Morphological filter mean-absolute-error representation theorems and their application to optimal morphological filter design

Rochester Institute of Technology

RIT Scholar Works

Theses Thesis/Dissertation Collections

5-1-1993

Morphological filter mean-absolute-error

representation theorems and their application to

optimal morphological filter design

Robert P. Loce

Follow this and additional works at:http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please [email protected].

Recommended Citation

Morphological Filter Mean-Absolute-Error Representation

Theorems and Their Application to Optimal Morphological

Filter Design

Robert P. Loce

Centerfor_ImagingScience

Rochester Instituteof_Technology

Morphological Filter Mean-Absolute-Error Representation

Theorems and Their Application to Optimal Morphological

Filter Design

Robert P. Loce

Center for_ImagingScience

Advisor: E._Dougherty

Committee: A.Biles E._Dougherty R. Haralick

Morphological Filter Mean-Absolute-Error Representation Theorems and Their Application to Optimal Morphological Filter Design

by Robert P. Loce

Rochester Institute of Technology Rochester, New York

(1993)

A thesis submitted in partial fulfillment of the requirements for the degree of Ph.D. in the Center for Imaging Science

in the College of Imaging Arts and Sciences of the Rochester Institute of Technology

May 1993

Signature of Author Robert P. Lace

Accepted by JohnR.Shaft

COLLEGE OF IMAGING ARTS AND SCIENCES ROCHESTER INSTITUTE OF TECHNOLOGY

ROCHESTER, NEW YORK

CERTIFICATE OF APPROVAL

Ph.D. DEGREE THESIS

The Ph.D. Degree Thesis of Robert P. Loce has been examined and approved by the the

thesis committee as satisfactory for the thesis requirement for the

Ph.D. degree in Imaging Science

Dr. E.R. Dougherty, Thesis Advisor

ProfessorJ.A. Biles

Dr. R. M. Haralick

THESIS RELEASE PERMISSION

ROCHESTER INSTITUTE OF TECHNOLOGY COLLEGE OF IMAGING ARTS AND SCIENCES

Abstract

The presentthesis deriveserror representations and develops design meth

odologies for optimal mean-absolute-error _(MAE) morphological-based filters.

Four related morphological-based filter-types are treated. Three are

translation-invariant, monotonicallyincreasing operators, andour analysis is

basedontheMatheron (1975)representation. Inthisclass we analyzeconven

tional _binary, conventional _gray-scale, and computational morphological fil

ters. The fourth filter class examined is that of_binary translation invariant

operators. Ouranalysis isbased onthe Banon andBarrera (1991) representa

tion and hit-or-miss operator ofSerra (1982). A _startingpointwill be the op

timalmorphologicalfilterparadigm ofDougherty(1992a,b) whoseanalysisde

scribesthe optimalfilter _by a system of nonlinear inequalitieswith no known

method of_solution,andthusreducesfilter designtominimal searchstrategies.

Althoughthesearch analysisis_definitive,practicalfilter designremained elu

sive because the search space can be _{prohibitively} large if it not mitigated in

some way.

Thepresentthesisextendsfrom Dougherty's_startingpointinseveralways.

Centraltothe thesis isthe MAEanalysis forthe variousfilter _settings, where

ineach _case,atheoremis derivedthatexpresses overallfilter MAEas a sumof

MAE values of individual structuring-element filters and MAE of combina

canbe employed ina computer algorithm to _rapidly evaluate combinations of

structuringelements and searchforan optimalfilter basis.

AlthoughtheMAEtheoremsprovide a rapid meansfor_{examining the}filter

designspace, thecombinatoric natureofthis space_is, in_{general, too}large fora

exhaustive search. Another_keycontribution ofthis thesisconcernsmitigation

ofthe computationalburdenviadesignconstraints. The_resultingconstrained

filterwillbe _{suboptimal,but,} ifthe constraints are imposed ina suitable man

ner, there is little loss offilter performance in return for design tractability.

Three constraint approaches developed here are (1) limiting the number of

terms inthefilter expansion, (2) constraining the observation_window, and(3)

employing structuring element libraries from which to search for an optimal

basis.

Anothercontribution ofthis thesisconcernstheapplicationof optimal mor

phological filters to image restoration. Statistical and deterministic image

and degradation models for _binary and low-level _{gray images} were developed

here that relate to actual problems in the optical character recognition and

electronic _printing fields. In the filter design process, these models are em

ployed to generate _{realizations,} from which we extract single-erosion and

single-hit-or-miss MAE statistics. These realization-based statistics are uti

TABLE OF CONTENTS

Section Page No.

1: INTRODUCTION 1

1.1: StatementofProblemand Objectives 1

1.2: BackgroundandHistorical Perspective 5

1.3 OverviewofThesis 9

2: OPTIMAL BINARY MORPHOLOGIC AL FILTERS 10 2.1: Reviewof_BinaryMorphological_Filtering 10

2.2: ReviewofOptimal Estimation intheBinarySetting 14

2.3: _BinaryMorphologicalFilter MAE Theorem 17

2.4: Constrained Approachto MorphologicalFilter Design 30

2.4.1: Window Constraint 33

2.4.2: _LibraryOptimal Filters 35

2.4.2.1: ASpecific Expert_Library 37

2.4.2.2: First-Order_Library 47

TABLEOF CONTENTS cont.

Section Page No.

3: OPTIMAL GRAY-SCALE MORPHOLOGICAL FILTERS 73

3.1: ReviewofGray-Scale Morphological_Filtering 74

3.2: Reviewofthe Gray-Scale Optimal MorphologicalEstimation

Paradigm 76

3.3: Gray-Scale MorphologicalFilter MAE Theorem 80

3.4: ExamplesoftheFilter DesignMethod Appliedto Gray-Scale

Signal Restoration 85

3.5: ApplicationoftheGray-ScaleMAE Theoremto Order-Statistic

Filters 91

4: OPTIMAL COMPUTATIONAL MORPHOLOGICAL

FILTERS 96

4.1: ReviewofComputationalMorphological_Filtering 97

4.2: ReviewofOptimal Estimation Paradigm 100

4.3: ComputationalMorphologicalFilter MAE Theorem 103

4.3.1 Gray-Scaleto_BinaryMorphological Filter MAE

Analysis 103

4.3.2 AnalysisofGeneral Gray-Scale Filters 106

4.4: Application oftheMAE TheoremtoDesignofImage

TABLE OF CONTENTS cont.

Section Page No.

5: OPTIMAL HIT-OR-MISS FILTERS 126

5.1 Review ofHit-or-Miss_Filtering 127

5.2: Optimal Hit-or-Miss Filters- MorphologicalRepresentation

ofthe ConditionalExpectation 129

5.3: Mean-_{AbsoluteError AnalysisofHit-or-MissFiltering} ... 133

5.4: ApplicationofOptimalHit-or-MissFilterstoText

Restoration 137

6: CONCLUSION 140

Appendix1: Derivationof_BinaryMAETheorem 143

Appendix 2: Derivation ofGray-ScaleLemmas 148

Appendix3: DerivationofComputational_{MorphologyBinary}

MAE Theorem 153

LIST OF ILLUSTRATIONS

Figure No. _PageNo.

1 a)Unconstrained5X5Window_W,b)Constrained5X5

Window W 34

2 A Specific Expert_Library 39

3 IdealImage

-S,andSalt NoiseCorruptedImage

-T,

Certain Hole-Fillersofthe Expert_LibraryWould

Have Overfilled Upon_Filtering 44

4 Hole-FillersthatPreventOver_Filling 46

5 ForExample 1: a) RealizationofImage_Process,

b) RealizationofImage-NoiseProcess 53

6 ForExample 1: MAEvsBasis Size 53

7 ForExample 1: ImageFiltered_usinga)ExpertMethod,

b)First-OrderMethod 54

8 For Example 1: First-Order_Library 55

9 ForExample 1: Filter Bases fora)Expert_Method,

b)First-Order Method 56

10 For Example 2: RealizationofImage-Noise Process 57

11 For Example 2: MAEvsBasisSize 58

12 For Example 2: ImageRestored_usingExpert_Library

Method 59

LIST OF_{ILLUSTRATIONS (cont.)}

Figure No. _PageNo.

14 MAEvsBasis Size for Image-Noise_{ofExample2,with}

Noise Area Coverage of_5%, 10%and 15% 60 15 Optimal6-ErosionBasesforImage-NoiseofExample_2,

with_{Noise Area Coverage}of_5%, 10%and 15% 61 16 ForExample3: a) Ideal TextImage, b)Light-Copy

Degraded Text_Image, c)MAEvsBasisSize,

d)Optimal_Basis,e)Optimal Dilation_Element,

f) Restored Image 64

17 ForExample 4: a)Min-Noise Degraded_Image, b)MAEvsBasis_Size,c)Optimal_Basis,

d)Restored Image 66

18 ForExample5: a)Ideal CheckImage,b)Threshold

Degraded Check_Image, c)MAEvsBasis_Size,

d)Optimal_Basis,e)RestoredImage 68

19 ForExample 5: a)_{Statistically}Relevant_Structuring

Elements, b)GraphicalDepiction ofMAE_Theorem,

c)GraphicalDepictionofRecursive MAE_{Theorem, 69} 20 ForExample6: a)Ideal Address Image, b)Threshold

Degraded Address_Image,c)MAEvsBasisSize,

d)Optimal _Basis,e)RestoredImage 72

LIST OF_{ILLUSTRATIONS (cont.)}

Figure No. Page No.

22 _{For Example} 11: Strong-NeighborMedianBasis 93

23 For Example 11: 5-PixelWindow 116

24 For Example 11: (a) _BinaryImage, DifferenceImages

-(b)A2o ~

A128, (c)Ai28 ~

Auo, (d)Ai28 ~

A250 H8

25 For Example 11: MAEvsBasis Size 119

26 For Example 11:Optimal 4-ErosionBasis 120

27 For Example 11: Character Image(a) Before_Filtering

-Binary, (b) After_Filtering

-Quaternary 121

28 ForExample 11: SchematicoftheXerographicLaser

Printer Model 123

29 ForExample11: PrintSimulation for_(a)Unfiltered

and(b) Filtered Image 125

30 For Example 12: a)Ideal TextImage,b)Image Degraded

byText_Noise, c)MAEvsNumberofStructuring-Element

LIST OF TABLES

Table No . PageNo.

1 Elementsof aSpecific Expert_Library 45

2 MAEandFilter_EfficiencyfortheThreeOptimal

1. Introduction

1.1 Statementof_Problemand Objective

When_employinga statistical estimation rulefor imagerestoration,pattern

recognition, image _{transformation,} etc., it is usually desiredthat the estimate be optimal asjudged_by a chosen criterion. As an _example, consider the Wie

nerfilter [Wiener _(1942)], wherethe estimation rulecanbewritten as a linear

combination ofobserved data values. In this case _{the optimality} criterion is

minimum mean-square error _(MSE) between an idealized image model and a

collection of estimated image values. Mathematical formalisms for practical design ofnonlinear optimal filters are not _generally available [See Pitas and

Venetsanopoulos (1989) for a discussion ofvarious types of nonlinear filters.].

Images oftenhave_{sharpedges,binary}structures and a limitednumber ofbits

per pixel. Therestrictionthat the estimatationrulebe alinearcombination of

observed valuesis_typicallyunsuitable. Thepresentthesisderives error repre

sentations and develops design methodologiesfor optimal mean-absolute-error (MAE) morphological-basedfiltersthatare well suitedfor_operatingonimages

thatare_binaryorlow-level gray-scale (2 or3bits/pixel)innature.

Four related morphological-based filter types are analyzed in this thesis.

The first three are _{translation-invariant, monotonically} increasing _operators,

and our analysisis based onthe Matheron(1975) representationforthatfilter

and computational morphological filters. As will be explained _{subsequently,}

the conventional_binarytheory is not ipsofacto subsumed_bythe conventional

gray-scale _theorybecause_{binary morphology is}embeddedintogray-scalemor

phology in a mannerthat doesnot allow a _{0-1 structuring} elementto betreat

edthesame inboththeorieswhen_consideringestimationerrorfordiscreteim

ages _possessingfinite _gray ranges. Computational _morphologyis a recent at

tempt _[Dougherty and Sinha _{(1992, 1993a,b)] to unify the binary} and _gray

scale morphological_filteringtheoriesand requiresananalysis distinct from its

conventional counterparts. The fourth filter class examined is that of_binary

translation invariant operators. Our analysis is based on the Banon and Barrera (1991)representationandthehit-or-missoperator ofSerra(1982).

Aswehave _{stated, this thesis}willprovide error representation and_develop

design methodologies for optimal nonlinear estimators. A _starting point will

be the optimal p-norm morphological filter paradigm of_{Dougherty (1990a,b,}

1992a,b). _Dougherty performed an analysis of discrete morphological filters

based on a finite number ofobservations in the context ofthe

mean-square-error _(MSE) criterion. There are two key aspects to the analysis. _First, it is

based on the Matheron representation for _{translation-invariant,}

monotonically increasing filters; namely, that a _binary (or _{gray-scale)filter} is

represented by aunion (maxima inthe gray-scale case) oferosions over struc

turing elementsin thefilter basis. Second,the analysis describes the optimal filter _bya system ofnonlinearinequalities with noknownmethod of_solution,

strat-egies over subsets of _{Af-dimensional discrete Cartesian} space. Although the search analysisis_definitive, inthe sensethatitderivesminimal search spaces

for optimalfilter _design,the search space canbe _{prohibitively}large ifit is not

mitigatedinsome way.

ThepresentthesisextendsfromDougherty's_startingpointinseveral ways.

Central to the thesisis the MAE analysisforthevarious filter _settings,where

ineach _case, atheoremisderivedthatexpresses overallfilter MAEas asum of MAE values of individual structuring-element filters and MAE of combina

tions of unions _(maxima) ofthose elements. Recursive forms ofthe theorems

can be employed in a computer algorithm to rapidlyevaluate combinations of

structuring elements and search for an optimal filter basis. In _{addition, the} theorems provide insight into the performance of nonlinear filters. For in

stance, we analyse order-statisticfilters in termsofthe gray-scale morphologi

calfilter MAEtheorem.

AlthoughtheMAEtheoremsprovidearapidmeansfor_examiningthefilter design space, the combinatoric nature ofthisspace _is, in _{general, too} large for

an exhaustive search. To render the search _tractable, we have developed a

constraint _{methodology that} is employed in conjunction with the computer search. The design_{methodology involves constraining the} class offilters from

whichtheoptimal morphologicalfilter istobechosen;hence,fromtheperspec

tive of the original optimization paradigm of _{Dougherty (1992a,b), this ap}

imposed torenderthefilter designprocesstractable: (1) Limiting

structuring-element size and _shape, which we refer to as window _constraint; (2) Limiting

the number of_structuring elements _formingthe _filter, termed basis size con

straint; (3) Limitingthe search to _structuring elements derived from some li

brary that has been chosen in a expert manner or _by means of some prelimi

nary statistics, which we call _libraryconstraint. Thisconstraint_methodology isanother_keycontributionofthepresentthesis.

Image restoration is _{the primary} application ofthe morphological estima

tors developedinthis thesis. Imagerestoration involves application ofafilter

to anobserved _image, assumedto be _degraded,withthe intent_beingto restore

theobservedimageto someidealform. The _methodologyisbasedon statistical

analysis ofsignal andnoise_processes,and theconsequentdevelopmentof afil

ter (statisticalestimator) to take the observed image as an input and yield an

estimateofthe ideal (uncorrupted)image as anoutput. Interms ofthe _model,

the _{image, noise,} and_noisy image arerandom_processes, so that the estimator

is a function of random inputs, but in actual practice the filter is applied to a

realizationofthe image-noiseprocess.

Anothercontribution ofthis thesis concerns_modelingfortheapplication of

optimal morphological filters to image restoration. Not muchcan be found in

the literature on statistical and deterministic degradation models for _binary

andlow-level_grayimages,so several suchmodelshavebeen developedthatre

printing fields. In the filter design process, these image and image-noise models are employed to generate realizations from which we extract

single-erosion and single-hit-or-miss MAE statistics. These realization-based statis tics are utilized in the search for the optimal combination of_structuring ele

ments.

1.2 BackgroundandHistoricalPerspective

We begin _{by referencing} several seminal works and general publications

thatprovide abasisof_{understanding}ofmathematical_morphology, estimation theory, and image restoration. We then reference general papers on various applications and properties ofmorphologicalfilters. _Last,and most relevantto the_thesis,wefocus onmorphologicalfiltersappliedto imagerestoration.

Morphological analysis of _binary images was originated _by Matheron

(1967, 1972, 1975) for _{the study} of porous images. The method is based upon

Minkowskialgebra [Minkowski (1903)] as developed_by Hadwiger (1957). For

a general description of mathematical _morphology see Serra _{(1982, 1986,}

1988), Haralick, Sternberg and Zhuang (1987), Giardina and _Dougherty

(1988), and_Dougherty (1992c). Anoverview of estimation_theory as employed in linearsystems is given_by Lewis (1986), itsapplicationto digital imageres

nonline-ar _filtering methods can be found in the book _by Pitas and Venetsanopoulos (1989).

The _early applications ofmorphological_filteringwere aimed at image _seg

mentation and patterndescriptionthroughgranulometricanalysis [Matheron

(1975)] and operationssuchas theHit-or-Misstransform [Serra (1982)]. _Prop

ertiesforbothgray-scale and_binarymorphologicalfilters arewelldescribed in

theliterature. _Alternatingsequentialfilters aredescribed_byLougheed₍₁₉₈₃₎

and _Sternberg (1986). Serra _(1988), and Serra and Vincent ₍₁₉₈₉₎ present a lattice theoretic approach to morphological filtering. Maragos _(1985), and

Maragos and Schafer (1987a,b) give a set-theoretic analysis of morphological

filters and describe _{their relationship to linear, median,} order-statistic and

stackfilters. _Doughertyand Giardina (1986b)alsodescribe_{the relationship to}

linear filters. _Dougherty andKraus(1990a,b,c,d,1991) describemorphological filters with convolution like properties. The present _study is concerned with

employing the Matheron expansion in filter design. Matheron _(1975), Serra

(1982), Dougherty and Giardina _(1986a, 1988) and Maragos ₍₁₉₈₉₎ describe

thefilter basisofMatheron.

Concerning morphological-based restoration, we _{take the optimality} para

digmof_Dougherty (1990a,b, 1992a,b)as our_startingpoint. _Dougherty formu

designof actual filters remained elusive dueto the combinatoric nature ofthe

problem. In this thesis we renderthe design process practical through use of

themorphologicalfilter MAE_theorems, design_constraints, andtheutilization

ofimageandimage-noise models. Muchofthisworkhasbeenpublishedinthe

course ofthe thesis research_[Dougherty and_{Loce(1991,1993a,c)} and Loceand

Dougherty (1991, 1992a,b,c,d,e,f)] and has beensummarizedinabook chapter [DoughertyandLoce(1993a)]. Inadditionto_developingdesign_{methodologies,} DoughertyandLoce _{(1992, 1993b) have} performedan analysis onthe goodness

of the estimators produced _by their morphological filter design methods. In particular, they examined the realization-based nature ofdesign andthe con straintonbasissize.

For _{binary images,} there have been other strategies employed to facilitate

an efficient search within the paradigm of _Dougherty (1992a). _Dougherty,

Mathew, and Swarnakar (1991) and _{Mathew, Dougherty} and Swarnakar

(1991, 1992), have developedanalgorithmic approachthatderivestheoptimal

morphological filter from the conditional expectation. For certain types of

noise conditions this technique has been demonstrated to be quite

computationallyefficient.

Various other approaches have beentaken tofacilitate filter design in the

binarysetting. One has been to design filtersbased upon conducive imagere

the Wiener approach in the _frequency domain. As in the Wiener approach, this requires constraints onthe image-noise model. Progress in this direction

has been promising, but atthe present stage remains quite theoretical. Dou

gherty and Haralick _{(1991, 1992)} have proposed a spectral decomposition based upon a hole _spectrum, which in effect models an image _{by considering}

holes in the image. In a _closely related _{filtering paradigm, Haralick,} Dou

gherty, and Katz _(1991a,b) have taken an approach _very similar to Wiener

spectral design _{by defining} an _opening spectrum. Also note that _Dougherty, Haralick, Chen, Agerskov Jacobi and Sloth _{(1991, 1992)} utilize

pattern-spectrainformation todesignoptimal x-openings.

Other approaches to optimal morphological restoration have been taken thatdonotfollowthe Matheron-representation_paradigm,that_is,optimization

is notover all _{monotonically increasing,} translation invariantmappings rela

tive to the MAEor MSEgoodness criterion. Schonfeld andGoutsias _(1989a,b,

1990, 1991a,b) describe an _alternating sequential morphological filter opti mized for _binary image restoration based on a set-difference metric and the

morphologicalpattern spectrum.

Koskinen, Astola, andNeuvo have analyzed the input-output statistics for certain flat morphological filters _operating on i.i.d. random variables. _They have obtained both exact and asymptotic representations for the output dis

1.3 OverviewofThesis

After this _{introduction,} the thesis describes optimal estimation in four fil

tering settings: conventional _{binary morphological,} conventional gray-scale

morphological, computational _{morphological,} and hit-or-miss. Each _settingis analyzed _{independently} in detail in Sections 2

-5, respectively. The sections arewritten in a parallelfashion. _Theyeach start with abriefreview offilter

ing in _{that setting; then,} in general _terms, discuss the respective approach of

optimal estimation. _Followingthat_{introductorymaterial,}probabilistic_analy sis is performed that leads to an MAE expression or theorem for that filter type. For _{the monotonically increasing} type of_filters, recursive forms ofthe MAEtheorems are given. Several ofthe more lengthy proofs are given in _ap

pendices. It is this analysis that is central to the thesis and _key to our _ap proachtofilter designvia efficient search. Atthispointineach_section,wedis

cuss additionaldesign considerations, such asconstraintsthat mustbeusedto renderthedesign spacetractable. Whenthisconstrained approachtodesignis first introduced _{(binary filtering,} Section _3), we discuss the methodology and

2.Optimal_{BinaryMorphological Filters}

This sectionpresents the _theoryand design _methodology of optimal _binary

morphological filters. We first review _{binary morphology} and the optimal estimation paradigm of _{Dougherty (1990a, 1992a). We then perform a} probabilistic analysis thatallows us to derive a_binarymorphological_filtering MAEtheorem. It isthenshownhowtherecursiveform ofthis theorem maybe

employed in a computer search algorithm that determines the optimal filter

(optimal subject to constraints). The constrained design _{methodology that}

renders the computer search realizable is also _described in detail. _Lastly, we

apply thedeveloped designmethodto_binaryimagerestoration problems.

2.1 Reviewof_BinaryMorphological_Filtering

A review of _binary morphological _filtering is presented _{here. We first}

describethe fundamental morphological _{operations, then}proceed onto general

morphological filters and filter optimization. The _definitions of the

morphologicaloperations and ofgeneral morphologicalfilters are _independent

of_{dimension, applying to}both images (2-D) and signals(1-D). For _simplicity,

We _employfour_elementarymorphological operations in thepresent paper:

erosion, dilation, opening, and closing. In the _binary setting, these are

respectivelydefinedby

S9 B = {z:_{B +z C S}}

, (1)

S B = _{U{B + x: x S}}

, (2)

SOB = (S 0B) B

, (3)

SB =

[S(-B)]0(-B), (4)

where B + z = {b ₊ z: b _B), B = {b:b _{B}, and}B is _{called a}

structuring element. In _designing and _analyzing morphological _filters, the operation of erosionplays a _key role. Akey _property of erosion employed throughout this paper is the following: ifA C _B, then S A D S B. Both erosion/dilation and_{opening/closingsatisfyduality}relationsrelativetocomplementation:

SB =

[SC e(-B)]c

,

(5)

SB = _(ScOB)c. (6)

More generally, we consider a binary morphological filter to be a set

mapping W that is increasing [S C T implies W(S) C ^(1)] and translation

filter to be called "morphological."] The kernel of an _increasing, translation invariant mapping W, KerPFI, is the class of all images S such that W(S)

contains the origin. The dual is defined _{by W*(S)} = _{W(SC)C, it is also a} morphologicalfilter,thedualofthedualistheoriginalfilter(*** ₌

W),and

KerfY*] ={B: Bc

KerfV]}.

(7)

From Eqs. ₍₅₎ and ₍₆₎we seethatdilation _byB is thedualof erosion_by Band

closingbyB isthedualof_openingbyB. Forthe design_methodology developed

in the present _paper, it is important to recognize that Matheron ₍₁₉₇₅₎ gave

expressions forthe kernels ofthe morphological mappings ofEqs. (1) through

(4):

Ker[.0B] = {A:_{A DB}}

, (8)

Kerf. _B] =_{{A:A n}(- _B) * _0} , (9)

Kerf.O_B] = {A:_B

-zCA forsome_zCB}, (10)

Kerf. B] = _{{A:An (B}

-z) * 0forall_ziB) , (11)

As noticed _{by Dougherty} and Giardina (1986a) and Maragos and Schafer (1987a,b), excludingcertainpathological _cases, a morphological filter Whas a basis, BasTO, of_structuring elements such that the Matheron expansion can betakenoverBasfY] insteadofKerPF]:

W(S) = _{U{S0 B:B BasPF]}}

. (12)

The basis is a minimal _{(nonredundant)} class of _structuring elements within thekernel: for_anyB _KerFP],there exists

B'

BasPF] such thatB'C _B; and

there does not exist a pair of_structuringelements in Bas[*F] properly related by set inclusion. If B is the basis for a _filter, we will sometimes denote the filter _by ^b- _{By duality,} there exists a morphologicalfilterrepresentation in terms of dilation [Matheron (1975)]. For lattice extensions of the Matheron

representation, see Serra _(1988a,b,c), Matheron (1988), Heijmans and Ronse (1990), and Heijmans (1991). Generalization to nonmonotonic mappings is

given _by Banon and Barrera (1991). Finally, the characterization of translation-invariant mappings between _arbitrary complete lattices is

2.2 Optimal Estimation inthe_BinaryMorphological_Setting

We now turn to the application of morphological filtering to statistical

estimation. Relativeto statistical _{optimization,}erosionplaysthekeyrole. To

adapt its definition to a digital environment and to statistical estimation,

consider N _binary observation random variables X\, X2, , Xn. Each

realization oftherandom vectorX = _{(Xi, X2,} . . . _,Xrf)is a 0-1 AT-tuple denoted

byx = _(x\,X2,

, xn). Ifwe let 1 and 0denotepoints ofthe domainofXthat lie within or outside the domain _{{1, 2,} . . .

, N}, then each realization x of X

constitutes a subset of{1, _2, ...

, N\, and we can erode x by a deterministic

structuringelementb = _{(61, 62,}

, *w), bjbeing0or 1. Theerosionx b is

abinary functional, itsvalue_beingeither0or 1:

x b = min{xy. _bj= _1}.

j

(13)

Using the Matheron representation [Eq. (12)] as a _guide, in Dougherty

(1990a, 1992a) an N-observation digital morphologicalfilter is defined to be a

functionaloftheform

W(x) = max{x _bj =_max{min {xy. _bu=

1}}. (14)

i i J

where x and b; are deterministic binary N-vectors, and assuming

Given N observation random variables _{X\, X2,..., Xn} and a random

variable Yto be estimated, theoptimal mean-absolute-error _(MAE) estimator

isthefunctiong(X\,X2, . . .

,Xn)thatminimizestheexpected value,

MAE =E I Y-^X1,X2,...,XN)\

(15)

In the _{binary setting,} mean-absolute error equals mean-square error _(MSE),

andtherefore the analysis of_{Dougherty (1990a, 1992a)} applies _directly to the

presentstudy. _Furthermore, it is wellknownthat theoptimal MSEestimator

is given by the conditional _expectation; however, rather than find the best

estimator, it iscommonpractice tolook forthe bestestimator _amonga classof

estimators, _thereby restricting the nature ofthe estimation rule g. Here we

require_gtobemorphological.

For a random vector X and fixed _structuringelement_b, erosion defines an

estimator that can be used to estimate another random variable Y. The

optimal mean-absolute erosion filter is the one defined by the structuring

elementB_minimizing

MAE(b)= _{E[\ 7-(Xb)|} ]=E[\ Y-mm{Xf.bj = l}\ .

(16)

MAE(W) over all possible choices of _{AT-observation} morphological filters W.

Theoptimal mean-absolute_{N-observationmorphologicalfilter isgiven}

by

MAE{W) = E\\ Y-W(X)\

}

(17)

(see Eq. 14). Since W is _fully determined _by its _{basis, finding} the optimal

N-observation filterreducesto _selectingthesubset ofthe2N

structuringelements

thatyields minimumMAE(W).

A _strikingfeature ofthe optimal MAEerosion filter is_that, relative to the

entire_image, it isnot a morphologicalfilter. Because_filtering isconsidered as

point-wise _estimation, each pixel possesses its own optimal basis. _Thus, it is not_{spatially invariant,} and therefore nottranslationinvariant. Inthecase of

optimizing a linear _estimator, wide-sense _stationarity yields spatial

invariance; however, morphological estimators require strict-sense

2.3. _BinaryMean-Absolute-ErrorTheorem

Thepresent section states a _binarymorphologicalfilter MAEtheoremthat

can be employed in a computer algorithm to search for an optimal basis. In

effect, the theorem states that the MAE of a morphological filter can be

expressedas alinearcombination oftheMAEs ofits individualbasiselements

and their unions. Under the assumption of _{stationarity,} a computer search

approach could _{directly employ} Eqs. ₍₁₆₎ and _(17), which give the numerical

MAE expressions for single-erosion and multiple-erosion _{filters, respectively,}

to estimate MAE from image realizations _{by comparing} filtered

noise-corrupted realizations _{to corresponding} uncorrupted realizations. _However,

rather _{than actually filtering} images to determine estimation _efficiency and

find theoptimal-filterbasis, it is_{computationally} more efficient_{to employ the}

general theorem regarding MAE for morphological filters. Employed over a

range ofbasis sizes (basis size refers to number of_structuringelements inthe

basis), to an allowed limit, an algorithm _{employing the} theorem can provide

thebasisthatyields minimumMAE. Thetheoremisproven intheappendix.

The theorem depends upon certain structuring-element

"fit,"

or

"subset,"

statistics based upon an image model and image degradation model (or more

generally, image transformation model), orrealizationsofsuchmodels. First,

we will describe the fit statistics, how they pertain to two types of estimation

expression for MAE ofthe single-erosion filter. To provide _{understanding} on

how the general theorem for n erosions will lead _to a basis-search _algorithm,

we will next derive the 2-erosion MAE formula. A prooffor _arbitrary n is in

the appendix. Strict-sense

stationarity ofthe image process and degradation

process isassumedthroughoutthederivation.

Toproceed, let S and S'

respectively denote theuncorrupted and corrupted

image, let_Bz denote the _structuring element B translated _bythe vector _z, Bz

= _B

+ z, and let_Kz denotethe set of observed pixels aboutthe pixel_{z, Kz} = S'

Pl _Wz, where Wz is an observation window W translated _by z. Under the

assumption of strict-sense _{stationarity,}we can speakoftheMAE forafilter as

beinganimage_error, sinceit ispixelindependent. Fora singleerosion_byB,

MAE(B) =_{E[\ S{z)} -(S'

B)(z)\

]

= _P[S(z) * _(S'eB)(z)}

(S(z)=l) n ((S'B)(z) =

0)

]

1)

]

As written, MAE(B)is evaluated relative to sets beingtreated as {0, l}-valued

functions. _However, each of the events in Eq. (18) possesses a random-set

formulation. For instance, the random-function event [(S'

B)(z) = _0]

Similarconsiderations _applyto more general multi-erosion_filters, Eq. ₍₁₈₎ holdingfor _MAE(W) with_W(S*)in place of

S'

B. If _W(S0 =

max{S' Bt},

then the _following random-function events and random-set events are

equivalent:

[W(S')(z) _=0] = V,

(S'

B.)(z) = _{0 (random functionformulation)} ,

(19)

[z _{ew)] =n. (B.)} (ZK l z z

(randomsetformulation). _C2SS\

In Eq. _(20), it must be kept in mind that the intersection is an intersection of

events. To avoid needless notational _pedantry, in the _following analysis we

will _frequently mix random-function and random-set _{formulations,} and

certain symbols _(e.g., S, B) _may be used to represent either functions or sets dependingontheformulation.

When _estimating S(z) _by (S' B)(z), one oftwo types of estimation error

can occur. We define type-0error asthatwhichoccurswhenS(z) = ₁but

(S'

B)(z) = _0;

eroding the observedimage resultsin a zero at alocationwherethe

ideal value is one. Type-1 error occurs when S(z) = 0 but (S'

B)(z) =

1;the

erosion estimate is one where the ideal state of the pixel is zero. _The

probabilities of type-0 and type-1 errors _occurring can be written as

pQ[5]= _p[(S'B)(z) =

0) n(S(z)=l)

= _p

BzdKz ] n ( z e s

(21)

Pif5]= _p[((S'B)(z) =

l) n(S(z)=0)

= _p

p^c

Kz ) n ₍ z e s

(22)

where_{pof-B] andpi[B]} arethe probabilities oftype-0 and type-1 estimation er

rors, respectively, when erosion _by B is the estimation rule. Note the

equivalent events

[(S'

B)(z) = _0] = _{[Bz <ZKz] and}

[(S'

)(z) = _1] =

[BzC

i<L2], where erosion is treated as set exclusion or inclusion for derivationpur

poses. The mean-absolute error ofestimation atzis thesum ofthese two mu

tuallyexclusive error probabilities:

MAE{B)= _pQ[B]+Pi[B]

= _P

(*2cztf2)n(zes)

z z

)

ri(zes)

ItisadvantageoustoviewEq. ₍₂₃₎froma_slightlydifferentperspective:

MAE(B)= P

H

A A B = (Ac_{nB) U (A nBc)} = _(AUB)

-(An₅₎ . (25)

The assumption ofstrict-sense _stationarity results in the MAE relative to

the entire image _beingequaltothe point-wiseMAEofEq. (24). Providedwith

a suitable image model and image transformation model we _may extract the

probabilities oftype-0 and _{type-1 errors,} and thus calculate MAE for a given

single-erosion filter. _Depending on the nature of_{the models,} it _may be more

convenientto state_poand_pi ina particular_form,suchas conditional probabil

ities, set_inclusions, orerosions. Also, forcomputational _{efficiencypurposes,} it

maybemoredirect_{to employ}an_"exclusive

or"

(XOR),whichisthelogicequiv

alent ofA.

Toward our goal of_derivingand _{understanding} a _{theorem relating}

single-erosion MAE and n-erosion _MAE, we next examine the two-erosion case. The

probability oftype-0 error can be written in terms ofthe max _(\y) oftwo ero

sions:

P [B ,B ]= P

yQ 1'

2

((S'eB^iz)

(S'b2)(z) = _o

) n _{ S(z) = _i

(-.

z z I V 2 z z

n [zts

Equation (26) can be simplified to a sum of single-erosion type-0 probabilities

byusingP[C D D] = _{F[C\ + P[D]}

-P[C U _D], with C= _[(Bi)z <Z Kz] n [z S]

8LndD=[(B2)z(lKz]n[zS]:

p [B ,B ] = P

*o r 2

(^V.**

)

zes + p _(B2)z <zx ₎ n ( z s

- p (BJ <Ztf u (BJ <ZK

1 2 2/ \ 2z 2 n

zcs

(27)

The eventthat atleast one _structuring elementis not a subset ofthe observa

tionsetisequivalentto theunionof_structuringelements not_fittinginKz:

((iVzC*

)

)

((B1US2}2C*2

)

(28)

SubstitutingintoEq._(27), weobtain

P0IB1,B2)=P

(Bi)z(LKz ) n ( zS + P (B) Q.K 2 2 2

)

n(zts)

-P _{(B1 VB2)2dKz} ] n ₍ ztS

(29)

andfinally,by_{recognizing the}formofEq. _(21),

From Eq. (30) we see that _knowing the _probability oftype-0 estimation_error,

individually, of_{any two structuring} elements andtheir _union,we _{may simply}

calculate the probability oftype-0 error when the two elements are used as a

morphologicalfilter basis.

Probabilityoftype-1 error canbecalculatedsimilarly:

P1[B1,B2]=P

(S'

BJiz) V (S'QBJiz) = ₁

) D ₍ S(z) = ₀

= _P (BJ CK

)

1 2 2/ \ 2 2 2 n

zts

(31)

Apply P[C U D] = _{PVC] +} P[D)

-P[C n _D],where C= [(Bi)z C X2] n [z S]

andD = _[(B2)zCKz] n [z _f? S]:

P^^.^l = _p

- P

)n(.

(Bi)zCJf ) n ( zS +p

(B ) ck n (BJ cii: 1 2 2 / V 2 Z 2

(Bj ex n _[ zS

2 z z

n zts

(32)

The event where both structuringelements are subsets ofthe observation set

isequivalentto the union ofthestructuringelementsbeingasubsetofKz:

{(B1)zcKz)

)

(UVJ*AC*.

Substituting into Eq. _(32), we obtain an expression that can be writtenin the

sameformasEq. (30):

P1[Bi,B2]=pi[Bi] +P1lB2]-pi[BiUB2]. ₍₃₄₎

Mean-absolute error ofthe 2-erosion filter is the sum of_{the two mutually}

exclusive errorprobabilities:

MAE{BVB2)= _{P0fBi;B2] +}

p^B^BJ

=

*<W + P0^-VBiUB2l + _{VBi] +P1[B2]-p1[B1UB2],}

(35)

whichitself isasum ofmean-absolute _errors,

MAE(BVB2)=

MAEiBJ + MAEiBJ-MAEiB^BJ.

The _key result ofthisprobabilistic analysis isthat the MAEofa 2-erosion fil

ter is equal to a linear combination ofthe MAEs ofthe_{individual structuring}

elements andtheir union. We found a similarresult withthe individual error

probabilities [Eqs. ₍₃₀₎ and (34)]. Equation (36) leads us to a minimum MAE

filter basis search strategy. _Knowingthe MAEvalues for individualstructur

ingelements, which_may have beenchosen_{for, say,} theirfilter_properties, and

knowing the MAE for _{corresponding} unions, one _may calculate all 2-erosion

It isalso worth_interpretingEq. (36)fromamorphological _filteringperspec

tive. Equation _(36), in_essence, counts estimation errors incurredupon filter

ingwith a 2-erosion filter. Theterms_MAE(B\) and_MAE{B}countthe errors

incurred upon _{eroding individually} with_B\ and_B2, respectively. When over

all estimation error _{(MAE(B\, B2))} is writtenin terms ofindividual _errors, a subtraction of the mean-absolute error of the structuring-element union (MAE{B\ U_B2)) mustbeperformed sothaterrors arenot countedtwice.

To provide a _key intermediate _{step to the understanding}and derivation of

the general ^-erosion mean-absolute error_theorem, we endthe 2-erosion ana

lysis _{by stating the} MAE in terms of the symmetric difference. We _employ

DeMorgan's Law to show that Eqs. (26) and (31) contain _{the complementary}

events

(B) CK n (B) CK

1 z z I \ 2 z z

(B) CK u (B) CK

1 2 Z I \ 2z 2 (37)

Using this relationship and _combining Eqs. (24), and _(26), we can express

2-erosion mean-absolute error as the probability ofthe symmetric difference of events:

MAE(B ,B >= P

1 2

(B ) CK U (B ) CK

1 2 z I V 2 2

a[z _es]

We now turn to the general _{binary case, stating the} morphological filter

MAEtheorem. Thetheorem statesthat the MAEof a morphologicalfilter can

be expressed as a linear combination ofthe MAEs ofits individual basis ele

ments andtheirunions. AproofisgiveninAppendix 1.

Theorem 1

-BinaryMorphologicalFilter MAE Theorem: An n-erosion

binary morphological filter Y_possessingbasisBas^] = {B\, B2, . .

,Bn}will provide a point estimatewith mean-absoluteerror givenby

n MAE(W) =

53

Yl

/=1 l<i,<i0<...<i.sn k

J

1 2 _j

Besides _{providing insight} into MAE incurred upon morphological filter

ing, the _binary morphological filter MAE theorem provides a mechanism for

searchingfor an optimal filter basis. For computer searchefficiency, it is de

sirabletorewrite an expressionfor MAE inrecursiveform. Corollary 1isare

cursive form ofthe rc-erosion MAEtheorem canbestbe understood intermsof

asingle-erosionfilterBnandtwo(71-l)-erosionfilters,Y-iand

On-l-Corollary 1 - Recursive Form ofthe BinaryMorphological Filter MAE

Bas[W]

-{Bi, B2, . . . _, Bn} will provide _{a point estimate}

with mean-absolute

error given_by

MAE{W ) = MAE(W

) + MAE(B ) - MAE(<f> )

n n-l n x n-Y

whre

Bas[^_i]={Pi,P2)...>Bn_i}(

Bas[Vn] =Bas[Wn_1]U{Bn}= _{By,B2, . . .

,BJ ,

Bas[3> _{1={BUB,BUB,...,B UB}.}

n_{1 1}

n 2 n n

1 n

Proof: TheMA"

theorem maybewritten

MAE(W ₎ =_MAE(B

,B,...,B > + MA(B )

n 12 71₁

n

i-l

+ V (-1/ V _MAE(U{ , B. U B ) . (39)

^ ' z ' _{* 1 1 n}

7=1 l<i: <i <...<.. Sn-1 *

12 _^

where we seethat thefirsttermistheMAE foran(nl)-erosionfilterwith_ba

sis_{Bast^H-].],the} secondterm corresponds to thesingle erosion filter_Bn,and

the summation ofthe third term is ofthe same form as the MAEtheorem for

To more_clearly see the formofthe_binary morphologicalfilterMAEtheo

rem and its recursive _formulation,we write the expressions in expanded form fora specificbasissize. The MAEof a3-erosionfilterisgiven_by

MAE(B ,B BJ= MAE(B,> + MAE(B) + MAE (B )

A ^ o 1 2 3

-MAE(B, UB)

-MAE(B UB ₎- MAE{B _U B ₎

12 1 3 2 3

+MAE(Bx UB UB ₎ ,

1 2 3' '

or,inrecursiveform,

MAE(W) =_{MAE(B,,B0) + MAE(B0)}

-MAE(B,l)B0, BUB,) .

o 1 Z o 1 o 1 o

(40)

(41)

Single-erosion-filterMAEfor_structuringelements_{B\, B2, 3} andtheirunions

may be obtained through image models andimage degradationmodels. MAE of_{the corresponding 3-erosion}filter_maybecalculatedthrough thelinearoper ations indicated inthe MAEtheorem. An optimal 3-erosion filter _may be de signed _{by examining} libraries (Section 2.4.2) of structuring-element candi

dates, where, knowing single-erosion statistics, all _3-tuples, are evaluated _by

the_theorem, andthe combination _yieldingminimumMAE is optimal. Ingen

eral, the _binary morphological filter MAE theorem _may be employed over a range ofbasis _{size, to} an allowed limit _n, thereby providing the n-optimal fil

(greaternumberof_{structuringelements) to approachafilter}

with suitable low MAE,therecursiveformoftheMAEtheoremshouldbeemployed.

To see the manner in which the recursion formula can be employed in filter _{optimization,} suppose we wish to optimize _{by selecting}bases from some

structuring-element collection B = _{{Bi, B2,..., Bq}. For}

p =

1, 2,..., q, let _Bp denotetheclosure ofBunder unions_ofpelements withinB. Then_Bi = _{B, B2}

=

[Bh U _{Bi2}, B3} =

{Bh U _Bi2 U _Bi3},..., and_Bi C _B2 C . . . C Bq. Suppose we know the MAE for each _structuring element in Bq. We can proceed in the

followingrecursive manner. For _any 2-erosion filter _^2 withBasf1!^] =

{Biv

MAE (W> = MAE(B. )+ MAE(B. )-MAE(B. UB. ).

2

li l2 '1 l2 (42)

Now suppose_W3 is a 3-erosion filter withBas[TP3] =

{BivBiv B^. _MAE(Ws)

canbeobtained viaEq. _(41),withare_{the terms coming}fromtheprevious stage

oftheiteration. Wecan continue recursively.

Note _{that the generality} ofthe MAEtheorem doesallowfor evaluation of

redundant structuring-element combinations, but reduces to nonredundant forms. For example, consider _{the structuring} elements _{B\, B2,} and _B3 as a

2.4. Constrained ApproachtoMorphologicalFilterDesign

Rather than select the optimal filter W over an AT-pixel window _W, we

may wish to constrain W_byrequiringthat some extra properties be satisfied.

We might wish to impose some algebraic constraints on W. For _instance, we

mightdesire thatWbe antiextensiveor_idempotent, orperhaps_both, sothatit

is a x-opening. Each constraint translates into some constraint on the filter

basis, and conversely. In the case of _{antiextensivity, the} requirement is that

each basis element must contain the origin. When _applying algebraic

constraints we gain some desirable filter _{property in} exchange for a possible

increase in _MAE, because _{requiring the} filter to possess certain properties is

tantamount _{to restricting the} setofpossiblefilter basesto some subcollection

of all potential filterbases. This type of algebraic constraint wasdiscussed at

length in _Dougherty (1992a) with respect to _binary x-openings and in

Dougherty(1992b)with respecttogray-scalelinearoperators.

Our concern in the present paper is design tractability, and to that end

we are concerned _mainly with three constraint paradigms, each of which

involves moreefficientfilter design inreturnfor _{suboptimality}(an increasein

MAE). While we will analyze each ofthe three individually, tractable design

may verywellinvolve_usingallthreeconstraintsinconjunction.

First, we might wish to constrain the number ofbasis elements to some

we obtain the optimal single-erosion filter. _{By fixing} a limit m we do not

require that there be _exactly merosions in the_{filter, only that the} number be

bounded _by m. This convention is crucial because if B and E are two

structuring-elementclasses suchthatBC _E,then^b^^e. _Thus,forcingtoo

large a basis can cause overestimation: there will be too _many erosions inthe

Matheron expansion. We call this type of constraint size constraint. The

impactofbasis-sizeconstraint onthegoodness of morphological estimatorshas

beenexamined_{by Dougherty}andLoce_(1992, 1993b).

Ratherthanfixthemaximum number ofbasiselements at_{the outset,}we

often choose to _dynamically limit the number _{by plotting} MAE against the

number of_structuring elements. Whenthis size-MAEcurvebeginsto _flatten,

we recognize thatit is _{highly unlikely that}largerbaseswill havea significant

effect. We _could, of _course, wait for a definitive increase in the size-MAE

curve; however, computation _{time increases combinatorically} with_increasing

basis size and therefore it is pragmatic to choose some heuristic decision

mechanism to judge when the basis is sufficiently large to preclude further

significantreductioninMAE.

A second constraint _arising from our desire to reduce computation in

filter designinvolvesobservation restriction. Ratherthanconsider allpossible

structuring elements in a window W, we might restrict our attention to

structuring elements in some subwindow W in W. Strictly _speaking,

unconstrained optimization over _{W; however, assuming that} we would

actually like to optimize over _W, we can view restriction to Was a constraint

that will allow us _only to consider a subclass of all _{W-bases, thereby likely}

yielding a filter with _{increased MAE. In effect, this type of constraint is a}

special case of the general constraint problem in which basis restriction is

relativetowindow geometry. Wetermitwindow constraint.

A third type of constraint is _library constraint. _Here, rather than

optimize over all possible _{structuring elements,} we restrict ourselves to some

predetermined _subset, or _subsets, of _structuring elements in the window.

Specifically,we might postulate mcollections of_{structuringelements,Li, L2,} .

. .

, Lm, each of which is suited for accomplishing a certain type offiltering

task. _Letting L = _{U Li, we constrain our} basis _{selection process to}

L, or

perhaps to some _{subfamily L;p L;2,} . . . _, L;r. The manner in which such a

restriction is developed is _key to the goodness of the _filter, where (as in

common statistical usage) goodness refers to the accuracy and precision of

estimation.

2.4.1. Window Constraint

Suppose we wish to reduce the computational burden _{by only employing}

structuring elements in a subwindow W ofthe window W. _{Heuristically,} we willdelete pixelsthatwebelievetoform _structuringelementsthatdo not_play

a major role in _reducing MAE. If our assumptions are reasonable, we will

obtain a suboptimal filter that is _{only marginally} less efficient than the

optimal filter over W. For _instance, we might desire a 5 X 5 square _window;

however,such awindowpossesses225 = _33,554,432

structuringelements,and

therefore there are _{33, 554,432!/m!(33,}554,432 m)\ m-element _structuring

element combinations. Of_course, a multitude ofthese are redundant due to

basisminimality; nevertheless, even ifwelimitourselvesto small _m, a search

throughallpossible structuring-element combinationsisstill_{computationally}

burdensome. Rather than use the 5X5window, we can _greatly decreasethe

search space_{byeliminating}8pixelstoformthe subwindow WshowninFig. 1.

Thecostis a constraintthat impinges ultimatelyonfilter_properties,as well as

uponMAE.

Window constraint_can, in_general, beformulated intermsofthe Matheron

representation, Eq. (12). Suppose W is a filter with Bas[^] over window W,

which means that all basiselements lie in W. If Wis a subwindow of _W, the

window-constrainedfilter isgivenby

(a) (b)

Fig. 1. a)Unconstrained 5x5 WindowW,b)Constrained5x5Window W.

wheretheconventionis adoptedthatBfl Wisdeletedfromtheexpansionifit

isnull. If ithappensthatB n W *0 forallB in_Bas[W],then, sinceBDW'C

B,eachimage inthe constrained expansionisasuperset oftheoriginal_image,

so that W(S) D ^(S). Here one mustbe prudent: ifthe constraint eliminates

some ofthebasiselements_altogether,then therearefewer imagesformingthe

union and the constrained filter might not dominate the original filter.

Moreover, note that Bas[*F1 need not equal {B D W: B Z Bas[W]}, since

intersection with W might create redundancy; however, Basf^H can be

derived _by simply eliminating redundancy. Note that we have published

additional material on window constraint, in particular, on

2.4.2 Library-Optimal Filters

A_keymethod of_achieving design_tractability over a given window Wis to

limit the set of potential basis elements: rather than optimize over all

nonredundant collections of _structuring elements within W, we preselect a

library L from which to form _bases, and we _{say that the} optimal filter from

among those whose bases are formed from L is ~L-optimal. _Library

optimization constraintischaracterized_bythelimitationsofL. In_selectingL,

two conditions must be met: ₍₁₎ bases formed from L must produce a class of

filters that provides good _{suboptimality} over the imagerange of_interest, and

(2) the size ofL must be _sufficiently small to yield design tractability. Aside

fromthese global_constraints,there areother propertiesthatwillprove_useful,

andthesewillbecomeapparentaswe proceed.

While one might approach the construction of a library in a number of

ways, herewe willdescribetwomethods. Onewillbebaseduponknowledge of

important filter bases. Because knowledgeoffilter behavior willbecontained

inthe final _library, we call it an expert library. Wewill also discussa method

of_library constructionbased onfirst-order statistical properties of_structuring

elements. Thiswecallafirst-order library.

Concerning properties of libraries constructed by either method or a

G =

{yltV2,...,WJ, (44)

with bases _Bast^;], i = _{l,2,...,n. If C is a class of}

structuring elements

such that C D U _Bas[^;], and Wis _C-optimal, then _MAE(W) < _{MAE(Wt) for}

all _i,sothatW isbetterthan_Wiforall i. In_particular, ifwe let L = _{U Bast1?;],}

thentheL-optimal filter is betterthan_Wiforalli.

Taking a_slightly more general_approach, let_{Gi, G2,..., Gm}be collections of

filters, let

Li = _{U{Bas[Wy]:Wy-6Gi},} (45)

and letL = _{U L;. If W is L-optimal, we say that}it is generated_byG = _{U G;,}

andforeach iwe _{say the}Lj-optimal filterWiisgeneratedbyG;. Thefilter W is

better than _Wi, for each _i, and each Wi is better than any of the filters

comprisingG;. Ifwe let

L(k i2...ir) = Lfl UL^U ... uL,v (46)

Milh-Js> =

L/, u_L,-2 u ... uLJs, (47)

where {i, _i2, . . .

, ir} C {/^j2, . . . J8}. then the LO'i/2 js>-optimal filter is

betterthanthe _L(iii2. . . ir>-optimalfilter. Inparticular,the L-optimalfilteris

filter is better_{than any} sublibrary-_{optimal filter.}

Generally, if W is _anyfilter

whose basis lies in _L<y . . . ), then Wis saidto be_generated

by L(y . . . ), orby

G(ij...) = _{Gt U}

G/ U . . . . Notethatinour

notation,L(i) = _L;andG(i) = Gt.

Given the _{preceding methodology,} ourtask is to form an appropriate set of

Gi, or_{equivalently,} an appropriate setofsublibrariesL;.

2.4.2.1. A Specific Expert_Library

For an expert _library, our approach is to select important filter classes G;.

Keep in mind that a particular _{sublibrary labeling} will no longer be _general,

butinstead willberelativeto thespecificlibrarythatwedevelop. Throughout

the present _study, we will be _focusing our attention on a centered 5X5

window, but the actual _library will involve the constrained 5X5 window of

Fig. 1. _Thus, we will _actually be _{employing two} constraint criteria in

conjunction. Henceforth, we will denote the constrained 5X5 window_by

W5andthefullcentered3X3window_by W3.

We will now describe the particulars ofa specific expert library. To begin

with, there are9 singleton elements in W3 and 8 in W5

-W3 [Fig. 2(a)]. Each

ofthese representsatranslation:ifB =

{z}isa_singleton,then S Q B = _S

-z.

Consequently, ifB = _{{B\, B2,} . . .

, Bq} isa collectionofq singletons, Bi

= _{zj},

*PB(S) = _U{S

Bt} = _U{S

-zj= _S (- _U{Bt})

. (48)

LetLi and _L2 denotethesingletons in _W3and _W5

-W3, respectively. _Gi and

G2 denote the collections of translations _by pixels in _W3 and _{W5 W3,}

respectively. The filtersgenerated_byGi aredilations_{bystructuring}elements

in _W3, and the filters generated _{by Gi} U _G2 are dilations _{by structuring}

elementsin W5.

There are four length-two linear openings: _{vertical, horizontal,} and two

diagonal. These comprise G3. Each possesses a basis _consisting of two

elements, the two translates ofthe opening structuring elementthat contain

the origin. These 8 _structuring elements comprise _L3 and are depicted in the

first two rows of Fig. 2(b). Since _{any x-opening} formed from the length-two

linear-openingstructuringelements isaunionofthe _openings,andthereforea

union oferosionsformed from L3,these x-openings aregenerated_byG3. The4

length-three linear openings comprise G4. Each has a basis with three

elements,andthe 12elements ofL4areshown in thelast fourrows ofFig. 2(b).

Taken _{together, L3} and _L4 generate all x-openings with linear _structuring

elements oflengthstwoandthree.

In addition to linear openings, we also desire square openings. _Letting

G5 consist of the single 2X2 opening, L5 consists of the 4 _structuring

a) c-ft C-3 Cl.-.! c_:"D c::!d it:-3 nr_{1 P}

uLJu CT--3 C..-3 cz-.j czz 3

C--.3

n rn n

c::i:j

lILJu -+P

c:::d cHtd

n I _p uLJU

C.Z.t c:::d &-.-3

c n_p c:::d d-tri

c:::d i":_d

n r1 P J LJu n n_p :::d nrt-D

m

M

ffi

5Hf ci' =* d^ JUL .TJu

f) c:;:d g) c!:jd C.D m---j z:: 3 &T3

H

nri p uLJ L

CT1 lib CIIIJ C"."D h) ci! i^ i) c!I!d tl c

is

n DP TJ_ip n

r]p n r₁ zz, ^i

n

iu J L

in r

J ll J I

]p n r Ju lTL

in nr

] u ib lJLJ b tJL111 dl

nnn

c

tt

uuhu lTHu uDu irlrt

n r1 P i 1

JTJTj

n ri p

i i J LTu C .-3

observation window can hold _{the 9 basis elements for the 3 X 3}

opening;

however, here we must contend with the constrained window W5.

Consequently, L6 consists of the 9 elements depicted in Fig. _2(d), these

constituting thebasisofthewindow-constrainedopening.

An erosion _{structuring element comprises its own single element basis.}

Letting G7 consist of the 4 centered length-three linear _{erosions, L7} is composed ofthe 4 _structuring elements shown in the first row ofFig. 2(e). Eachoftheseis a subsetofW3. _Lsconsists ofthe 8_length-four,noncentered

linear _structuring elements shownin the second andthird rows ofFig. 2(e). L9 consists ofthe 4 _length-five, centered linear structuring elements in the

lastrow ofFig. 2(e). _Gs and _G9 are composedofthe erosions _{corresponding}

to _L8 and _L9,respectively. For both _Ls and _L9, window constraint plays no

role.

As for nonlinear antiextensive _erosions, we consider _{two classes, the}

elements of which are to some extent

"ball-like."

L10 has 3 _elements, each

lying in _W3, and these are shown in the first row of Fig. 2(f). _Ln has 2

elements,each_requiring W5. Theseareshownin the secondrowofFig. 2(f).

The nextthreesublibraries correspondto hole-filling_erosions,and are of

particular importance for _{restoring images} degraded by salt noise. We call

6_i, 6q, b\, . . . , bq)represents the

structuringelementwith value _6/at_{(i, 0)}

in thegrid,and_havingall otherpixelvalues zero.

Consider _{closing by} B = _{(1 1). There are two}

translates B

-z, z 6 _B,

these_beinggiven_byBand(1 1). Thesingleton set₍₁₎andthe two-pointset

(1 0 1) both lie in _KerPF], and _{it is readily} seen _{that any kernel element}

must contain one ofthese. _Thus,

Basl^] =_{{(1),(1 0}

1)}, (49)

and,since erosion_bythe originisthe_{identity map,}

SB = _{SU[S(1 0 1)]. (50)}

In _effect, for a horizontal two-point linear _{element, the closing} maintains S

while _{filling any} inactivated pixel with activated pixels bothto the left and

to theright. In our_terminology,theelement (1 0 ₁₎isahole-filler.

As a second illustration relevant to our _study, suppose we consider

closing by a horizontal linear three-point element B. This _closing fills

inactivated pixels _{lying horizontally} between two activated pixels and fills

horizontal pixel pairs _{lying horizontally<}