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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 11, NOVEMBER 2006 3579

Spatially Variant Morphological Restoration andSkeleton Representation

Nidhal Bouaynaya, Student Member, IEEE, Mohammed Charif-Chefchaouni, andDan Schonfeld, Senior Member, IEEE

Abstract—The theory of spatially variant (SV) mathematicalmorphology is used to extend and analyze two important imageprocessing applications: morphological image restoration andskeleton representation of binary images. For morphologicalimage restoration, we propose the SV alternating sequential filtersand SV median filters. We establish the relation of SV median fil-ters to the basic SV morphological operators (i.e., SV erosions andSV dilations). For skeleton representation, we present a generalframework for the SV morphological skeleton representation ofbinary images. We study the properties of the SV morphologicalskeleton representation and derive conditions for its invertibility.We also develop an algorithm for the implementation of the SVmorphological skeleton representation of binary images. Thelatter algorithm is based on the optimal construction of the SVstructuring element mapping designed to minimize the cardinalityof the SV morphological skeleton representation. Experimentalresults show the dramatic improvement in the performance ofthe SV morphological restoration and SV morphological skeletonrepresentation algorithms in comparison to their translation-in-variant counterparts.

Index Terms—Adaptive morphology, alternating sequentialfilter, kernel representation, median filter, morphological skeletonrepresentation, spatially variant mathematical morphology.

I. INTRODUCTION

OVER the past few decades, morphological operators havegained increasing popularity in the implementation of

signal and image processing systems [1]–[12]. The inherentparallelism of the class of morphological filters allows theimplementation of very efficient and low-complexity algo-rithms for signal and image processing applications [1], [2],[6], [9]–[11], [13]–[16].

The basic concepts and analytic tools in mathematicalmorphology can be found, for binary images, in set theoryand integral geometry [6], [9]. In mathematical morphology,a binary image is represented as a subset of the two-dimen-sional (2–D) Euclidean space, or its digitized equivalent

, and image processing transformations are represented asset mappings between collections of subsets. Erosions anddilations are the two fundamental morphological operators

Manuscript received June 3, 2005; revised January 5, 2006. The associate ed-itor coordinating the review of this manuscript and approving it for publicationwas Dr. Giovanni Ramponi.

N. Bouaynaya and D. Schonfeld are with the Department of Electrical andComputer Engineering, University of Illinois at Chicago, Chicago, IL 60607USA (e-mail: [email protected], [email protected]).

M. Charif-Chefchaouni is with the Institut National des Postes et Télécom-munications, Rabat, Morocco (e-mail: [email protected]).

Digital Object Identifier 10.1109/TIP.2006.877475

[1], [2], [17]–[21]. They are characterized by a subset calledthe structuring element that is used to probe the image. Math-eron has captured the ubiquity of morphological operatorsby demonstrating that any increasing (i.e., operators whichpreserve signal ordering) and translation-invariant operatorscan be represented as unions (respectively, intersections) oferosions (respectively, dilations) [6].

In most applications of mathematical morphology, the struc-turing element remains constant in shape and size as the image isprobed. Hence, the focus of mathematical morphology has beenmostly devoted to translation-invariant operators. An extensionof the theory to spatially variant (SV) operators has emerged dueto the requirements of some applications, such as traffic spatialmeasurements [22] and range imagery [23]. For example, in theanalysis of images from traffic control cameras in [11], vehi-cles at the bottom of the image are closer and appear larger thanthose higher in the image. Hence, the structuring element sizeshould vary linearly with the vertical position in the image. Inrange imagery, the value of each pixel is related to the distanceto the imaging device. Consequently, the apparent height of anobject is a function of the object’s intensity range. Hence, onecan process (e.g., extract or eliminate) differently scaled objectsof interest in the image by adapting the size of the structuringelement(s) to the local intensity range [23].

Different techniques and algorithms to spatially adaptthe structuring element in an image have been proposed in[23]–[28]. Roerdink [29], [30] introduced the theoretical back-ground for a mathematical morphology that is not based ontranslation-invariant transformations in the Euclidean space.The proposed approach, however, is restricted by variousrigid algebraic constructions such as polar morphology andconstrained perspective morphology. In [31], a new class ofmorphological operations, which allow one to select varyingshapes and orientations of structuring elements, is presented.However, the sweep erosion and dilation do not satisfy the basicproperties of mathematical morphology. In particular, they arenot increasing operators in general and the sweep dilationoperator does not commute with the union. A unified theoryof SV mathematical morphology requires a further abstrac-tion of the basic notions of translation-invariant mathematicalmorphology. A fundamental result in lattice morphology1

that provides the representation of a large class of nonlinearand nonnecessarily translation-invariant operators in terms oflattice erosions and dilations has been presented in [33]. This

1Lattice morphology, introduced by Serra [11], is a powerful tool for the ab-straction of mathematical morphology based on lattice theory, a topic devotedto the investigation of the algebraic properties of partially ordered sets [32].

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3580 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 11, NOVEMBER 2006

representation, however, does not posses the geometric inter-pretation, captured by the structuring element, that is crucial insignal and image processing. A separate approach to the rep-resentation of SV mathematical morphology, which preservesthe geometrical structuring element, in the Euclidean space,has been introduced by Serra in [11]. Serra defined the conceptof a structuring function, which associates to each point in thespace a structuring element. Charif-Chefchaouni and Schon-feld [34], [35] pursued the investigation of SV mathematicalmorphology in a systematic way. They introduced the basic SVmorphological operators and investigated their properties. Acomprehensive development of the theory of SV mathematicalmorphology in the Euclidean space has been presented in [36].

This paper elaborates on the theory and applications of SVmathematical morphology presented in [36]. We illustrate thepower of the theory of SV mathematical morphology for twoimportant image processing applications: morphological imagerestoration and skeleton representation.

Morphological Image restoration is an important problem inimage processing and analysis applications. It requires the de-velopment of an efficient filtering procedure which restores animage from its noisy version [4], [5], [8], [11], [37], [38]. Inorder to devise such a filtering procedure, we have to considertwo fundamental issues: 1) the restoration filter should be ef-fective in eliminating the noise degradation and 2) it should beable to restore various important aspects of the shape-size con-tent of the noise-free image under consideration, as well as pre-serve its crucial geometrical and topological structure. The ef-fectiveness of morphological filters in the restoration of noisyimages has been demonstrated repeatedly [4], [8], [9], [11],[22], [37]–[41]. For example, Maragos and Schafer [4] havedemonstrated a strong relationship between the alternating filterand the median filter. Sternberg [37] introduced alternating se-quential filters (ASF) and experimentally showed their noise re-moval capability. Moreover, Schonfeld and Goutsias [8] haveshown that the class of alternating sequential filters is a set ofsmoothing morphological filters which best preserve the crucialstructure of input images, in the least-mean-difference sense.Translation-invariant morphological filters can be used to re-move noise structures that are smaller than the size of the struc-turing element. However, some important features of the signalmight be removed as well. Many researchers considered adap-tive morphological filtering in order to deal with this problem[24]–[26]. For example, Chen et al. [24], developed an algo-rithm for adaptive signal smoothing by spatially varying the fil-tering scales depending on the local property of each point inthe signal. To achieve this goal, they introduced the progres-sive umbra-filling (PUF) procedure. Their experimental resultshave shown that this approach can successfully eliminate noisewithout oversmoothing the important features of a signal. It canbe easily established that the morphological operators proposedin [24] are the gray-level extensions of the SV binary morpho-logical operators [42]. In this paper, we show that the theory ofSV mathematical morphology unifies many classical morpho-logical filters that use SV structuring elements, which are suit-able for adaptive image processing applications. We introduceSV alternating sequential filters and SV median filters for therestoration of images from their noisy versions. We derive the

basic properties of SV alternating sequential filters. We also de-rive a kernel representation of SV median filters, which estab-lishes the relation between the SV median filters and the basicSV morphological operators.

We also demonstrate the power of the theory of SV mathe-matical morphology by generalizing the morphological skeletonrepresentation to the SV morphological skeleton representation.The morphological skeleton has been investigated by many re-searchers [9], [43]–[46], mainly for the purpose of image codingand shape recognition. In [44], a general theory for the morpho-logical representation of discrete binary images was presented.The basis of this theory relies upon the generation of a set ofnonoverlapping segments of an image, which produce a de-composition that guarantees exact reconstruction of the originalimage. Decreasing the cardinality of the morphological skeletonrepresentation by reducing its redundancy has been explored bymany authors [43], [44], [46]. In this paper, We extend the mor-phological skeleton representation framework presented in [44]to the SV case. We study the properties of the SV morpholog-ical skeleton representation and derive conditions for its invert-ibility. We also propose an algorithm for the implementation ofthe SV morphological skeleton. This algorithm minimizes thecardinality of the image representation by deriving an optimaluniversal algorithm for the construction of a SV structuring el-ement mapping.

This paper is organized as follows. In Section II, we pro-vide a brief overview of SV mathematical morphology. Specif-ically, we present the basic SV morphological operators. Wesummarize the SV kernel representation, which shows that anyincreasing operator that fixes the entire space can be representedas a union (respectively, intersection) of SV erosions (respec-tively, SV dilations). This result demonstrates the ubiquity ofthe basic SV morphological operators (i.e., SV erosions andSV dilations). In Section III, we illustrate the impact of SVmathematical morphology on image restoration applications.In particular, we address morphological image restoration byintroducing the SV alternating sequential filters and SV me-dian filters. We also provide an example for selection of the SVstructuring element mapping for image restoration applications.We experimentally show the superior noise removal capabilitiesof these filters compared to their translation-invariant counter-parts. In Section IV, we demonstrate the power of SV mathemat-ical morphology in skeleton representation applications. Specif-ically, we propose the SV morphological skeleton representa-tion, develop some of its properties and derive conditions for itsinvertibility. We further propose a universal algorithm for theimplementation of the SV morphological skeleton representa-tion. We provide simulation results which demonstrate the enor-mous improvement of the SV morphological skeleton represen-tation in comparison to its translation-invariant counterpart. Fi-nally, in Section V, we present a brief summary of our resultsand discuss our plan for future work.

The proofs of all theoretical results that are new contributionsin this paper are presented in Appendices A and B.

II. SPATIALLY VARIANT MATHEMATICAL MORPHOLOGY

Notation: We consider a nonempty set or . The setdenotes the set of all subsets of . Elements of the set

BOUAYNAYA et al.: SV MORPHOLOGICAL RESTORATION AND SKELETON REPRESENTATION 3581

will be denoted by lower-case letters; e.g., , , . Elements ofthe set will be denoted by upper-case letters; e.g., , , .An order on is imposed by the inclusion . We use and

to denote the union and intersection in , respectively.denotes the complement of . The set difference

of sets and is defined by . Thetranslation also denoted by of a set by

is defined by .The cardinality of a set is the total number of elementscontained in the set. We use to denote the setof all operators mapping into itself. Elements of the setwill be denoted by lower case Greek letters; e.g., , , . Anorder on is imposed by the inclusion ; i.e., if andonly if , for every . We shall restrictour attention to nondegenerate operators; i.e., and

for some and , for every(the set is used to denote the empty set).

A mapping is:• increasing if , for all

;• translation-invariant if , for every

and every ;• extensive (respectively, anti-extensive) if (re-

spectively, ), for all ;• idempotent if , for all .

The mapping is the dual of the mapping iff

A. SV Morphological Operators

We shall now present an overview of the basic definitionsand properties of SV mathematical morphology introduced byCharif-Chefchaouni and Schonfeld [34], [35]. For a more com-prehensive introduction to SV mathematical morphology referto [36].2

1) SV Erosions and SV Dilations: The SV structuring ele-ment is a mapping from into . The transposed SV struc-turing element is a mapping from into given by

(1)

In translation-invariant mathematical morphology, is the trans-lation mapping by a fixed set , i.e., ,

. Hence, the transposed structuring element correspondsto the translation by the transposed set .

The SV erosion is given by

(2)

2Some of the results presented in this section have been extended to the gray-level case by Chen et al. [24], [25].

The SV dilation is given by

(3)

The SV erosions and dilations satisfy the basic properties oftranslation-invariant erosions and dilations. Below we list themain properties that we need in the sequel.

2) Properties of SV Erosions and SV Dilations:a) Adjunction: For every mapping from to the pair

defines an adjunction on . In other words

(4)

This result states that and are an erosion and adilation, respectively, in the sense that these operators aredistributive over intersection and union, respectively, i.e.,

(5)for an arbitrary collection of sets

. These identities can also be derived easily withoutreference to the framework of adjunctions.

b) Duality: The SV erosion and the SV dilation aredual operators, i.e.,

(6)

This relation states that dilating a set by the mappingis equivalent to eroding its complement by the samemapping and complementing the result.

c) Increasing: For a given mapping , the SV erosion andthe SV dilation are increasing operators, i.e.,

(7)

Increasing operators preserve order (contrast) in the sensethat they prohibit extraction of information from occludedregions. This property is consistent with the models ofthe human visual system which have been investigated inthe field of cognitive psychology. Specifically, the high-level vision models of gestalt psychology state that theperceptual processes underlying the visual interpretationof a scene are increasing operators [9], [47], [48].

d) Extensivity and Anti-Extensivity: If for every, then the SV erosion is anti-extensive and the

SV dilation is extensive, i.e.,

(8)


e) Serial Composition: Let us use andto denote the mappings from into given by

and, for every . Successively SV eroding

(respectively, SV dilating) a set first by and then byis equivalent to SV eroding (respectively, SV dilating) bythe dilated mapping , i.e., we have

(9)

and

(10)

3) SV Openings and SV Closings: We shall now present theSV openings and closings and review their main properties. TheSV opening is given by

(11)

The SV closing is given by

(12)

The SV opening and SV closing have nice geometrical interpre-tations. Let us refer to by the local structuring element atpoint . The SV opening of a set is the domain swept out byall local structuring elements which are included in . By du-ality, a point belongs to the SV closing if and only if all thelocal structuring elements containing hit . Formally, we havethe following equivalent definitions for the SV opening and SVclosing:

(13)

(14)

for every .4) Properties of SV Openings and SV Closings:a) Duality: The SV opening and the SV closing are

dual operators, i.e.,

(15)

b) Increasing: For a given , the SV opening and the SVclosing are increasing operators on , i.e.,

(16)

c) Extensivity and Anti-Extensivity: For every mapping , theSV opening (respectively, SV closing) is anti-extensive(respectively, extensive). We have

(17)

d) Idempotence: The SV opening and SV closing areidempotent operators, i.e., for every ,

(18)

e) Fixed Points: It was shown in [6] and [49] that the trans-lation-invariant opening and closing can be completelyspecified from their fixed points. The framework of fixedpoints can be extended to the SV case by defining -openand -closed sets as follows.

5) Definition 1: is -open (respectively, -closed)if [respectively, ].

A useful characterization of -open and -closed sets is givenby the following proposition.

Proposition 1: [36] is -open [respectively,-closed] if and only if there exists such that

(respectively, ).The concept of -open and -closed sets was further extended

to mappings as follows. Consider mappings and frominto . The mapping is -open (respectively, closed)if is -open (respectively, -closed), for every .We then have the following result: If is -open, then

(19)a) Sieving structure: If is open, then we have

(20)

and

(21)

for every .

B. SV Kernel Representation

An important notion related to set mappings is that of thekernel, introduced by Matheron [6] for translation-invariantmappings. Matheron subsequently showed that every increasingand translation-invariant operator can be written as a union oftranslation-invariant erosions, or, alternatively, as an intersec-tion of translation-invariant dilations. This result was extendedto SV operators in [36]. To state the corresponding theorem,we need the notion of a SV kernel. Let . The SV kernel

of is given by

(22)

An operator is a covering operator (abbreviated as -op-erator) if is increasing and satisfies . An importantproperty of -operators is that their kernel is nontrivial and isunique. In other words, the mapping which associates to each

-operator its kernel is a one-to-one mapping. [36].The kernel representation of -operators is given by the fol-

lowing theorem;


Theorem 1: [36] An operator is a -operator if andonly if

(23)

for every .Theorem 1 is not only interesting theoretically but also for

practical applications. For some -operators, a subset of thekernel, called a basis, is sufficient for its representation in termsof SV erosions or SV dilations. Therefore, if the kernel of a

-operator or one of its basis has a finite number of mappingsthen the -operator can be exactly represented as a finite unionof SV erosions, or equivalently, as a finite intersection of SV di-lations. This representation can tremendously simplify the anal-ysis and implementation of the nonlinear -operator. We willshow later that the adaptive median filter has a finite basis rep-resentation. Hence, it can be exactly expressed via a closed for-mula involving only intersections and unions of sets without re-quiring any sorting.

In Sections III–V, we apply the general theory of SV math-ematical morphology to two image processing applications:morphological restoration of noisy images and morphologicalskeleton representation.

III. SPATIALLY VARIANT MORPHOLOGICAL RESTORATION

We shall consider SV alternating sequential filters (SVASF)and adaptive median filters for SV or adaptive image restoration.

A. SVASFs

1) Theoretical Aspects: Alternating sequential filters (ASFs)were introduced by Sternberg [37] and were extensively studiedby Serra [11]. Basically, an alternating sequential filter is a com-position of openings and closings by structuring elements of in-creasing sizes. In this section, we shall extend the class of ASFto the SV case. In our work and denote the trans-lation-invariant erosion and dilation, respectively, between thesets and . We use AF and ASF to denote the trans-lation-invariant alternating filter and translation-invariant alter-nating sequential filter, respectively. SVAF and SVASF will de-note the SVAF and SVASF, respectively.

Given a binary image , let us assume that the transformationproduces a degraded binary image given by

(24)

where

(25)

The model given by (25) is known as the germ-grain model [8],[9], [50]. In this case is a sequence of sets,known as the primary grains, whereas is a

sequence of sites, known as the germs, which are randomly dis-tributed in ; e.g., a Bernouilli point process.3 Observe that thesequence of germs indicates the “centers”of the primary grains in the noise process

, .The SVAF by the structuring element mapping is defined as

the compound SV open-close

(26)

A SV alternating sequential filter is an iterative application ofSV alternating filters

(27)

where , is the order of the filter and thesequence is increasing i.e., , for all

, for all .An operator is called a morphological filter if it is

increasing and idempotent. This terminology is different fromthe word “filter,” which is commonly used by the signal andimage processing community to denote an “operator.” Althoughthe class of increasing transformations is closed under compo-sition, the class of idempotent transformations is not. The fol-lowing proposition gives a sufficient condition on the sequenceof structuring element mappings for the SV alternating se-quential filter to be a morphological filter.

Proposition 2:(a) For every structuring element mapping , the SV alter-

nating filter is a morphological filter.(b) If is -open (i.e., , ),

for , then the SV alternating sequential filterof order , is a morphological filter.

In the translation-invariant case, the sequence of structuringelement mappings is usually chosen to be

, where ( times) and[8]. Then, we have

; that is, is open. Hence, thecondition of Proposition 2 is satisfied in the translation-invariantcase. Since for a special choice of the structuring element map-ping , the SVASF reduces to the translation-invariant ASF, theclass of SVASFs is still the subset of smoothing morphologicalfilters which best preserve the crucial structure of input imagesin the least mean difference sense [8].

2) Simulations: As in the translation-invariant case, SVmathematical morphology theory is not constructive, in thesense that it does not build a systematic algorithm to find the“optimal” structuring element at each point of the image. Thechoice of the structuring element mapping obviously dependson the considered application.

We propose a general selection rule for image restoration.We assume that the noise model degradation is characterizedby the germ-grain model given by (24) and (25). In translation-

3Often, the sequence of germs fx ; n = 1; 2; . . .g is distributed accordingto a Poisson point process. In this case, the germ-grain model is known as theBoolean model [9], [50], [51].


invariant morphology, a priori knowledge of the noise modelleads to the selection of a structuring element of size larger thanthe largest noise-grain [8]. This approach ensures that the noiseis eliminated from the image. However, the geometric and topo-logical structure of the restored image is degraded during therestoration process. Denoising should be performed so that thenoise is reduced sufficiently and the geometric and topologicalcharacteristics of the image are preserved. This can be achievedby spatially varying the structuring element depending on thelocal properties at each pixel in the image. The idea of the pro-posed algorithm is to use, at each pixel, a structuring elementof size slightly bigger than the size of the noise-grain at thatpoint. The mappings of the are selected as shownin the equation at the bottom of the page, where , isthe 3 3 square structuring element and . Ob-serve that the above definition of the s does not imply thatthese mappings are equal, since the noise is reduced as the it-erations increase. For instance, assume that at the first iteration,a point is detected as the center of a noise grain . So,

. Assume further that the latter noise-grain issmall enough so that it has been removed by the filter .Hence, at the second iteration, no noise-grain is detected at point

and thus , i.e., the point is not filtered at the seconditeration. Moreover, observe that the constructed s do not sat-isfy Proposition 2 and thus the , for , is notidempotent. The detection of the presence of a noise-graincentered at the pixel is determined by selecting the largestpossible grain in the germ-grain model given by (24) and(25), which is present or absent in the degraded image (i.e.,

or ) [8].The performance of the algorithm is measured by the

signal-to-noise ratio (SNR). Let denote the original imageof size and the restored (denoised) image. The SNRis defined by

Consider the binary image depicted in Fig. 1(a). The de-graded binary image , obtained by using (24) and (25), with

, formed by the overlapping of square structuringelements distributed according to a Bernouilli process, is de-picted in Fig. 1(b). Fig. 1(c) [respectively, Fig. 1(d)] shows theoutput image of the AF using the rhombus structuring elementin [43] (respectively, 3 3 square structuring element) dilatedeight times. A slight improvement is obtained by the use ofan ASF of order 8 using the rhombus structuring element (re-

Fig. 1. Denoising of a binary image using morphological filters and SNR com-parison. (a) Original binary image. (b) Degraded image by a germ-grain noisemodel. (c) AF using the rhombus SE dilated eight times. (d) AF using a 3� 3square SE dilated eight times. (e) ASF using the rhombus SE. (f) ASF usinga 3� 3 square SE. (g) SVAF. (h) SVASF .

spectively, 3 3 square structuring element) in Fig. 1(e) [re-spectively, Fig. 1(f)]. Although most of the noise is removed bythe ASF, the original image is highly smoothed and its originaltopology is lost. We immediately see the drastic improvementof the SVAF and SVASF over their translation-invariant coun-terparts in Fig. 1(g) and (h), respectively. The SVASF removesthe noise while preserving the edges and the geometric structureof the image. The SNR of the different experiments are providedbelow their corresponding images. The SVASF achieves a SNR80-dB higher than its invariant homologue.

The simulation results obtained illustrate that, given an imagedegraded by noise characterized by the germ-grain model, evenif one were to select the optimal parametrization [8] and theoptimal structuring element [38] for the translation-invariant AFand ASF morphological filters, the results would be inferior tothose obtained by using SV morphological filters (see [8, Fig.3]).

B. Adaptive Median Filter

1) Theoretical Aspects: Consider . Let bea mapping from into such that and

is odd, for every . Themedian of with respect to the adaptive window

is given by

(28)

It is easy to show that the median filter is a self-dual -operator,i.e., it is a -operator such that it is its own dual. Therefore, it can


Fig. 2. Denoising using median filters and SNR comparison: (a) Original grayscale image. (b) Image degraded by a germ-grain model. (c) Eight iterations of me-dian filtering by a fixed 3� 3 window. (d) Eight iterations of median filtering by a fixed 5� 5 window. (e) Median filtering by a fixed 17� 17 window. (f) Medianfiltering by a fixed 19� 19 window. (g) Alternating sequential median filter ASMF of size eight using initial window of size 3� 3 incremented by two at eachiteration. (h) Alternating sequential median filter ASMF using initial window of size 5� 5 incremented by two at each iteration. (i) SV median filtering withadaptive window size corresponding to the germ grain size incremented by one. (j) SV median filtering with adaptive window size corresponding to the germ-grainsize incremented by two. (k) Second iteration of SV median filtering with adaptive window size corresponding to the germ-grain size incremented by one. (l)Second iteration of SV median filtering with adaptive window size corresponding to the germ-grain size incremented by two.

be shown from the definition of the kernel and from Theorem 1that

(29)

where and if and only if is any subsetof of cardinality .

Equation (29) establishes the relation between the adaptivemedian filter and the basic SV morphological filters, i.e., SVerosion and SV dilation. The implication of (29) are profoundbecause they enable us to express the adaptive median filter viaa closed formula involving only unions and intersections of sets,without requiring any sorting. For small adaptive window sizes,it was shown that the implementation of the median filter viaits kernel representation is more attractive than sorting schemes[48].

Although the theory of SV mathematical morphology hasbeen presented for the binary case, its extension to gray-scaleis straightforward and follows the steps used to extend transla-tion-invariant binary morphology to translation-invariant gray-scale morphology [37], [42]. Therefore, we will apply adaptivemedian filtering to gray-scale images.

2) Simulations: The key idea of the implementation of theadaptive median filter is identical to the SVASF. Specifically, thesize of the local window at a given point is selected to be slightlylarger than the size of the germ-grain at this point. In our sim-ulations, we consider the original image depicted in Fig. 2(a).Its corrupted version by a germ-grain noise model appears inFig. 2(b).

In the first experiment, we applied the median filter iterativelyto the degraded image using a fixed square window of size 3 3.The output image is shown in Fig. 2(c). Notice that since someof the germ-grains have size larger than 3, the median filter usingthe 3 3 window fails to remove the larger noise structures. Werepeat the same experiment using a 5 5 square window. Theresulting image is depicted in Fig. 2(d). While the median filter

using the 5 5 window removes most of the noise in iteration8, the output image is more smoothed.

In a second experiment, we applied a Median filter using afixed square window of size 17 17 [see Fig. 2(e)]. Althoughmore noise has been removed than the iterative median filtersin the first experiment, the restored image is overly smoothed.The same experiment is repeated using a square window of size19 19 [Fig. 2(f)]. The filtered image is cleaner but equallysmoothed.

In a third experiment, we applied an alternating sequentialmedian filter (ASMF) of order 8, which is composed of eightmedian filters of increasing window sizes. The window size isincremented by two at each iteration. Fig. 2(g) shows the re-sult of ASMF using an initial square window of size 3 3. Al-though the noise-grains are totally removed, the restored imageis over smoothed and its features (e.g., windows, edges) arecompletely lost. We reach the same conclusion if we apply anASMF using an initial square window of size 5 5 [Fig. 2(h)].Observe that the largest window of the ASMF with an initialwindow of size 3 3 (respectively, 5 5) has size 17 17 (re-spectively, 19 19).

In the last experiment, we applied the SV median filter(SVMF). Fig. 2(i) and (j) is obtained by SV median filteringusing, at each point, a local window size equal to the size ofthe noise-grain, at the same point, incremented by one andtwo, respectively. The restored images preserve the edgesand geometric structure of the noise-free image. Notice thatusing a local window size equal to the size of the noise-grainincremented by two performs better in terms of noise removalcapability, than using a local window of size equal to the size ofthe noise-grain incremented by one. The reason is that some ofthe noise-grains overlap and merge to form bigger noise-grains.Therefore, a larger local window size is needed to ensure thatthese noise-grains are suppressed. However, if the local windowsize is too large, the median value computed in that windowwill not provide a good estimate of the noise-free pixel becauseonly pixels in a small neighborhood are strongly correlated.Therefore, there is a trade-off between the noise removal


capability and the accuracy of the estimation. Fig. 2(k) and (l)shows two iterations of the SVMF using a local window sizeequal to the size of the noise-grain incremented by one andtwo, respectively. All of the noise has been removed in thesecond iteration of the SVMF without altering the topologicalcharacteristics of the noise-free image. The SNR of the aboveexperiments are provided below their corresponding images inFig. 2. Notice that the SVAMF achieves a SNR which is 20 dBhigher than its translation-invariant counterpart.

IV. SPATIALLY VARIANT MORPHOLOGICAL

SKELETON REPRESENTATION

Let , , and denote the translation-invarianterosion, dilation and opening, respectively, of the set by thestructuring element [9]. Let ( times).

A. Theoretical Aspects

Consider a sequence of mappings from intosuch that , for every , for all and

, for all . Consider the sequence of mappings frominto given byfor , and , for every . We definethe integer by for a given

.Let denote a collection of operators

in such that

(30)

for every , every and , forevery .

Definition 2: Consider . The SV morphologicalskeleton representation of is given by

(31)

where is the SV morphological skeleton representationsubset of order given by

(32)

Definition 3: The SV morphological skeleton representationof is invertible if there exists a sequence

of operators in such that

(33)

The following theorem establishes a restriction on the choicesof the sequences as a direct consequence ofconstraint (30) and establishes the invertibility of underthis restriction.

Theorem 2: If the sequence of operatorsin satisfies constraint (30), then

(34)

for and for every . Moreover, isinvertible and

(35)

From Theorem 2, we observe that the SV morphological rep-resentation of , , obtained by the sequence of transfor-mations which satisfy restriction (30), de-composes into a sequence of subsets

which uniquely characterizes , thereby allowing fora SV morphological representation which permits the exact re-construction of . In the remainder of this section, we shallinvestigate some properties of .

The following proposition shows that the resultingmorphological image representation subsets , for

, are disjoint and anti-extensive.Proposition 3: We have

(36)

(37)

In the following proposition we show that, under a certaincondition on the mappings , a repeated appli-cation of the transformation does not influence the imagerepresentation.

Proposition 4: If

(38)

then

(39)

When condition (38) is not satisfied, the repeated applicationof may result in a further reduction of the total cardinalityof the representation. This is a desirable result in many applica-tions of interest (e.g., image coding [52], [53]).

1) Example (Generalized Morphological Skeleton [3]): Thefollowing example is an important special case of the generalSV morphological image representation . Consider

(40)

The following proposition shows that the above sequencesatisfies constraint (34).


Proposition 5:

(41)

for and .Therefore, the representation , given by (31) and (32),

is invertible and the reconstruction formula is given by (35).

B. Algorithmic Analysis

In this section, we develop an algorithm for the implemen-tation of the SV morphological skeleton representation studiedabove. We compare the SV skeleton representation with thetranslation-invariant representation. The performance of themorphological skeleton representations is assessed by thenumber of points, in the image representation, required forexact reconstruction of the original image.

From our perspective, the purpose of morphological skeletonrepresentations is coding, compression and storage. In a com-munication framework, our goal is to minimize the averagecode length of the compressed binary image. It has been shownthat efficient encoding of the skeleton representation usingrun-length type codes can be used to provide an efficient com-pression routine for binary images [43]. This approach relieson the sparse representation of skeletons to efficiently encodelong run-lengths corresponding to pixels that do not lie in theskeleton. Therefore, it is not surprising that investigators havedetermined that lower cardinality skeleton representations yieldsuperior compression ratios [43]. Our goal is thus to minimizethe cardinality of the source image under the constraint oflossless compression. For the purpose of this presentation, weassume the channel to be noiseless. Therefore, the receiverwill be able to reconstruct the original image perfectly withouterror.

The optimal SE, in the sense of minimizing the cardinality ofthe morphological skeleton representation, would be the imageitself. The translation-invariant and SV morphological skeletonrepresentations would be identical and consist of one point.However, this is a trivial solution since it is impractical and as-sumes that the image to be transmitted by the sender is alreadyknown by the receiver and is stored in its library. We will as-sume a fixed library of structuring elements at the encoder anddecoder and construct the optimal SV structuring element map-ping to minimize the cardinality of the morphological skeletonrepresentation. The idea of the proposed algorithm is similarto the matching pursuit algorithm [54]. The matching-pursuitalgorithm adaptively decomposes a signal into waveformsthat are the dilations, translations, and modulations of a singlefunction; thus, providing an interpretation of the signal struc-ture. In the proposed algorithm, let denote the originalimage and a fixed structuring element. Table I describes auniversal algorithm to construct the optimal structuring elementmapping for the SV morphological skeleton representation.The algorithm is an iterative process. At each iteration, thealgorithm selects the center of the dilated structuring element

that maximally intersects the image, for some integer .The union of these center points constitutes the SV morpho-logical skeleton representation. The exact reconstruction of the

TABLE IUNIVERSAL ALGORITHM TO CONSTRUCT THE OPTIMAL STRUCTURING

ELEMENT MAPPING FOR THE SV-MORPHOLOGICAL

SKELETON REPRESENTATION

Fig. 3. Morphological skeleton representation: (a) Original image. (b) SV mor-phological skeleton representation (162 points). (c) Translation-invariant mor-phological skeleton representation (502 points).

original image is guaranteed given the set of center points andtheir corresponding integer .

It is easy to show that the resulting SV morphologicalskeleton representation using the proposed algorithm is com-pact, in the sense that the set is notredundant; i.e., the reconstruction based on any partial subset ofthe resulting SV morphological skeleton representation wouldform a strict subset of the original image. It is also interestingto observe that the sets are notnecessarily disjoint. Thus, the optimal morphological skeletonrepresentation is not derived by a decomposition of the originalimage into nonoverlapping shapes. Instead, overlapping shapesare exploited in order to reduce the number of shapes requiredto cover the image. This approach allows for a substantialreduction in the cardinality of the morphological skeleton rep-resentation. Moreover, as we have seen earlier in the exampleof the SV skeleton representation of a synthetic image, theresulting SV morphological skeleton representation using theoptimal SV structuring elements outlined in the algorithm isidentical to the SV erosion using the same structuring elementmapping. The SV and translation-invariant morphologicalskeleton representations are shown in Fig. 3(b) and (c), re-spectively. The SV morphological skeleton representation hasa compression ratio which is more than three times higher


than its translation-invariant counterpart. We shall once againdetermine the number of parameters needed for representationof the skeleton and ignore the effects of the use of encodingschemes using run-lengths type techniques in computing thestorage capacity requirements. If the morphological skeletonrepresentation is used for storage or communication, then weneed parameters to transmit the SV morpho-logical skeleton representation. On the other hand, the totalnumber of parameters required for transmission of the transla-tion-invariant morphological skeleton representation is equalto . Therefore, the SV morphological skeletonrepresentation has a storage capacity gain that is 2.32 timeshigher than the translation-invariant morphological skeletonrepresentation.

V. CONCLUSION

In this paper, we presented a general theory of SV mathe-matical morphology and showed its enormous potential throughtwo important image processing applications. First, we intro-duced SV alternating sequential filters and SV median filtersfor SV morphological denoising of degraded images. Simula-tion results demonstrated that, not only is the noise removalcapability of the SV morphological filters dramatically higherthan their translation-invariant counterparts, but also the topo-logical and geometrical structure of the original image are pre-served. Second, we extended the translation-invariant morpho-logical skeleton representation to the SV case. We have also de-veloped a universal algorithm for optimal selection of the SVstructuring element mapping for skeletonization by minimizingthe cardinality of the SV morphological skeleton representation.This approach has been shown to yield a substantial reductionin the cardinality of the SV morphological skeleton representa-tion in comparison to its translation-invariant counterpart. As aresult of this investigation, we have complemented the eleganttheory of SV mathematical morphology with powerful prac-tical algorithms for image processing applications. The SV mor-phological framework presented can be applied to many clas-sical image processing problems related to nonlinear filtering. Inthe future, we plan to further explore the power of the SVMMframework by developing more sophisticated and faster algo-rithms for non linear filtering in various image processing ap-plications. We also plan to investigate the optimal SV struc-turing element mapping for morphological image restorationapplications by extending the work of Schonfeld [38] for trans-lation-invariant morphological filters. We further plan to explorethe robustness of the SV morphological skeleton representa-tion to noise degradation by extending the work of Schonfeldand Goutsias [45] for the translation-invariant morphologicalskeleton representation. A formal analysis of the optimal struc-turing element mapping and investigation of the robustness ofthe morphological structuring element require the use of randomset theory [6], [9], [50], [51].

APPENDIX APROOF OF PROPOSITIONS

Proof [Proof of Proposition 2]: From Section II-A4, weknow that and are increasing operators. Since the class ofincreasing operators is closed under composition, we concludethat the and the are increasing.

a) Let us first prove the idempotence of the SVAF. By theanti-extensivity property of , we have

for all . Applying to the above inequalityand, since is increasing, we obtain

where the last equality follows from the idempotence of. Therefore, we have

(42)

By the extensivity property of , we have

Applying to the above inequality and since is in-creasing, we obtain

(43)

where the last equality follows from the idempotence of.

Applying to (43), and since is increasing, we obtain

Therefore, we have

(44)

Using (42) and (44), we establish the idempotence of the.

b) The idempotence of the follows from the moregeneral result established by Schonfeld and Goutsias in[8, Proposition 1]. All the necessary conditions to applyProposition [8, Proposition 1] are satisfied.

Proof [Proof of Proposition 3]: From (32) and constraint(30), we have

(45)


Assume that are such that . From thedefinition of , we have

Therefore,. From (45), we obtain . Equation (37) is a direct

consequence of (32) and the fact that .This completes the proof.

Proof [Proof of Proposition 4]: Let such that. From the definition of and , we have

, for every . Therefore, .From (32) and the fact that , we have

. From the constraint(30) and since , we have . Therefore,from (32), we have for . Finally, we see that

(46)

Now, we will show that satisfies .Let us take . From (32), we see thatand . From constraint (30), we seethat . From thedefinition of , there exists such that

. Sinceand by hypothesis for ,

we have . Thus, for every. Consider now, and consider

the set . By definition of the mappings , we havethat . Since , we conclude that

for every . Thus,for every or . Finally, by using (46)with , we prove (39).

Proof [Proof of Proposition 5]: By the extensivityproperty of the SV dilation, we have that

. By the extensivity propertyof SV closing, we have

(47)

Using (10), we have that .From the construction of , we have that

, for and . Therefore

Therefore, ,. Finally, (47) becomes

.Therefore, the sequence sat-

isfies constraint (34) for every . Hence, by The-

orem 2, is invertible and.

APPENDIX BPROOF OF THEOREMS

Proof [Proof of Theorem 2]: Since constraint (30) must besatisfied for every , we can choose .Therefore, we obtain

(48)

for . From (12), we observe that. From the fact

that is -closed, we have, using Proposition 1

(49)

Substituting the above equation into (48), we obtain (34).From (32), we have

(50)

where satisfies restriction (30).Let . Applying to (50) and using (34) and

the anti-extensivity of , we obtain

(51)

Observe that and . By iter-ating (51) for , , we obtain

(52)

From (32), we have ; thus, since the SV dila-tion is increasing, , i.e.,

(53)

for . We observe that is -open. There-fore, from (19), we have , which results in

(54)

From (53) and (54), we have

(55)


Equations (52) and (55) prove that

(56)

Observe that . Therefore, we obtain (35).

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Nidhal Bouaynaya received the B.S. degree inelectrical engineering and computer science from theEcole Nationale Supérieure de l’Electronique et deses Applications (ENSEA), France, the M.S. degreein electrical and computer engineering from theIllinois Institute of Technology, Chicago, in 2002,and the Diplôme d’Etudes Approfondies in signaland image processing from ENSEA in 2003. She iscurrently pursuing the M.S. degree in mathematicsand the Ph.D. degree at the Department of Electricaland Computer Engineering, University of Illinois at

Chicago.Her research interests are in signal, image, and video processing; mathemat-

ical morphology; and genomic signal processing.Ms. Bouaynaya won the Best Student Paper Award in Visual Communication

and Image Processing in 2006.

Mohammed Charif-Chefchaouni received the B.S.and M.S. degrees in electrical engineering and com-puter science from the Ecole Nationale Supérieure del’Aéronautique et de l’Espace, Toulouse, France, in1978 and 1980, respectively, and the Ph.D. degree inelectrical and computer engineering from the Univer-sity of Illinois at Chicago in 1993.

From 1980 to 1990, he was a Lecturer of appliedmathematrics and control at the Institut des Posteset Télécommunications, Rabat, Morocco. He is cur-rently the Director of the Institut National des Postes

et Télécommunications, Rabat. His research interests include image modelingand analysis.

Dan Schonfeld (M’90-SM’05) was born in Westch-ester, PA, in 1964. He received the B.S. degree inelectrical engineering and computer science from theUniversity of California, Berkeley, and the M.S. andPh.D. degrees in electrical and computer engineeringfrom the Johns Hopkins University, Baltimore, MD,in 1986, 1988, and 1990, respectively.

In 1990, he joined the University of Illinois atChicago, where he is currently an Associate Pro-fessor in the Department of Electrical and ComputerEngineering. He has authored over 100 technical

papers in various journals and conferences. His current research interests arein signal, image, and video processing; video communications; video retrieval;video networks; image analysis and computer vision; pattern recognition; andgenomic signal processing.

Dr. Schonfeld was coauthor of a paper that won the Best Student Paper Awardin Visual Communication and Image Processing 2006. He was also coauthorof a paper that was a finalist in the Best Student Paper Award in Image andVideo Communication and Processing 2005. He has served as an Associate Ed-itor of the IEEE TRANSACTIONS ON IMAGE PROCESSING (Nonlinear Filtering) aswell as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING

(Multidimensional Signal Processing and Multimedia Signal Processing).

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