ � �������� ������� ����� ���� �������� �������� �������� ����� �� ���������� �������� �������� �������� �������� ��������� �������� ����� �� ����� �� �������� ������ �� �������� �������� �������� �������� ���������� ����� �� �������� �������� �������� ���������� �������� ���� ��� ���� ��� �������� ���� ����� ���� ��� ���� ��� ���� � � ���� ��� ���� ���� ���� ��� ���� � � ���� ��� ���� ��� ���� ����� ���� ��� ���� ��� ���� ��� ���� �� ���� ����� ���� ��� ���� �� ���� ��� ���� �� ���� ���

Table ���� The values of normalized reconstruction errors for the �ve reconstructedChinese characters�

��

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Figure �� � Five Chinese characters and their reconstructed patterns via Legendremoments�

0 10 20 30 40 50 60Moment Order

0

10

20

30

40

50

Rec

onst

ruct

ion

Err

or (

x10-3

)

C1C2C3C4C5

Figure ���� Normalized reconstruction errors for the �ve reconstructed Chinese char�acters�

��

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Clearly� the numerical results shown in Table ��� and Figure ��� are concordant

with the visual results presented in Figure �� �

��� Method of Zernike Moments

����� Theory of Image Reconstruction from ZernikeMoments

As discussed in Chapter � the Zernike polynomials

Vnm�x� y� � Vnm� sin� cos� � Rnm� � exp�jm�� ������

where the Radial polynomial Rnm� � is de�ned as

Rnm� � ��n�jmj ��X

s�

����s �n� s��

s� �njmj�

� s�� �n�jmj�

� s�� n��s� ������

are a complete set of complex�valued functions orthogonal on the unit disk

x� � y� � �� Z Zx�y���

�Vnm�x� y���Vpq�x� y� dxdy �

n � ��np �mq� ������

Subject to the discussion of orthogonal functions for the Legendre moments� the

image function f�x� y� can be expanded in terms of the Zernike polynomials over

the unit disk as

f�x� y� ��Xn

Xm

Anm Vnm�x� y�� ������

where m takes on positive and negative integers subject to the conditions n� jmj �even� and jmj � n�

We rewrite the de�nition of Zernike moments here for convenience�

Anm �n� �

Z Zx�y���

f�x� y�V �nm� � �dxdy� ������

If terms only up to the maximum Zernike moment Nmax are taken� then the

truncated expansion is the approximation to f�x� y��

f�x� y� bfNmax�x� y� �NmaxXn�

Xm

bAnm Vnm�x� y�� ������

��

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where bAnm and bfNmax�x� y� are the Zernike moment numerically computed from

f�x� y� and the reconstructed image from f�x� y� with the maximum Zernike mo�

ment Nmax�� while m is subject to the conditions n� jmj � even� and jmj � n�

Note that V �nm� � � � Vn��m� � �� ������ can be expanded as

bfNmax�x� y� �NmaxXn�

Xm

bAnm Vnm� � �

�NmaxXn�

Xm��

bAnm Vnm� � � �NmaxXn�

Xm��

bAnm Vnm� � �

�NmaxXn�

Xm��

bAn��m Vn��m� � �

�NmaxXn�

Xm��

bAnm Vnm� � � � bAn� Vn�� � �

bfNmax�x� y� �NmaxXn�

Xm��

� bA�nm V �

nm� � � �bAnm Vnm� � ��

� bAn� Vn�� � �� ��� ��

considering that

Vnm� � � � Rnm� � �cos�m� � j sin�m��

and

V �nm� � � � Rnm� � �cos�m� � j sin�m���

Then ��� �� becomes

bfNmax�x� y� �NmaxXn�

Xm��

f�Re� bAnm�� j Im� bAnm��Rnm� � �cos�m� � j sin�m��

� �Re� bAnm� � j Im� bAnm��Rnm� � �cos�m� � j sin�m��g

� �Re� bAn�� � j Im� bAn���Rn�� �

�NmaxXn�

Xm��

Rnm� � f�Re� bAnm�� j Im� bAnm�� �cos�m�� j sin�m��

� �Re� bAnm� � j Im� bAnm�� �cos�m� � j sin�m��g

� �Re� bAn�� � j Im� bAn���Rn�� �

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bfNmax�x� y� �NmaxXn�

Xm��

Rnm� � �Re� bAnm� cos�m�� Im� bAnm� sin�m��

� �Re� bAn�� � j Im� bAn���Rn�� �� ��� ��

The formula ��� �� is the basic equation employed in image reconstruction via the

Zernike moments�

����� Reconstruction Error Analysis

In this section� a similar de�nition as in����� is adopted to measure the error between

the original image and its reconstructed version from Zernike moments�

Error� bfNmax � �Z Z

x�y���j bfNmax�x� y�� f�x� y�j�dxdy� ��� �

Since

bfNmax�x� y�� f�x� y� �NmaxXn�

Xm

bAnmVnm�x� y�

��NmaxXn�

Xm

AnmVnm�x� y� ��X

nNmax��

Xm

AnmVnm�x� y��

�NmaxXn�

Xm

Vnm�x� y�� bAnm �Anm�

��X

nNmax��

Xm

AnmVnm�x� y�� ��� ��

therefore it follows that

Error� bfNmax � �Z Z

x�y���j bfNmax�x� y�� f�x� y�j�dxdy

�Z Z

x�y���

NmaxXn�

Xm

jVnm�x� y�j�j bAnm �Anmj�dxdy

� Z Z

x�y���

NmaxXn�

Xm

Vnm�x� y�� bAnm �Anm� dxdyZ Zx�y���

�XNmax��

Xm

AnmVnm�x� y� dxdy

�Z Z

x�y���

�XnNmax

Xm

jV �nmx� y�j�jAnmj� dxdy� ��� ��

��

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It is clear that the second term on the right side of ��� �� is zero� So� the recon�

struction error for Zernike moments consists of two terms�

Error� bfNmax � �Z Z

x�y���

NmaxXn�

Xm

jVnm�x� y�j�j bAnm �Anmj� dxdy

�Z Z

x�y���

�XnNmax�

Xm

jVnm�x� y�j�jAnmj� dxdy� ��� ��

By recalling ������� we obtain�

Error� bfNmax � � NmaxXn�

Xm

j bAnm �Anmj�n� �

� �X

nNmax

Xm

jAnmj�n� �

� ��� ��

where the �rst part is from the approximation error in Zernike moment comput�

ing and the second term is due to truncating the higher order moments in image

reconstruction�

����� Experimental Results

The same set of �ve Chinese characters used in the case of Legendre moments is

employed in this section as the test images as well� Figure ��� illustrates the �ve

characters on unit disks�

Traditional Zernike Moment Method

The traditional Zernike moment method� which de�nes the Zernike moments

Anm �n� �

Z Zf�x� y�V �

nm� � �dxdy

on the unit disk

x� � y� � ��

is implemented in the image reconstruction� The simplest ��dimensional formula�

along with the two dierent types of ��dimensional formulas� and the ���dimensional

formula with two dierent sets of coecients are employed in the experiment�

��

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Figure ���� Five original Chinese characters used in image reconstruction viaZernike moments� From left to right are C�� C�� C�� C�� and C��

The normalized mean square errors

e� �

R R jf�x� y�� bf�x� y�j�dxdyR R�f�x� y���dxdy

� x� � y� � �� ��� ��

are adopted here to measure the qualities of the reconstructed images via theZernike

moments�

Order ��D ��D�I� ��D�II� ���D�I� ���D�II� ����� �� �������� �������� �������� ��������� �������� �������� �������� �������� ��������� �������� �������� �������� �������� ��������� �������� �������� �������� ����� �� ����� ���� ������� �������� �������� �������� ��������� �������� �������� �������� �������� ���������� �������� �������� �������� �������� ���������� �������� ���� ��� ������ � �������� ������ �� �������� �������� �������� ������ �������� � �������� ����� �� ���� ��� ���� �� �� �� � ����� � �� ����� ����� � ������� �������� � ��� ��� ����� �� �������� �������� ������� � �� ����� ������� �� ����� � �������� ���������� ���� ��� ��� �� �� �������� ����� ���� �������� ���������� �������� ���������� ������ � ���������� �������� ��������

Table �� � Values of the normalized mean square errors from appling �ve dierentformulas to character C��

The Chinese character C� is used as the test image for all �ve dierent formu�

las� Table �� and Figure ��� show the e� values for dierent formulas when the

��

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Figure ���� The Chinese character C� and its reconstructed patterns via Zernikemoments�

reconstruction orders go up�

Only the simplest ��dimensional formula and the ��dimensional formula II� which

is shown in Figure ���� provide relatively lower computing errors� The remaining

three formulas certainly are not good candidates for image reconstruction because

of the excessive computing errors�

Figure ��� illustrates the reconstructed images of C�� The �rst row and the second

row display the results from using the ��dimensional formula and the ��dimensional

formula II� respectively� The patterns in the �rst column are reconstructed from

order ��� then from left to right� they are the reconstructed images from order ���

��� �� � �� �� �� and ��� respectively�

When the order is �� Table �� indicates that in terms of the normalized mean

square error� the reconstructed image from the ��dimensional formula II has lower

error than that of ��dimensional formula� However� the visual results are contrary�

The reason is that the normalized mean square error treats all pixels equally� while

the individual key pixels which contain more features aect the visual results more

signi�cantly�

��

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0 5 10 15 20 25 30 35 40Moment Order

0

0.2

0.4

0.6

0.8

1

Squ

ared

App

roxi

mat

ion

Err

or

1-D5-D(I)5-D(II)13-D(I)13-D(II)

Figure ���� The normalized mean square errors from appling �ve dierent formulasto character C��

��

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Modi ed Zernike Moment Method

We introduced a modi�ed version of the Zernike moments in Chapter �� We re�

de�ned the Zernike moments as

bAnm �n� �

Xxi

Xyj

hAnm�xi� yj� f�xi� yj�� x�i � y�j � �� �� ��� ��

where

hAnm�xi� yj� �Z xi

�x�

xi��x�

Z yj�y�

yj��y�

V �nm� � �dxdy� ��� ��

and � is an adjustable factor� For example� in this research� we let

� ��x��y

�� �� ������

where � is an arbitrary small number�

The main reason to adopt ������ is that when we use the multi�dimensional for�

mulas to increase the approximate accuracy� we want to reduce the number of nodes

which fall outside the unit disk� The price of adopting the new version is that the

geometrical errors will become larger�

For the sake of convenient comparison� the same Chinese character C� is employed

as the test image� and the normalized mean square error de�ned in ��� �� is adjusted

to

e� �

R R jf�x� y�� bf �x� y�j�dxdyR R�f�x� y���dxdy

� x� � y� � � � �� ������

to measure this new method�

Table ��� and Figure ��� show the e� values of all �ve dierent formulas when the

orders of the reconstructed images go up�

Table ��� and Figure ��� indicate that all �ve dierent formulas perform bet�

ter than they did in the cases of traditional Zernike moments� Speci�cally� four

multi�dimensional formulas produce lower computing errors than the simplest ��

dimensional does� Among the multi�dimensional formulas� the ��dimensional formula

��

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Order ��D ��D�I� ��D�II� ���D�I� ���D�II� �������� �������� �������� �������� ��������� �������� ����� �� ������� ����� �� ����� ��� �������� �������� ������ � �������� ��������� �������� �������� �������� �������� ���������� �������� �������� �������� �������� ��������� �������� �������� �������� �������� ������ ��� �������� �������� �������� �������� ������� �� �������� �������� ������ � ������� ���������� ���� ��� ���� �� �������� ���� ��� ����� � � ������ � ��� ���� �������� ��� ���� ��� ���� �������� ��� ���� ��� ���� ��� �� � ��� �� � � ��� �� � ����� �� ��� � � ��� ���� ��� ���� � �������� �������� ��� ���� ��� �� � ��� ���� � �������� �������� ������� ��� ���� ��� ������ �������� �������� �������� �������� ��� ����� �������� ��� �� ������ � ��� ���� ��� ������ �������� �������� ��� ���� ���� �� ������� �� ��� ���� �������� ��� ���� �������� ���������� �������� �������� ���� ��� ����� �� ���������� ��� ���� ��� �� � �������� ���� ��� ��������� �������� �������� ����� �� �������� �� � ����� ����� �� �� ����� ���� ��� �� ����� ���������� ����� �� ���� ��� �������� ����� �� ���������� ������� �������� �� �� �� �� ����� ����� ��

Table ���� Values of the normalized reconstruction errors from the reconstructed �veChinese characters with the new proposed Zernike moment technique�

��

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Figure ���� The Chinese character C� and its reconstructed patterns via the modi�edZernike moments�

II which is shown in Figure ���� is superior to the other three formulas and is the

best candidate for the image reconstruction under this speci�c situation�

The reason that the ��dimensional formula II provides better result is that with

the new condition for the Zernike moments� all nodes used in this formula fall

inside the unit disk� For the ��dimensional formula I and the ���dimensional for�

mulas� however� �� and �� nodes used in the computing fall outside the unit disk�

respectively�

The reconstructed images from the Chinese character C� with �ve dierent for�

mulas are shown in Figure ���� The �rst row shows the reconstructed patterns from

the ��dimensional formula� and the second� third� fourth� and �fth show those of

��dimensional formula I� II� ���dimensional formula I� and II� respectively� All im�

ages in the �rst column are reconstructed from order ��� then from left to right� are

results from order ��� �� �� ��� ��� and ���

��

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0 5 10 15 20 25 30 35 40 45 50Moment Order

0

0.2

0.4

0.6

0.8

1

Squ

ared

App

roxi

mat

ion

Err

or

1-D5-D(I)5-D(II)13-D(I)13-D(II)

Figure ���� Normalized reconstruction errors from the reconstructed �ve Chinesecharacters via the new proposed Zernike moment technique�

��

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Figure ���� The �ve Chinese characters and their reconstructed patterns via themodi�ed Zernike moments with ��dimensional formula II�

By using the ��dimensional formula II� we reconstructed all �ve Chinese characters

with the Zernike moment order �� �� �� � � ��� ��� ��� and ��� respectively�

Figure ��� shows the reconstructed images while Table ��� and Figure ��� list and

illustrate the normalized mean square reconstruction errors of these patterns�

��� Conclusions

In this chapter� the image reconstructions via the Legendre moments and Zernike

moments have been discussed�

����� Image Reconstruction via Legendre Moments

Since we have solved most of accuracy and eciency problems related to the Le�

gendremoment computing in Chapter �� image reconstruction from the higher order

Legendre moments results in a successful task�

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Order C� C� C� C� C�

� �������� �������� ���� �� �������� ���� ��� � ��� � � ��� ���� ��� ���� ��� ���� ��� ���� � ������� �������� ��� � � ��� ���� ��� ����� ������ � �������� ������ � ��� ���� ���������� ��� ���� ��� �� � �������� ���� �� ���������� �������� �������� �������� ����� �� ���������� ���� ��� �������� ������� ����� �� ���������� �� �� �� �������� �� ����� �� ����� �� �����

Table ���� Values of the normalized reconstruction errors from the reconstructed �veChinese characters via the new proposed Zernike moment technique�

Five similar Chinese characters are used as the test images in the image recon�

struction procedure� Numerical and visual results both show that the reconstructed

images from the high order Legendre moments are very close to the original ones�

When the order goes higher� the dierence between the original image and its recon�

structed pattern becomes smaller�

����� Image Reconstruction via Zernike Moments

In this chapter� we discussed the image reconstruction via the traditional Zernike

moments and a proposed new Zernike moment method as well�

Traditional Zernike Moment Method

Five dierent cubature formulas are applied in the image reconstruction via tradi�

tional Zernike moment method� However� most of multi�dimensional formulas em�

ployed to increase the accuracy of the Zernike moment computing cannot provide

the expected results in the image reconstruction procedure� The reason for the fail�

ure is that the pixels on the unit disk boundary may contain nodes falling out of the

unit circle and which bring in excessive computing errors�

In this task� the simplest ��dimensional formula provides relatively better recon�

structed patterns than all the other multi�dimensional formulas do�

��

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New Proposed Zernike Moment Method

The same �ve formulas used in the traditional Zernike moment method are em�

ployed in the proposed new Zernike moment technique� As expected� all multi�

dimensional formulas can reconstruct images with better qualities than the ��dimen�

sional formula does� and the ��dimensional formula II obviously is the best approach�

In terms of image reconstruction� however� compared with the Legendremoment

method� the Zernike moment method is severely handicapped� The reason is that

the two major problems in the Zernike moment computation� geometrical error

and approximation error� cannot be solved completely� Though carefully selecting

the multi�dimensional cubature formulas can indeed reduce the computing errors

and improve the quality of the reconstructed image� it is very unlikely that the

performance of the Zernike moment method can reach the same level as that of the

Legendre moment method�

��

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Chapter �

Reconstruction of Noisy Images via

Moments

��� Introduction

Image reconstructions based on the orthogonal moments under noise free condition

have been discussed in Chapter �� However� in the presence of noise� the image

reconstruction is expected to be more complicated�

It is interesting to consider how close we can recover the original image from

a �nite set of moments computed from the noisy data� Certainly� the higher order

moments suer greater degradation due to noise� On the other hand� higher moments

are able to supply the detail information about the image �������� These two opposite

factors working against one another imply that there exists an optimal number of

moments yielding the best possible representation of the image�

��� Legendre Moments

Two commonly used orthogonal moments for image reconstruction are Zernike

moments and Legendre moments� In this chapter� the Legendre moments are

employed for discussion� However� the results presented can be extended straight�

forwardly to other types of orthogonal moments�

��

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As discussed in Chapter �� if only Legendre moments �mn of order �Mmax are

given� the original image function f�x� y� can be approximated by a truncated series�

bfMmax�x� y� �MmaxXm�

mXn�

b�m�n�n Pm�n�x�Pn�y�� �����

Clearly� the square reconstruction error

Error� bfMmax � �Z Z

� bfMmax�x� y�� f�x� y��� dxdy ��� �

goes to zero as Mmax� � �� see formula ���� �� That is� by employing higher

order moments one can make the reconstruction error arbitrarily small� However�

this scheme breaks down if the image is contaminated by noise� The noise aects the

higher order moments greater than it does to the lower order moments����� Therefore�

given the minimal value of the reconstruction error� an optimal number of moments

exists� In other words� when noise is involved� the square reconstruction error will

initially decrease �not necessarily in a monotonic way� down to a certain number of

moments and then increase to in�nity as N� ���

��� The Reconstruction Error

Let g�x� y� be the noisy degraded version of f�x� y� and adopt the following simple

image observation model

g�x� y� � f�x� y� � z�x� y�� �����

where z�x� y� is a Gaussian random process with zero mean and �nite variance ���

From the discussion in Chapter �� the Legendre moments of the noisy version

g�x� y� of f�x� y� can be obtained numerically by the formula

e�mn �� m� ��� n � ��

mXi�

nXj�

he�mn�xi� yj� g�xi� yj� �����

��

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where e�mn presents the Legendre moments obtained from the noisy image g�x� y�

and

he�mn�xi� yj� �Z xi

�x�

xi��x�

Z yj�y�

yj��y

Pm�x�Pn�y� dxdy� �����

Then� if the order �Mmax is given� the noisy image g�x� y� can be reconstructed by

bgMmax�x� y� �MmaxXm�

mXn�

e�m�n�n Pm�n�x�Pn�y�� �����

Since

Eg�xi� yj� � f�xi� yj�� �����

we have

Ee�mn � b�mn� �����

If we write e�mn as e�mn � e�mn �Ee�mn �Ee�mn�

then� it follows

e�mn � �e�mn �Ee�mn� � b�mn� �����

Similar to the case of non�noise in ���� �� Error�bgMmax � can be writen as

Error�bgMmax � �Z �

��

Z �

���bgMmax�x� y�� f�x� y��� dxdy

� �MmaxXm�

mXn�

�m� n� � �

n� ��e�m�n�n � �m�n�n�

���X

mMmax�

mXn�

�m� n� � �

n � ���m�n�n� ������

Therefore� E�Error�bgMmax �� has the form of

E�Error�bgMmax �� � �MmaxXm�

mXn�

�m� n� � �

n� �E�e�m�n�n � �m�n�n�

���X

mMmax�

mXn�

�m� n� � �

n � �E���m�n�n�� ������

��

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Considering ������ we have

E�e�m�n�n � �m�n�n�� � E��e�m�n�n �Ee�m�n�n� � �b�m�n�n � �m�n�n��

� E��e�m�n�n �Ee�m�n�n�� � �b�m�n�n � �m�n�n��

� E�e�m�n�n �Ee�m�n�n� �b�m�n�n � �m�n�n�� ���� �

Since

E�e�m�n�n �Ee�m�n�n� �b�m�n�n � �m�n�n� � ��

it follows

E�e�m�n�n � �m�n�n�� � E�e�m�n�n �Ee�m�n�n�� �E�b�m�n�n � �m�n�n�

� var�e�m�n�n� � �b�m�n�n � �m�n�n��� ������

Then� ������ becomes

E�Error�bgMmax �� � �MmaxXm�

mXn�

�m� n� � �

n� �var�e�m�n�n�

��MmaxXm�

mXn�

�m� n� � �

n� ��b�m�n�n � �m�n�n�

���X

mMmax�

mXn�

�m� n� � �

n � ���m�n�n� ������

The �rst term on the right�hand side of ������ depends on the noise added to the

original image� When the noise increases� it increases too� The second term on

the right side can be viewed as a matching measure between b�m�n�n and �mn based

on the total �N� �N� � moments� while the last term comes from truncating higher

order moments in reconstruction�

Comparing ���� � with ������� the main dierence between the cases of the absence

and presence of noise is focused on the �rst term on the right side of ������� In terms

of the sensitivity to noise� the higher order Legendre moments are more sensitive�

From ������ we can see that when the order of Mmax increases� the sums of

var�e�m�n�n� and �b�mn � �mn�� increase� On the other hand� however� the third

��

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Error

���

���

����

� ��

����

����

Order� �� �� � � �� �� �� �� �� ��

tt

t

t

ttt

t

t

t

tt

t

t

ttt

tt

tt

t

tttt

tttt

ttttttttt

tttt

tttt

ttt

tt

Figure ���� Square error Error�egMmax �� �� � ����

��

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term on the right�hand side of ������ decreases when the order of Mmax increases�

These two factors against each other� indicating that the square reconstruction error

Error�eg� will initially decrease down to an optimal number of moments and thenincrease�

In order to verify such properties� the Chinese character C� is employed as the

testing pattern in our experiment� Figure ��� shows the trend of the squared recon�

struction error Error�egMmax � averaged on �� runs with �� � ���� As expected� the

error decreases �rst� reaches minimum at N � ��� then increases� Table ��� lists the

numerical values of Error�bgMmax �� Figure �� illustrates the noisy image of C� and

its reconstructed versions from order � up to order ��� from left to right� �rst row

to last row� respectively�

Order I�N� Order I�N� Order I�N�� ��� �e��� � ��� �e��� �� �����e���� �����e��� �����e��� �� �����e���� �����e��� � �����e��� �� �� ��e���� �����e��� � �����e��� � ��� �e���� �����e��� � ��� �e��� �� �����e���� �����e��� � �����e��� �� ��� �e���� �����e��� � �����e��� �� �����e����� ��� �e��� � �����e��� �� �����e����� �����e��� � �����e��� �� �����e���� �����e��� �� �����e��� �� �����e����� �� ��e��� �� �����e��� �� �� �e����� �� ��e��� � �����e��� �� �����e����� �����e��� �� �� ��e��� �� �����e����� �����e��� �� �� ��e��� � �����e����� �����e��� �� �����e��� �� �����e����� �����e��� �� �����e��� �� ���� e����� ��� �e��� �� �����e��� �� �� ��e��� � �����e��� �� �����e��� �� �����e���

Table ���� Square reconstruction error Error�egMmax � with �� � ����

Obviously� the second and third terms in ������ are not aected by the noise�

therefore� when the noise increases or decreases� the sum of �b�mn � �mn�� is the

��

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Figure �� � Noisy version of C�� with �� � ���� and its reconstructed versions�

��

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only factor deciding the position of the optimal number of moments� Since the noise

aects the higher order Legendre moments more than it does to the lower ones� the

reconstruction error Error�bgMmax � will increase faster when the higher level noise

is involved� These discussions lead to the conclusion that when the level of noise

increases� the optimal number of moments for the least reconstruction error becomes

smaller� An experiment was designed to verify this assumption and the result is

illustrated in Figure ���� The same Chinese character C� and the noise model shown

in ����� are employed� The result is averaged on �� runs and the noise varies from

�� � ��� to �� � ���� Due to the nature of this experiment� the computation

involved is very large� It took �� hours for a �� MHz ��� computer to complete the

task� It is fair to say that the amount of computation involved in this experiment

has reached the limitation of a current personal computer�

��� Data�Driven Selection of the Optimal Number

It is very interesting to consider how to select a "good# optimal number N directly

from the available g�xi� yj�� Ideally� it is expected to have N�� to minimize the square

reconstruction error� Notice that N�� is a function of the data at hand�

This� in turn� is equivalent to taking the minimizer of the following criteria

Error�bgMmax ��ZZ

f��x� y�dxdy � �MmaxXm�

mXn�

�m� n� � �

n � ��e�m�n�n � �m�n�n�

���X

mMmax�

mXn�

�m� n� � �

n� ���m�n�n

���X

m�

mXn�

�m� n� � �

n� ���m�n�n

Error�bgMmax ��ZZ

f��x� y�dxdy � �MmaxXm�

mXn�

�m� n� � �

n � ��e�m�n�n � �m�n�n�

��MmaxXm�

mXn�

�m� n� � �

n� ���m�n�n� ������

However� the solution of equation ������ is not feasible since the �mn�s are unknown�

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N

��

��

��

��

����� ��� ���� ���� ��� ���

t

t t

t t t t

t

t t t t t

t

t t t t

t tt t

Figure ���� Optimal moments numbers�

��

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If �mn is replaced by e�mn� equation ������ will yield the unacceptable solution

N ���To overcome this diculty� a resampling technique utilizing the cross�validation

methodology has been introduced and the asymptotic optimality of such a selection

has been proved ���������

Other possible techniques to solve this problem include the utilization of discrete

measures with penalty factors� For instance� other than the Error�bgMmax � criterion�

its discrete approximation

ED�M� � �x�ynXi�

mXj�

�g�xi� yj� � bgMmax�xi� yj��� ������

can be used� ���

The empirical selectors corresponding to ED�M� are of the form

dED�M� � �x�ynXi�

mXj�

�g�xi� yj�� bgMmax�xi� yj���$�N�� ������

i�e�� it is a penalized version of the residual error

�x�ynXi�

mXj�

�g�xi� yj�� bfMmax�xi� yj����

see � ���

In the case of Gaussian noise� the prescription we proposed for the penalty factor

$�N� is

$�N� � ��� F ����L�N��x�y��p� ������

where L�N� is the total number of moments used in bgMmax � e�g�� L�N� ��N� �N�

for Legendre moments� With carefully selected p and F ����� signi�cant simulating

results of Figure ��� can be expected�

Generally speaking� the automatic selection of the optimal number N�� from the

data at hand is still an open problem� Though some initial experimental results

are quite positive on this task��������� due to the extreme amount of computation

��

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involved� we could not provide the full scale research on this issue based on the

available equipment� For a related problem in the context of image restoration� we

refer to � ���

��

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Chapter �

Character Recognition via Moments

��� Introduction

Character recognition is believed to be typical of many other practical problems

that depend on general shapes rather than details of the image� The recognition

of characters from imagery may be accomplished by identifying an unknown char�

acter as a member of a set of known characters� Various character recognition

techniques have been utilized to abstract characterizations for ecient character

representations� ���� �� Such characterizations are de�ned by measurable features

extracted from the characters� Therefore� the eectiveness of the technique for a

given application is dependent on the ability of a given technique to uniquely repres�

ent the character from the available information� Since no one single technique will

be eective for all recognition problems� the choice of character characterization is

driven by the requirements of a speci�c recognition task�

Based on theUniqueness Theorem����� the double moment sequence is uniquely

determined by an image function f�x� y�� This nature makes the method of moments

a proper candidate in character recognition�

��

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��� Character Recognition via Central Moments

In consideration of the fact that there is no inverse problem involved in the clas�

si�cation of visual patterns and characters� and the property of invariance under

translation� the classical moment is discussed in this chapter for the purpose of

pattern recognition�

As mentioned in Chapter � the central moments �pq are de�ned in � ���

�pq �Z �

��

Z �

���x� �x�p �y � �y�q f�x� y� dxdy�

where

�x �M��

M��� �y �

M��

M���

and Mpq are the classic moments de�ned in � ���

Mpq �Z �

��

Z �

��xp yq f�x� y�dxdy�

Hu demonstrated the utility of moment techniques through a simple pattern re�

cognition experiment����� The �rst two moment invariants were used to represent

several known digitized patterns in a two�dimensional feature space� The experi�

ment was performed by using a set of � capital letters as input patterns� In the

two�dimensional feature space� all the points representing each of the characters were

fairly distinct except those of M and W�

Compared with the set of English letters� the Chinese character set is large� and

in terms of character recognition� is more dicult to classify� In this section� similar

to Hu�s experiment� a simulation program of a character recognition model using

two moment invariants� has been proposed� The following two moment functions

X� �p��� � ��� �����

and

X� �q���� � ������ � ����� � ����� ��� �

��

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are used to compute the representations of all known characters in the feature space�

Therefore� each point of �X��X�� represents one Chinese character in the image

plane �x� y��

Considering the similarity� �rst� we employ the set of Chinese characters used

before� Figure ��� shows these characters�

Figure ���� Five original Chinese characters used for testing�

The values of X� and X� are given in Table ��� and the representations of the �ve

Chinese characters are shown in Figure �� �

Characters X� X�

C� ���� � ������C� ������ ������C� ������ ������C� ������ ��� ��C� ������ ������

Table ���� Values of the �ve Chinese characters in the central moment feature space�

Figure �� shows that the representations of the �ve Chinese characters are quite

close to each other in the two dimensional �X��X�� feature space� From the classi�

�cation point of view� this disadvantage certainly will limit the usage of the central

moment method in Chinese character recognition tasks�

Then� randomly� we selected �� Chinese characters as the testing samples� Fig�

ure ��� shows these �� Chinese characters� In Figure ���� we call the �rst sample

from the left in the �rst row S�� the second sample from the left on the same row

S�� and so on� For example� S�� will be the second sample from the left on the �fth

row� and S�� is the �fth character from the left on the ninth row�

��

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���

��

���

���

���

���

X�

��� ��� �� ��� ��� X�

ttt

tt

Figure �� � Representations of the �ve Chinese characters in the central momentfeature space�

��

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Figure ���� Ninety Chinese characters�

��

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The values of X� and X� are listed in Table �� and the representations of these

�� Chinese characters in the central moment feature space are plotted in Figure ����

Obviously� similar to the results shown in Table ��� and Figure �� � the two central

moment functions� X� and X�� cannot recognize those Chinese characters success�

fully�

��� Character Recognition with Legendre Moments

The Legendre moments do not have the property of invariance under translation�

However� compared with the classical moments� the same order of the Legendre

moment contains more terms than that of the central moment does� Therefore� in

terms of classi�cation� the Legendre moments contain more information than the

central moments do�

Similar to the two classi�cation measures de�ned in ����� and ��� �� the following

two Legendre functions

Y� �q��� � ��� �����

and

Y� �q���� � ������ � ����� � ����� �����

are employed in our new recognition model� where �mn�s are the Legendremoments

de�ned in � �����

First� we use the same set of �ve Chinese characters shown in Figure ���� The val�

ues of all �ve Chinese characters in the two�dimensional Legendre moment feature

space �Y�� Y�� are listed and illustrated in Table ��� and Figure ���� respectively�

We can see that the �ve Chinese characters are well separated in the two�dimen�

sional Legendre moment feature space �Y�� Y��� In other words� in this particular

Chinese character recognition task� the Legendre moment technique is superior�

��

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Sample X� X� Sample X� X� Sample X� X�

S� ������ �� ��� S�� ����� �� ��� S�� ���� � ������S� ������ ����� S�� ������ ������ S�� ������ ������S� ������ ���� � S�� ������ ������ S�� ������ ������S� ������ ������ S�� ������ ������ S�� ������ ����� S� ������ ������ S�� ������ ������ S�� ��� �� ��� ��S� ������ �� ��� S�� ������ ������ S�� ������ ������S� ����� ��� �� S�� ������ ������ S�� ������ ������S� ���� � ������ S�� ����� ������ S�� ������ ������S� ��� �� ������ S�� ������ ������ S�� ���� � �� ���S�� ������ ������ S�� ������ ������ S�� ������ ����� S�� ������ ������ S�� ������ ������ S�� ���� � �� ��S�� ������ ������ S�� ������ ��� �� S�� ������ ������S�� ������ �� ��� S�� ��� �� ������ S�� ��� �� �� � �S�� ������ ������ S�� ������ ������ S�� ������ ������S�� ������ ������ S�� ������ ������ S�� ������ ������S�� ���� � ������ S�� ������ ������ S�� ������ ������S�� ������ ������ S�� ������ ���� � S�� ������ ������S�� ������ ������ S�� ������ ��� �� S�� ������ ������S�� ������ ������ S�� ������ ������ S�� ������ ���� �S�� ������ ������ S�� ������ ������ S�� ���� � ������S�� ������ ������ S�� ������ ��� �� S�� ������ ������S�� ������ ���� � S�� ������ ������ S�� ������ ��� ��S�� ������ ������ S�� ������ ���� � S�� ������ �� ���S�� ���� ������ S�� ���� � ������ S�� ������ ������S�� ��� �� ������ S�� ������ ������ S�� ������ ������S�� ������ ������ S�� ������ ����� S�� ������ ������S�� ������ ������ S�� ������ �� ��� S�� ���� � ������S�� ������ ������ S�� ������ �� �� S�� ������ ������S�� ������ ������ S�� ����� ���� S�� ����� �� ���S�� ���� ��� �� S�� ������ �� ��� S�� ������ ������

Table �� � Values of the ninety Chinese characters in the central moment featurespace�

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���

��

���

���

���

���

X�

���� ��� �� ��� �� ��� X�

r

rr rr

r

r

r

r

r

rr r

r

r

r

rr r

r

r

r

r

r

rr

r

r

r

r

r

r

rr

r rr

r

r

rr

r

r

r

r

r

r

r

r

r

r

r r

rrr

r

rr

rr rr

r

r

r

rr r

r

rr

rr

r

rr

r

r

r

rr

r

r

r

r

r

r

r

r

Figure ���� Representations of the ninety Chinese characters in the central momentfeature space�

��

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Characters Y� Y�C� �� �� ������C� ����� �� ���C� ����� ������C� � �� �� ��C� ����� �� ���

Table ���� Values of the �ve Chinese characters in the Legendre moment featurespace�

���

���

���

��

��

��

Y�

��� ��� �� ��� ��� Y�

t

C�

t

C�

t

C�

t

C� t

C�

Figure ���� Representations of the �ve Chinese characters in the Legendre momentfeature space�

��

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Then� the ninety randomly selected Chinese characters shown in Figure ��� are

employed as the testing patterns� Table ��� displays the values of these �� Chinese

characters in the Legendre moment feature space� and Figure ��� plots these rep�

resentations in the two�dimensional �Y�� Y�� plane�

���

���

���

���

��

��

Y�

���� �� �� ��� ��� ��� Y�

qS�

qS�

qS�

qS�

qS�

qS�

qS�

qS� q

S

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS�

qS��

qS��

qS��q

S��

qS��

qS��

qS��

qS�� q

S�

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS�

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS��

qS�

qS

Figure ���� Representations of the ninety Chinese characters in the Legendre mo�ment feature space�

Figure ��� shows that most of Chinese characters are well separated� However�

��

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Sample Y� Y� Sample Y� Y� Sample Y� Y�S� ������ ������ S�� ��� �� ����� S�� ������ ��� ��S� ������ ����� S�� ������ ����� S�� ������ ������S� ������ ������ S�� ������ ����� S�� ������ ���� S� ��� �� ����� S�� ������ ������ S�� ������ ������S� ����� ������ S�� ������ ���� � S�� ������ ���� S� ����� ��� �� S�� �� � ���� � S�� ������ ��� ��S� ������ ����� S�� ���� ������ S�� ����� �� ���S� ������ ����� S�� ������ ������ S�� ��� �� ���� S� ��� �� � ��� S�� ������ ������ S�� ������ ������S�� ������ ����� S�� ��� �� ����� S�� ������ ������S�� ������ �� ��� S�� ������ ������ S�� ��� � ����� S�� ������ ������ S�� ��� �� ����� S�� �� �� �����S�� ������ ������ S�� ������ �� �� S�� ������ ������S�� ����� ����� S�� ������ ����� S�� ���� ������S�� ���� ������ S�� ������ ������ S�� ������ ������S�� ������ ������ S�� ������ ������ S�� ������ ��� ��S�� ����� ����� S�� ������ �� � S�� ������ �� � �S�� ������ �� ��� S�� ������ ����� S�� ������ ������S�� ������ ������ S�� ����� ��� �� S�� ������ ����� S�� ����� ����� S�� � ��� ������ S�� ������ ������S�� ������ ������ S�� ������ ����� S�� ������ ������S�� ������ ������ S�� ���� � ����� S�� ������ ������S�� ������ ������ S�� ������ ������ S�� ������ �����S�� ����� ����� S�� � �� ��� �� S�� ��� � ��� �S�� �� �� ������ S�� ������ ������ S�� ���� ������S�� ������ �� ��� S�� ��� � ������ S�� ����� ��� �S�� ��� � ����� S�� ������ ����� S�� ��� � ������S�� ������ ������ S�� ����� �� �� S�� ������ �����S�� ������ ����� S�� ������ ����� S�� ������ ��� ��S�� ����� ������ S�� ��� �� ������ S�� ������ ������

Table ���� Values of the ninety Chinese characters in the Legendre moment featurespace�

��

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it is observed that two characters� S�� and S��� are very close to each other in the

Legendre moment feature space� Although the results shown in Figure ��� are

indeed better than those of Figure ���� yet the Legendre moment two�dimensional

feature space cannot be used as a successful technique to recognize a speci�c Chinese

character from the whole Chinese character set�

One option to improve the Legendre moment technique is to add a new feature

to the feature space� We use the following equation� which is based on � ����� as the

third feature�

Y� �q���� � ����� � ������ �����

In this new three�dimensionalLegendremoment feature space� characters S�� and

S�� have Y� values ������ and ������� respectively� Therefore� all Chinese characters

shown in Figure ��� can be separated successfully� Table ��� displays the values of

Y�� Y�� and Y� for all ninety Chinese characters�

��� Conclusions

In this chapter� we have discussed character recognition via moment methods and

compared the well known central moment feature space with the proposed new Le�

gendre moment feature spaces for Chinese character recognition�

The two�dimensional central moment feature space was used by Hu���� to recog�

nize � English capital letters� The experiment performed fairly well except that the

distance between two points representing letters M and W in the feature space is

very close�

Compared with the set of English letters� however� the set of Chinese characters

is larger and more dicult to classify� Two sets of Chinese characters� one including

�ve similar Chinese characters and the other containing ninety randomly selected

characters� are used as the input patterns to a simulation program based on the

��

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Sample � � � Sample � � �

� ������ ������ ���� �� ����� ���� �����

� ���� ���� ���� �� ������ ����� �����

� ����� ���� ���� �� ����� ����� �����

� ������ ���� ����� �� ������ ����� ������

������ ����� ���� ������ ���� �����

� ����� ���� ����� � ����� ������ �����

� ������ ����� ����� � ������ ������ ����

� ���� ���� ����� � ������ ������ ����

� ����� ������ ���� � ���� ����� ������

� ������ ������ ����� ������ ��� �����

�� ���� ������ ������ � ������ ����� ������

�� ����� ����� ������ � ����� ������ �����

�� ����� �� ������ � ������ ������ ������

�� ��� ����� ������ � ���� ����� �����

� ����� ����� ����� � ������ ����� �����

�� ����� ������ ������ �� ������ ��� ������

�� ����� ����� ������ �� ����� ���� ������

�� ����� ����� ���� �� ������ ����� ������

�� ����� ����� ������ �� ����� ���� ����

� ����� ����� ������ � ���� ����� ����

�� ����� ����� ������ �� ����� ���� ������

�� ����� ������ ����� �� ������ ����� �����

�� ������ ����� ����� �� ������ ������ ������

�� ����� ������ ������ �� ����� ������ ������

� ������ ������ ����� � ����� ����� ������

�� ����� ������ ����� �� ����� ����� �����

�� ���� ���� ���� �� ����� ����� ������

�� ���� ���� ���� �� ��� ����� �����

�� ������ ����� ������ �� ����� ������ ������

� ����� ���� ����� � ������ ����� ������

�� ����� ����� ����� �� ���� ���� ������

�� ����� ������ ���� �� ������ ����� ������

�� ���� ����� ���� �� ���� ���� ������

�� ���� ������ ����� �� ������ ����� ���

� ������ ����� ���� � ����� ����� �����

�� ����� ����� ������ �� ������ ���� ������

�� ������ ���� ����� �� ������ ������ �����

�� ������ ��� ������ �� ������ ���� �����

�� ����� ������ ������ �� ����� ������ ������

� ����� ����� ������ � ����� ������ �����

�� ���� ������ ������ �� ��� ����� ������

�� ����� ������ ����� �� ����� ������ �����

�� ����� ����� ����� �� ����� ����� ������

�� ��� ������ ����� �� ������ ������ ������

� ������ ������ ��� � ������ ������ ������

Table ���� Values of the ninety Chinese characters in the Legendre moment three�dimensional feature space�

��

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central moment technique� The results show that most of the representations of the

Chinese characters in the Central moment feature space are crowded to a small area

in the two�dimensional central moment feature plane� Therefore� in both cases� it is

impossible to recognize those Chinese characters successfully�

We proposed some new Legendre moment feature spaces in this chapter� First� a

two�dimensional Legendre moment feature space was developed and applied� The

same two sets of Chinese characters are employed as the input patterns� For the set

of �ve similar characters� the experiment demonstrated that all �ve representations

in the Legendre moment feature space are well separated� The performance of re�

cognizing ninety randomly selected Chinese characters with the Legendre moment

feature space is much more re�ned than that of the central moment feature space

as well� However� the distance in the two�dimensional Legendre moment feature

space between two characters� S�� and S��� is quite small� This can be a potential

problem in a full scale Chinese character recognition application�

To improve the recognizing ability� we added one new feature to the two�dimen�

sional Legendre moment feature space� The new three�dimensional feature space is

able to separate all ninety randomly selected Chinese characters easily�

It is noted that the highest order Legendre polynomials involved in the three�

dimensional Legendre moment feature space is �� With the development of the

better VLSI moment generator chips���� a hardware device for Chinese character

recognition becomes possible�

Because of some technical reasons� we cannot obtain the whole set �more than

������� of Chinese characters and test all of them individually� However� with a

possible fourth feature being added to the three�dimensional Legendre moment

feature space� we are very optimistic to say that the Legendre moment technique

can solve the Chinese character recognition problem�

��

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With the discussions and the experimental results we had in this chapter� we are

con�dent that feature spaces based on Legendre moments are the right direction

to solve the practical Chinese character recognition problem�

���

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Chapter �

Conclusions and Recommendations

�� Conclusions

We have been concerned here with moment methods in image analysis� We found

that a fundamental element of moment methods� accuracy in moment computing�

had not attracted the attention it deserved� We have proposed and implemented

several procedures to increase the accuracy in Legendre and Zernike moments

computing�

Eorts made to reduce computing errors in Legendre moments turned out to be

very successful� Primarily� we have solved the problem of computation errors related

to the Legendre moment computing� Meanwhile� by working out up to order ��

Legendre polynomials� we reduced the moment computation time dramatically and

made the utilization of higher order Legendre moments practically possible�

Based on these improvements� we performed image reconstruction via Legendre

moments� We found that the reconstructed images were very close to the original

image numerically and visually� The quality of reconstructed images is superior to

all published results�

The computation errors of Zernike moments have been investigated as well� Be�

cause of the nature of the Zernikemoments computing� there are two types of major

errors� geometric and approximate� in the computation� Adopting the result from a

���

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classical problem in Number Theory� The Lattice Points of a Circle� we concluded

that the geometric error in Zernike moment computing cannot be completely re�

moved� We also proposed several procedures to reduce the approximate errors in

Zernikemoment computing� Though improvement has been obtained� none of them

works �awlessly� We concluded that the lack of ecient measures to reduce both

geometric and approximate errors eectively would impede further utilization of the

Zernike moments�

Image reconstruction via Zernike moments was performed as well� Applying the

best formula proposed� we reconstructed some images from their original versions

with reasonable quality� The reconstructed images via Zernike moments indeed

have better qualities than the results published previously� but� they are simply not

as good as those images reconstructed via Legendre moments�

We have been also concerned here with reconstructing images from a �nite set of

moments computed from the noisy observed data� We conclude that there exists an

optimal number of moments yielding the best possible representation of the original

image without noise�

Finally� we discussed the recognition of Chinese characters via moment methods�

We concluded that the method of Legendremoment works quite well for the Chinese

character interpretation� Since the highest order Legendre polynomials involved in

the Chinese character recognition task is �� with the developments in the area of

VLSI moment generator chips� a hardware device for Chinese character recognition

becomes technically possible�

�� Recommendations

After reviewing the results from this research� we have a few recommendations for

further study�

��

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A visible extension of two�dimensional image reconstruction is the reconstruction

task in three�dimensional space� Since the prime accuracy and eciency problems of

computing high order of Legendre moments have been solved in this thesis� there

is no real technical diculty for reconstructing a three�dimensional image via the

Legendre moments�

Though we cannot reduce the geometric error in the Zernike moment computing

eectively� we can� however� reduce the approximation error further by developing

new formulas to calculate integrations for all pixels along the boundary of the unit

circle� This could be a challenging task� but must be solved before the further full

scale utilization of the Zernike moments�

Practically� we can build a database including all Legendre moment space fea�

tures covering the whole Chinese character set without real technical diculty� This

will be the �rst important step to develop a reading machine for the Chinese lan�

guage� which is one the most dicult languages in terms of arti�cial intelligence

reading�

���

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Appendix A

Some of the Higher Order Legendre Polynomials�

P���x� ��

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��

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P���x� ��

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P���x� �x

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P���x� ��

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P���x� �x

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P���x� �x

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P���x� ��

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