Spatially-Variant Morphological Restoration and Skeleton Representation

Transcription

1 IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Spatially-Variat Morphological Restoratio ad Skeleto Represetatio Nidhal Bouayaya, Studet Member, IEEE, Mohammed Charif-Chefchaoui ad Da Schofeld, Seior Member, IEEE. Abstract The theory of spatially-variat (SV) mathematical morphology is used to exted ad aalyze two importat image processig applicatios: morphological image restoratio ad skeleto represetatio of biary images. For morphological image restoratio, we propose the SV alteratig sequetial filters ad SV media filters. We establish the relatio of SV media filters to the basic SV morphological operators (i.e., SV erosios ad SV dilatios). For skeleto represetatio, we preset a geeral framework for the SV morphological skeleto represetatio of biary images. We study the properties of the SV morphological skeleto represetatio ad derive coditios for its ivertibility. We also develop a algorithm for the implemetatio of the SV morphological skeleto represetatio of biary images. The latter algorithm is based o the optimal costructio of the spatially-variat structurig elemet mappig desiged to miimize the cardiality of the SV morphological skeleto represetatio. Experimetal results show the dramatic improvemet i the performace of the SV morphological restoratio ad SV morphological skeleto represetatio algorithms i compariso to their traslatio-ivariat couterparts. I. INTRODUCTION Over the past few decades, morphological operators have gaied icreasig popularity i the implemetatio of sigal ad image processig systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. The iheret parallelism of the class of morphological filters allows the implemetatio of very efficiet ad low-complexity algorithms for sigal ad image processig applicatios [1], [2], [6], [9], [10], [11], [13], [14], [15], [16]. The basic cocepts ad aalytic tools i mathematical morphology ca be foud, for biary images, i set theory ad itegral geometry [6], [9]. I mathematical morphology, a biary image is represeted as a subset of the 2D Euclidea space, R 2 or its digitized equivalet Z 2, ad image processig trasformatios are represeted as set mappigs betwee collectios of subsets. Erosios ad dilatios are the two fudametal morphological operators [1], [2], [17], [18], [19], [20], [21]. They are characterized by a subset called the structurig elemet that is used to probe the image. Mathero has captured the ubiquity of morphological operators by demostratig that ay icreasig (i.e., operators which preserve sigal orderig) ad traslatio-ivariat operators ca be represeted as uios (resp., itersectios) of erosios (resp., dilatios) [6]. I most applicatios of mathematical morphology, the structurig elemet remais costat i shape ad size as the image is probed. Hece, the focus of mathematical morphology has bee mostly devoted to traslatio-ivariat operators. A extesio of the theory to spatially-variat operators has emerged due to the requiremets of some applicatios, such as traffic spatial- measuremets [22] ad rage imagery [23]. For example, i the aalysis of images from traffic cotrol cameras i [11], vehicles at the bottom of the image are closer ad appear larger tha those higher i the image. Hece, the structurig elemet size should vary liearly with the vertical positio i the image. I rage imagery, the value of each pixel is related to the distace to the imagig device. Cosequetly, the apparet height of a object is a fuctio of the object s itesity rage. Hece, oe ca process (e.g., extract or elimiate) differetly-scaled objects of iterest i the image by adaptig the size of the structurig elemet(s) to the local itesity rage [23]. Differet techiques ad algorithms to spatially adapt the structurig elemet i a image have bee proposed i [23], [24], [25], [26], [27] ad [28]. Roerdik [29], [30] itroduced the theoretical backgroud for a mathematical morphology that is ot based o traslatio-ivariat trasformatios i the Euclidea space. The proposed approach, however, is restricted by various rigid algebraic costructios such as polar morphology ad costraied perspective morphology. I [31], a ew class of morphological operatios, which allow oe to select varyig shapes ad orietatios of structurig elemets, is preseted. However, the sweep erosio ad dilatio do ot satisfy the basic properties of mathematical morphology. I particular, they are ot icreasig operators i geeral ad the sweep dilatio operator does ot commute with the uio. A uified theory of spatially-variat mathematical morphology requires a further abstractio of the basic otios of traslatio-ivariat mathematical morphology. A fudametal result i lattice morphology 1 that provides the represetatio of a large class of oliear ad o-ecessarily traslatio-ivariat operators i terms of lattice erosios ad dilatios has bee preseted i [33]. This represetatio, however, does ot posses the geometric iterpretatio, captured by the structurig elemet, that is crucial i sigal ad image processig. A separate approach to the represetatio of spatially-variat mathematical morphology, which preserves the geometrical structurig elemet, i the Euclidea space, has bee itroduced by Serra i [11]. Serra defied the cocept of a structurig fuctio, which associates to each poit i the space a structurig elemet. Charif-Chefchaoui ad Schofeld [34], [35] pursued 1 Lattice morphology, itroduced by Serra [11], is a powerful tool for the abstractio of mathematical morphology based o lattice theory, a topic devoted to the ivestigatio of the algebraic properties of partially-ordered sets [32].

2 IEEE TRANSACTIONS ON IMAGE PROCESSING 2 the ivestigatio of spatially-variat mathematical morphology i a systematic way. They itroduced the basic spatially-variat morphological operators ad ivestigated their properties. A comprehesive developmet of the theory of spatially-variat mathematical morphology i the Euclidea space has bee preseted i [36]. This paper elaborates o the theory ad applicatios of spatially-variat mathematical morphology preseted i [36]. We illustrate the power of the theory of spatially-variat mathematical morphology for two importat image processig applicatios: morphological image restoratio ad skeleto represetatio. Morphological Image restoratio is a importat problem i image processig ad aalysis applicatios. It requires the developmet of a efficiet filterig procedure which restores a image from its oisy versio [4], [5], [11], [37], [8], [38]. I order to devise such a filterig procedure, we have to cosider two fudametal issues: (a) the restoratio filter should be effective i elimiatig the oise degradatio; ad, (b) it should be able to restore various importat aspects of the shape-size cotet of the oise-free image uder cosideratio, as well as preserve its crucial geometrical ad topological structure. The effectiveess of morphological filters i the restoratio of oisy images has bee demostrated repeatedly [4], [8], [38], [9], [11], [22], [37], [39], [40], [41]. For example, Maragos ad Schafer [4] have demostrated a strog relatioship betwee the alteratig filter ad the media filter. Sterberg [37] itroduced alteratig sequetial filters (ASF) ad experimetally showed their oise removal capability. Moreover, Schofeld ad Goutsias [8] have show that the class of alteratig sequetial filters is a set of smoothig morphological filters which best preserve the crucial structure of iput images, i the least-mea-differece sese. Traslatioivariat morphological filters ca be used to remove oise structures that are smaller tha the size of the structurig elemet. However, some importat features of the sigal might be removed as well. May researchers cosidered adaptive morphological filterig i order to deal with this problem [24], [25], [26]. For example, Che et al. [24], developed a algorithm for adaptive sigal smoothig by spatially varyig the filterig scales depedig o the local property of each poit i the sigal. To achieve this goal, they itroduced the progressive umbra-fillig (PUF) procedure. Their experimetal results have show that this approach ca successfully elimiate oise without oversmoothig the importat features of a sigal. It ca be easily established that the morphological operators proposed i [24] are the gray-level extesios of the spatially-variat biary morphological operators [42]. I this paper, we show that the theory of spatially-variat mathematical morphology uifies may classical morphological filters that use spatially-variat structurig elemets, which are suitable for adaptive image processig applicatios. We itroduce spatially-variat alteratig sequetial filters ad spatially-variat media filters for the restoratio of images from their oisy versios. We derive the basic properties of spatially-variat alteratig sequetial filters. We also derive a kerel represetatio of spatially-variat media filters, which establishes the relatio betwee the SV media filters ad the basic SV morphological operators. We also demostrate the power of the theory of spatiallyvariat mathematical morphology by geeralizig the morphological skeleto represetatio to the spatially-variat morphological skeleto represetatio. The morphological skeleto has bee ivestigated by may researchers [9], [43], [44], [45], [46], maily for the purpose of image codig ad shape recogitio. I [44], a geeral theory for the morphological represetatio of discrete biary images was preseted. The basis of this theory relies upo the geeratio of a set of o-overlappig segmets of a image, which produce a decompositio that guaratees exact recostructio of the origial image. Decreasig the cardiality of the morphological skeleto represetatio by reducig its redudacy has bee explored by may authors [43], [44], [46]. I this paper, We exted the morphological skeleto represetatio framework preseted i [44] to the spatially-variat case. We study the properties of the spatially-variat morphological skeleto represetatio ad derive coditios for its ivertibility. We also propose a algorithm for the implemetatio of the SV morphological skeleto. This algorithm miimizes the cardiality of the image represetatio by derivig a optimal uiversal algorithm for the costructio of a spatially-variat structurig elemet mappig. This paper is orgaized as follows: I Sectio II, we provide a brief overview of spatially-variat mathematical morphology. Specifically, we preset the basic spatially-variat morphological operators. We summarize the spatially-variat kerel represetatio, which shows that ay icreasig operator that fixes the etire space ca be represeted as a uio (resp., itersectio) of SV erosios (resp., SV dilatios). This result demostrates the ubiquity of the basic SV morphological operators (i.e., SV erosios ad SV dilatios). I Sectio III, we illustrate the impact of spatially-variat mathematical morphology o image restoratio applicatios. I particular, we address morphological image restoratio by itroducig the SV alteratig sequetial filters ad SV media filters. We also provide a example for selectio of the spatially-variat structurig elemet mappig for image restoratio applicatios. We experimetally show the superior oise removal capabilities of these filters compared to their traslatio-ivariat couterparts. I Sectio IV, we demostrate the power of spatiallyvariat mathematical morphology i skeleto represetatio applicatios. Specifically, we propose the SV morphological skeleto represetatio, develop some of its properties ad derive coditios for its ivertibility. We further propose a uiversal algorithm for the implemetatio of the SV morphological skeleto represetatio. We provide simulatio results which demostrate the eormous improvemet of the spatiallyvariat morphological skeleto represetatio i compariso to its traslatio-ivariat couterpart. Fially, i Sectio V, we preset a brief summary of our results ad discuss our pla for future work. The proofs of all theoretical results that are ew cotributios i this paper are preseted i the appedices.

3 IEEE TRANSACTIONS ON IMAGE PROCESSING 3 II. SPATIALLY-VARIANT MATHEMATICAL MORPHOLOGY Notatio: We cosider a o-empty set ξ = R or Z. The set P(ξ) deotes the set of all subsets of ξ. Elemets of the set ξ will be deoted by lower-case letters; e.g., a,b,c. Elemets of the set P(ξ) will be deoted by upper-case letters; e.g., A,B,C. A order o P(ξ) is imposed by the iclusio. We use ad to deote the uio ad itersectio i P(ξ), respectively. X c deotes the complemet of X P(ξ). The set differece X 1 X 2 of sets X 1 ad X 2 is defied by X 1 X 2 = X 1 X c 2. The traslatio X + {y} also deoted by X y of a set X P(ξ) by y ξ is defied by X + {y} = X y = {z : z = x+y, x X}. The cardiality X of a set X is the total umber of elemets cotaied i the set. We use O = P(ξ) P(ξ) to deote the set of all operators mappig P(ξ) ito itself. Elemets of the set O will be deoted by lower case Greek letters; e.g., α,β,γ. A order o O is imposed by the iclusio ; i.e., α β if ad oly if α(x) β(x), for every X P(ξ). We shall restrict our attetio to o-degeerate operators; i.e., α(x) ξ ad α(x) for some X P(ξ) ad α( ) =, for every α O ( the set P(ξ) is used to deote the empty set). A mappig ψ O is icreasig if X Y = ψ(x) ψ(y ), for all X,Y P(ξ). traslatio-ivariat if ψ(x a ) = (ψ(x)) a, for every X P(ξ) ad every a ξ. extesive (res. ati-extesive) if X ψ(x) (resp. ψ(x) X), for all X P(ξ). idempotet if ψ(ψ(x)) = ψ(x), for all X P(ξ). The mappig ψ O is the dual of the mappig ψ O iff ψ (X) = (ψ(x c )) c (X P(ξ)). A. Spatially-Variat Morphological Operators We shall ow preset a overview of the basic defiitios ad properties of spatially-variat mathematical morphology itroduced by Charif-Chefchaoui ad Schofeld [34] [35]. For a more comprehesive itroductio to spatially-variat mathematical morphology refer to [36]. 2 1) SV erosios ad SV dilatios: The spatially-variat structurig elemet θ is a mappig from ξ ito P(ξ). The trasposed spatially-variat structurig elemet θ is a mappig from ξ ito P(ξ) give by θ (y) = {z ξ : y θ(z)} (y ξ). (1) I traslatio-ivariat mathematical morphology, θ is the traslatio mappig by a fixed set B P(ξ), i.e., θ(z) = B + z, z ξ. Hece, the trasposed structurig elemet correspods to the traslatio by the trasposed set ˇB = { b : b B}. The spatially-variat erosio E θ O is give by E θ (X) = {z ξ : θ(z) X} = θ c (y) (X P(ξ)). y X c 2 Some of the results preseted i this sectio have bee exteded to the gray-level case by Che et al. [24], [25]. (2) The spatially-variat dilatio D θ O is give by D θ (X) = {z ξ : θ(z) X } = θ (y) (X P(ξ)). y X (3) The SV erosios ad dilatios satisfy the basic properties of traslatio-ivariat erosios ad dilatios. Below we list the mai properties that we eed i the sequel. 2) Properties of SV erosios ad SV dilatios: a) Adjuctio: For every mappig θ from ξ to P(ξ) the pair (E θ, D θ ) defies a adjuctio o P(ξ). I other words, D θ (X) Y X E θ (Y ) (X,Y P(ξ)). (4) This result states that E θ ad D θ are a erosio ad a dilatio, respectively, i the sese that these operators are distributive over itersectio ad uio, respectively, i.e., E θ ( X i ) = E θ (X i ), ad D θ ( X i ) = D θ (X i ), i I i I i I i I (5) for a arbitrary collectio of sets {X i P(ξ) : i I}. These idetities ca also be derived easily without referece to the framework of adjuctios. b) Duality: The SV erosio E θ ad the SV dilatio D θ are dual operators, i.e., E θ (X) = D θ (X) (X P(ξ)). (6) This relatio states that dilatig a set X by the mappig θ is equivalet to erodig its complemet X c by the same mappig ad complemetig the result. c) Icreasig: For a give mappig θ, the SV erosio E θ ad the SV dilatio D θ are icreasig operators, i.e., X Y = E θ (X) E θ (Y ), ad D θ (X) D θ (Y ). (7) Icreasig operators preserve order (cotrast) i the sese that they prohibit extractio of iformatio from occluded regios. This property is cosistet with the models of the huma visual system which have bee ivestigated i the field of cogitive psychology. Specifically, the high-level visio models of gestalt psychology state that the perceptual processes uderlyig the visual iterpretatio of a scee are icreasig operators [9], [47], [48]. d) Extesivity ad Ati-extesivity: If z θ(z) for every z ξ, the the SV erosio E θ is ati-extesive ad the SV dilatio D θ is extesive, i.e., E θ (X) X, ad X D θ (X) (X P(ξ)). (8) e) Serial Compositio: Let us use E θ1 (θ 2 ) ad D θ1 (θ 2 ) to deote the mappigs from ξ ito P(ξ) give by (E θ1 (θ 2 ))(z) = E θ1 (θ 2 (z)) ad (D θ1 (θ 2 ))(z) = D θ1 (θ 2 (z)), for every z ξ. Successively SV erodig (resp. SV dilatig) a set first by θ 1 ad the by θ 2 is equivalet to SV erodig (resp. SV dilatig) by the dilated mappig D θ (θ 2 ), i.e., we 1 have ad E θ2 (E θ1 (X)) = E Dθ (θ 2)(X) (X P(ξ)), (9) 1 D θ2 (D θ1 (X)) = D Dθ (θ 2)(X) (X P(ξ)). (10) 1

4 IEEE TRANSACTIONS ON IMAGE PROCESSING 4 3) SV opeigs ad SV closigs: We shall ow preset the spatially-variat opeigs ad closigs ad review their mai properties. The spatially-variat opeig Γ θ O is give by Γ θ (X) = D θ (E θ (X)) (X P(ξ)). (11) The spatially-variat closig Φ θ O is give by Φ θ (X) = E θ (D θ (X)) (X P(ξ)). (12) The SV opeig ad SV closig have ice geometrical iterpretatios. Let us refer to θ(z) by the local structurig elemet at poit z. The SV opeig of a set X is the domai swept out by all local structurig elemets which are icluded i X. By duality, a poit z belogs to the SV closig if ad oly if all the local structurig elemets cotaiig z hit X. Formally, we have the followig equivalet defiitios for the SV opeig ad SV closig: Γ θ (X) = {θ(y) : θ(y) X ; y ξ}, (13) Φ θ (X) = {z ξ : θ(y) X 0, for every θ(y) : z θ(y)}, (14) for every X P(ξ). 4) Properties of SV opeigs ad SV closigs: a) Duality: The SV opeig Γ θ ad the SV closig Φ θ are dual operators, i.e., Γ θ(x) = Φ θ (X) (X P(ξ)). (15) b) Icreasig: For a give θ, the SV opeig Γ θ ad the SV closig Φ θ are icreasig operators o P(ξ), i.e., X Y = Γ θ (X) Γ θ (Y ), ad Φ θ (X) Φ θ (Y ). (16) c) Extesivity ad Ati-extesivity: For every mappig θ, the SV opeig (resp. SV closig) is ati-extesive (resp. extesive). We have Γ θ (X) X, ad Φ θ (X) X (X P(ξ)). (17) d) Idempotece: The SV opeig Γ θ ad SV closig Φ θ are idempotet operators, i.e., for every X P(ξ), Γ θ (Γ θ (X)) = Γ θ (X), ad Φ θ (Φ θ (X)) = Φ θ (X). (18) e) Fixed poits: It was show i [6] ad [49] that the traslatio-ivariat opeig ad closig ca be completely specified from their fixed poits. The framework of fixed poits ca be exteded to the spatially variat case by defiig θ-ope ad θ-closed sets as follows: Defiitio 1: X P(ξ) is θ-ope (resp., θ-closed) if Γ θ (X) = X (resp., Φ θ (X) = X). A useful characterizatio of θ-ope ad θ-closed sets is give by the followig propositio; Propositio 1: [36] X P(ξ) is θ-ope (resp., θ-closed) if ad oly if there exists Y P(ξ) such that X = D θ (Y ) (resp., X = E θ (Y )). The cocept of θ-ope ad θ-closed sets was further exteded to mappigs as follows. Cosider mappigs θ 1 ad θ 2 from ξ ito P(ξ). The mappig θ 1 is θ 2 -ope (resp., θ 2 closed) if θ 1 (z) is θ 2 -ope (resp., θ 2 -closed), for every z ξ. We the have the followig result: If θ 1 is θ 2 -ope, the Γ θ1 (X) Γ θ2 (X), ad Φ θ2 (X) Φ θ1 (X) (X P(ξ)). (19) ad f) Sievig structure: If θ 1 is θ 2 ope, the we have Γ θ1 (Γ θ2 (X)) = Γ θ1 (X); Γ θ2 (Γ θ1 (X)) = Γ θ1 (X), (20) Φ θ1 (Φ θ2 (X)) = Φ θ1 (X); Φ θ2 (Φ θ1 (X)) = Φ θ1 (X), (21) for every X P(ξ). B. Spatially-Variat Kerel Represetatio A importat otio related to set mappigs is that of the kerel, itroduced by Mathero [6] for traslatio-ivariat mappigs. Mathero subsequetly showed that every icreasig ad traslatio-ivariat operator ca be writte as a uio of traslatio-ivariat erosios, or, alteratively, as a itersectio of traslatio-ivariat dilatios. This result was exteded to SV operators i [36]. To state the correspodig theorem, we eed the otio of a SV kerel. Let ψ O. The spatially-variat kerel Ker(ψ) of ψ is give by Ker (ψ) = {θ : z ψ(θ(z)), for every z ξ}. (22) A operator ψ O is a coverig operator (abbreviated as C-operator) if ψ is icreasig ad satisfies ψ(ξ) = ξ. A importat property of C-operators is that their kerel is o-trivial ad is uique. I other words, the mappig which associates to each C-operator its kerel is a oe-to-oe mappig. [36]. The kerel represetatio of C-operators is give by the followig theorem; Theorem 1: [36] A operator ψ O is a C-operator if ad oly if ψ(x) = E θ (X) = D θ (X), (23) θ Ker (ψ) θ Ker (ψ ) for every X P(ξ). Theorem 1 is ot oly iterestig theoretically but also for practical applicatios. For some C-operators, a subset of the kerel, called a basis, is sufficiet for its represetatio i terms of SV erosios or SV dilatios. Therefore, if the kerel of a C-operator or oe of its basis has a fiite umber of mappigs the the C-operator ca be exactly represeted as a fiite uio of SV erosios, or equivaletly, as a fiite itersectio of SV dilatios. This represetatio ca tremedously simplify the aalysis ad implemetatio of the o-liear C-operator. We will show later that the adaptive media filter has a fiite kerel represetatio. Hece, it ca be exactly expressed via a closed formula ivolvig oly itersectios ad uios of sets without requirig ay sortig. I the followig sectios, we apply the geeral theory of spatially-variat mathematical morphology to two image processig applicatios: morphological restoratio of oisy images ad morphological skeleto represetatio. III. SPATIALLY-VARIANT MORPHOLOGICAL RESTORATION We shall cosider SV Alteratig Sequetial Filters (SVASF) ad adaptive media filters for spatially-variat or adaptive image restoratio.

5 IEEE TRANSACTIONS ON IMAGE PROCESSING 5 A. SV Alteratig Sequetial Filters 1) Theoretical Aspects: Alteratig Sequetial Filters (ASF) were itroduced by Sterberg [37] ad were extesively studied by Serra [11]. Basically, a alteratig sequetial filter is a compositio of opeigs ad closigs by structurig elemets of icreasig sizes. I this sectio, we shall exted the class of ASF to the spatially-variat case. I our work A B ad A B deote the traslatio-ivariat erosio ad dilatio, respectively, betwee the sets A ad B P(ξ). We use AF ad ASF to deote the traslatio-ivariat Alteratig Filter ad traslatio-ivariat Alteratig Sequetial Filter, respectively. SVAF ad SVASF will deote the spatially variat Alteratig Filter ad spatially variat Alteratig Sequetial Filter, respectively. Give a biary image X, let us assume that the trasformatio Θ( ) produces a degraded biary image Y give by where N i = Y = Θ(X) = (X N 1 ) N 2, (24) =1,2, C i, + {x i, }, i = 1,2. (25) The model give by Eq. (25) is kow as the germ-grai model [8], [9], [50]. I this case {C i,, = 1,2, } is a sequece of sets, kow as the primary grais, whereas {x i,, = 1,2, } is a sequece of sites, kow as the germs, which are radomly distributed i Z 2 ; e.g., a Berouilli poit process. 3 Observe that the sequece of germs {x i,, = 1,2, } idicates the ceters of the primary grais {C i,, = 1,2, } i the oise process N i, i = 1,2. The SV Alteratig Filter by the structurig elemet mappig θ is defied as the compoud SV ope-close; SVAF θ (X) = Φ θ (Γ θ (X)) (X P(ξ)). (26) A spatially-variat alteratig sequetial filter is a iterative applicatio of spatially-variat alteratig filters; SVASF N (X) = SVAF θn SVAF θn 1 SVAF θ1 (X), (27) where X P(ξ),N N is the order of the filter ad the sequece {θ i } N i=1 is icreasig i.e., θ i(z) θ i+1 (z), for all z ξ, for all 1 i N 1. A operator ψ O is called a morphological filter if it is icreasig ad idempotet. This termiology is differet from the word filter, which is commoly used by the sigal ad image processig commuity to deote a operator. Although the class of icreasig trasformatios is closed uder compositio, the class of idempotet trasformatios is ot. The followig propositio gives a sufficiet coditio o the sequece of structurig elemet mappigs {θ i } for the spatially-variat alteratig sequetial filter to be a morphological filter. Propositio 2: (a) For every structurig elemet mappig θ, the spatially-variat alteratig filter SVAF θ is a morphological filter. (b) If θ i is θ i 1 -ope (i.e., Γ θi 1 (θ i (z)) = θ i (z), z ξ), for 3 Ofte, the sequece of germs {x i,, = 1, 2, } is distributed accordig to a Poisso poit process. I this case, the germ-grai model is kow as the Boolea model [51], [9], [50]. 2 i N, the the spatially-variat alteratig sequetial filter of order N, SVASF N is a morphological filter. I the traslatio-ivariat case, the sequece of structurig elemet mappigs is usually chose to be θ i (z) = (B i ) z = B i + z, where B i = B B B (i times) ad B P(ξ) [8]. The, we have ([(i + 1)B] z ib) ib = [((i + 1)B ib) ib] z = [(i + 1)B] z ; that is θ i+1 is θ i - ope. Hece the coditio of Propositio 2 is satisfied i the traslatio-ivariat case. Sice for a special choice of the structurig elemet mappig θ, the SVASF reduces to the traslatio-ivariat ASF, the class of SVASF s is still the subset of smoothig morphological filters which best preserve the crucial structure of iput images i the least mea differece sese [8]. 2) Simulatios: As i the traslatio-ivariat case, SV mathematical morphology theory is ot costructive, i the sese that it does ot build a systematic algorithm to fid the optimal structurig elemet at each poit of the image. The choice of the structurig elemet mappig obviously depeds o the cosidered applicatio. We propose a geeral selectio rule for image restoratio. We assume that the oise model degradatio is characterized by the germ-grai model give by (24) ad (25). I traslatioivariat morphology, a priori kowledge of the oise model leads to the selectio of a structurig elemet of size larger tha the largest oise-grai [8]. This approach esures that the oise is elimiated from the image. However, the geometric ad topological structure of the restored image is degraded durig the restoratio process. Deoisig should be performed so that the oise is reduced sufficietly ad the geometric ad topological characteristics of the image are preserved. This ca be achieved by spatially-varyig the structurig elemet depedig o the local properties at each pixel i the image. The idea of the proposed algorithm is to use, at each pixel, a structurig elemet of size slightly bigger tha the size of the oise-grai at that poit. The mappigs θ i of the SVASF N are selected as follows: θ i (z) = C(z) S, if z is detected as the ceter of a oise-grai C(z) at iteratio i;, otherwise, where z X, S is the 3 3 square structurig elemet ad i = 1,2,,N. Observe that the above defiitio of the θ i s does ot imply that these mappigs are equal, sice the oise is reduced as the iteratios icrease. For istace, assume that at the first iteratio, a poit z is detected as the ceter of a oise-grai C(z). So, θ 1 (z) = C(z) S. Assume further that the latter oise-grai is small eough so that it has bee removed by the filter Φ θ1 Γ θ1. Hece, at the secod iteratio, o oise-grai is detected at poit z ad thus θ 2 (z) =, i.e., the poit z is ot filtered at the secod iteratio. Moreover, observe that the costructed θ i s do ot satisfy Propositio 2 ad thus the SVASF N, for N 2, is ot idempotet. The detectio of the presece of a oise-grai C(z) cetered at the pixel z is determied by selectig the largest possible grai C i the germ-grai model give by (24) ad (25) which is preset or abset i the degraded image (i.e., C + {z} Y or C + {z} Y c ) [8].

6 IEEE TRANSACTIONS ON IMAGE PROCESSING 6 The performace of the algorithm is measured by the Sigal-to-Noise ratio (SNR). Let X o deote the origial image of size L C ad X r the restored (deoised) image. The Sigal to Noise Ratio is defied by L C i=1 j=1 SNR = X o(i,j) 2 L C i=1 j=1 (X o X r )(i,j). 2 Cosider the biary image X depicted i Fig. 1(a). The degraded biary image Y, obtaied by usig (24) ad (25), with C i,,i = 1,2, formed by the overlappig of square structurig elemets distributed accordig to a Berouilli process, is depicted i Fig. 1(b). Figure 1(c) (resp. 1(d)) shows the output image of the AF usig the rhombus structurig elemet i [43] (resp. 3 3 square structurig elemet) dilated 8 times. A slight improvemet is obtaied by the use of a ASF of order 8 usig the rhombus structurig elemet (resp. 3 3 square structurig elemet) i Fig. 1(e) (resp. 1(f)). Although most of the oise is removed by the ASF, the origial image is highly smoothed ad its origial topology is lost. We immediately see the drastic improvemet of the spatially-variat SVAF ad SVASF over their traslatio-ivariat couterparts i Figs. 1(g) ad 1(h), respectively. The SVASF removes the oise while preservig the edges ad the geometric structure of the image. The SNR of the differet experimets are provided below their correspodig images. The SVASF achieves a SNR 80 db higher tha its ivariat homologue. The simulatio results obtaied illustrate that, give a image degraded by oise characterized by the germ-grai model, eve if oe were to select the optimal parametrizatio [8] ad the optimal structurig elemet [38] for the traslatioivariat AF ad ASF morphological filters, the results would be iferior to those obtaied by usig spatially-variat morphological filters (see Fig. 3 i [8]). B. Adaptive Media Filter 1) Theoretical Aspects: Cosider ξ = Z 2. Let θ be a mappig from ξ ito P(ξ) such that y θ(y) ad θ(y) = (i,j) Z2[θ(y)](i,j) = is odd, for every y ξ. The media med (X,θ) of X with respect to the adaptive widow θ is give by med (X,θ) = {y ξ : X θ(y) + 1 }. (28) 2 It is easy to show that the media filter is a self-dual C- operator, i.e., it is a C-operator such that it is its ow dual. Therefore, it ca be show from the defiitio of the kerel ad from Theorem 1 that med (X,θ) = E λ (X) = D λ (X), (29) λ A λ A where λ : ξ P(ξ) ad λ A if ad oly if λ(y) is ay subset of θ(y) of cardiality ( + 1)/2. Equatio (29) establishes the relatio betwee the adaptive media filter ad the basic SV morphological filters, i.e., SV erosio ad SV dilatio. The implicatio of Eq. (29) are profoud because they eable us to express the adaptive media filter via a closed formula ivolvig oly uios ad itersectios of sets, without requirig ay sortig. For small adaptive widow sizes, it was show that the implemetatio of the media filter via its kerel represetatio is more attractive tha sortig schemes [48]. Although the theory of SV mathematical morphology has bee preseted for the biary case, its extesio to grayscale is straightforward ad follows the steps used to exted traslatio-ivariat biary morphology to traslatioivariat gray-scale morphology [37], [42]. Therefore, we will apply adaptive media filterig to gray-scale images. 2) Simulatios: The key idea of the implemetatio of the adaptive media filter is idetical to the SVASF. Specifically, the size of the local widow at a give poit is selected to be slightly larger tha the size of the germ-grai at this poit. I our simulatios we cosider the origial image depicted i Fig. 2(a). Its corrupted versio by a germ-grai oise model appears i Fig. 2(b). I the first experimet, We applied the media filter iteratively to the degraded image usig a fixed square widow of size 3 3. The output image is show i Fig. 2(c). Notice that sice some of the germ-grais have size larger tha 3, the media filter usig the 3 3 widow fails to remove the larger oise structures. We repeat the same experimet usig a 5 5 square widow. The resultig image is depicted i Fig. 2(d). While the media filter usig the 5 5 widow removes most of the oise i iteratio 8, the output image is more smoothed. I a secod experimet, we applied a Media filter usig a fixed square widow of size (see Fig. 2(e)). Although more oise has bee removed tha the iterative media filters i the first experimet, the restored image is overly smoothed. The same experimet is repeated usig a square widow of size (Fig. 2(f)). The filtered image is cleaer but equally smoothed. I a third experimet, we applied a Alteratig Sequetial Media Filter (ASMF) of order 8, which is composed of 8 media filters of icreasig widow sizes. The widow size is icremeted by 2 at each iteratio. Figure 2(g) shows the result of ASMF usig a iitial square widow of size 3 3. Although the oise-grais are totally removed, the restored image is over smoothed ad its features (e.g., widows, edges) are completely lost. We reach the same coclusio if we apply a ASMF usig a iitial square widow of size 5 5 (Fig. 2(h)). Observe that the largest widow of the ASMF with a iitial widow of size 3 3 (resp. 5 5) has size (resp ). I the last experimet, we applied the spatially-variat media filter (SVMF). Figures 2(i) ad 2(j) are obtaied by SV media filterig usig, at each poit, a local widow size equal to the size of the oise-grai, at the same poit, icremeted by 1 ad 2, respectively. The restored images preserve the edges ad geometric structure of the oise-free image. Notice that usig a local widow size equal to the size of the oise-grai icremeted by 2 performs better i terms of oise removal capability, tha usig a local widow of size equal to the size of the oise-grai icremeted by 1. The reaso is that some of the oise-grais overlap ad merge to form bigger oise-grais. Therefore, a larger local widow size is eeded to esure that these oise-grais are

7 IEEE TRANSACTIONS ON IMAGE PROCESSING 7 Fig. 1. Deoisig of a biary image usig morphological filters ad SNR compariso: (a) Origial biary image; (b) Degraded image by a germ-grai oise model (c) AF usig the rhombus SE dilated 8 times; (d) AF usig a 3 3 square SE dilated 8 times; (e) ASF 8 usig the rhombus SE; (f) ASF 8 usig a 3 3 square SE; (g) SVAF; (h) SVASF 3. suppressed. However, if the local widow size is too large, the media value computed i that widow will ot provide a good estimate of the oise-free pixel because oly pixels i a small eighborhood are strogly correlated. Therefore, there is a trade-off betwee the oise removal capability ad the accuracy of the estimatio. Figures 2(k) ad 2(l) show two iteratios of the SVMF usig a local widow size equal to the size of the oise-grai icremeted by 1 ad 2, respectively. All of the oise has bee removed i the secod iteratio of the SVMF without alterig the topological characteristics of the oise-free image. The SNR of the above experimets are provided below their correspodig images i Fig 2. Notice that the SVAMF achieves a SNR which is 20 db higher tha its traslatio-ivariat couterpart. IV. SPATIALLY-VARIANT MORPHOLOGICAL SKELETON REPRESENTATION Let A B,A B,A B deote the traslatio-ivariat erosio, dilatio ad opeig, respectively, of the set A by the structurig elemet B [9]. Let B = B B B ( times). A. Theoretical Aspects Cosider a sequece of mappigs {λ : 0} from ξ ito P(ξ) such that z λ (z), for every z ξ, for all ad λ (z) {z}, for all. Cosider the sequece of mappigs θ from ξ ito P(ξ) give by θ +1 (z) = t λ θ (z) (t) = D θ (λ (z)) for > 0,z ξ ad θ 0 (z) = {z}, for every z ξ. We defie the iteger N X by N X = max{ : E θ (X) } for a give X P(ξ). Let {ψ : = 0,1,,N X } deote a collectio of operators i O such that E θ+1 (X) ψ (E θ+1 (X)) E θ (X), (30) for every [0,N X ], every X P(ξ) ad ψ ( ) =, for every [0,N X ]. Defiitio 2: Cosider X P(ξ). The spatially-variat (SV) morphological skeleto represetatio R(X) of X is give by R(X) = {R 0 (X),R 1 (X),,R NX (X)}, (31) where R (X) is the spatially-variat morphological skeleto represetatio subset of order give by R (X) = E θ (X) ψ (E θ+1 (X)). (32) Defiitio 3: The SV morphological skeleto represetatio of X P(ξ) is ivertible if there exists a sequece {G : [0,N X ]} of operators i O such that N X X = G (R (X)). (33) =0 The followig theorem establishes a restrictio o the choices of the sequeces {ψ : [0,N X ]} as a direct cosequece of costrait (30) ad establishes the ivertibility of R(X) uder this restrictio. Theorem 2: If the sequece {ψ : [0,N X ]} of operators i O satisfies costrait (30), the E θ+1 (X) ψ (E θ+1 (X)) E θ (Γ θ+1 (X)), (34) for [0,N X ] ad for every X P(ξ). Moreover, R(X) is ivertible ad N X X = R 1 (R(X)) = D θ (R (X)). (35) =0 From Theorem 2, we observe that the SV morphological represetatio of X, R(X), obtaied by the sequece of trasformatios {ψ : [0,N X ]} which satisfy restrictio (30), decomposes ito a sequece of N X + 1 subsets {R (X) : [0,N X ]} which uiquely characterizes X, thereby allowig for a SV morphological represetatio which permits the exact recostructio of X. I the remaider of this sectio, we shall ivestigate some properties of R(X). The followig propositio shows that the resultig morphological image represetatio subsets R N (X), for = 0,1,,N X, are disjoit ad ati-extesive. Propositio 3: We have R 1 (X) R 2 (X) =, (36)

8 IEEE TRANSACTIONS ON IMAGE PROCESSING 8 Fig. 2. Deoisig usig media filters ad SNR compariso: (a) Origial gray scale image; (b) Image degraded by a germ-grai model; (c) 8 iteratios of media filterig by a fixed 3 3 widow; (d) 8 iteratios of media filterig by a fixed 5 5 widow; (e) Media filterig by a fixed widow; (f) Media filterig by a fixed widow; (g) Alteratig Sequetial Media Filter ASMF 8 of size 8 usig iitial widow of size 3 3 icremeted by 2 at each iteratio; (h) Alteratig Sequetial Media Filter ASMF 8 usig iitial widow of size 5 5 icremeted by 2 at each iteratio; (i) SV media filterig with adaptive widow size correspodig to the germ grai size icremeted by 1; (j) SV media filterig with adaptive widow size correspodig to the germ-grai size icremeted by 2; (k) secod iteratio of SV media filterig with adaptive widow size correspodig to the germ-grai size icremeted by 1; (l) secod iteratio of SV media filterig with adaptive widow size correspodig to the germ-grai size icremeted by 2. for 1 2 ad R (X) X, (37) for [0,N X ]. I the followig propositio we show that, uder a certai coditio o the mappigs {λ : N}, a repeated applicatio of the trasformatio R( ) does ot ifluece the image represetatio. Propositio 4: If λ (z) λ 0 (z), for every z ξ ad for [0,N X ], (38) the R(R (X)) = {R (X)}, (39) for [0,N X ]. Whe coditio (38) is ot satisfied, the repeated applicatio of R( ) may result i a further reductio of the total cardiality of the represetatio. This is a desirable result i may applicatios of iterest (e.g., image codig [52], [53]). 1) Example (Geeralized Morphological Skeleto [3]): The followig example is a importat special case of the geeral SV morphological image represetatio R(X). Cosider ψ (X) = D λ (X), = 0,1,,N X, X P(E). (40) The followig propositio shows that the above sequece ψ satisfies costrait (34). Propositio 5: E θ+1 (X) D λ (E θ +1 (X)) E θ (Γ θ+1 (X)), (41) for N ad X P(ξ). Therefore, the represetatio R(X), give by Eqs. (31) ad (32), is ivertible ad the recostructio formula is give by Eq. (35). B. Algorithmic Aalysis I this sectio, we develop a algorithm for the implemetatio of the SV morphological skeleto represetatio studied above. We compare the SV skeleto represetatio with the traslatio-ivariat represetatio. The performace of the morphological skeleto represetatios is assessed by the umber of poits, i the image represetatio, required for exact recostructio of the origial image. From our perspective, the purpose of morphological skeleto represetatios is codig, compressio ad storage. I a commuicatio framework, our goal is to miimize the average code legth of the compressed biary image. It has bee show that efficiet ecodig of the skeleto represetatio usig ru-legth type codes ca be used to provide a efficiet compressio routie for biary images [43]. This approach relies o the sparse represetatio of skeletos to efficietly ecode log ru-legths correspodig to pixels that do ot lie i the skeleto. Therefore, it is ot surprisig that ivestigators have determied that lower cardiality skeleto represetatios yield superior compressio ratios [43]. Our goal is thus to miimize the cardiality of the source image uder the costrait of lossless compressio. For the purpose of this presetatio, we assume the chael to be oiseless. Therefore, the receiver will be able to recostruct the origial image perfectly without error. The optimal SE, i the sese of miimizig the cardiality of the morphological skeleto represetatio, would be the image itself. The traslatio-ivariat ad spatially-variat morphological skeleto represetatios would be idetical ad cosist of 1 poit. However, this is a trivial solutio sice it is impractical ad assumes that the image to be trasmitted by the seder is already kow by the receiver ad is stored i its library. We will assume a fixed library of structurig elemets at the ecoder ad decoder ad costruct the optimal spatially-variat structurig elemet mappig to miimize the

9 IEEE TRANSACTIONS ON IMAGE PROCESSING 9 TABLE I A UNIVERSAL ALGORITHM TO CONSTRUCT THE OPTIMAL STRUCTURING ELEMENT MAPPING FOR THE SPATIALLY-VARIANT MORPHOLOGICAL SKELETON REPRESENTATION. 1. Choose N 0 = max{ : X B }. 2. Let X e = X N 0 B. 3. Choose z 0 X e such that {z 0 } N 0 B is maximal. 4. Let X = X ({z 0 } N 0 B). 5. Store the value of z 0 ad N Let M N0 = 0; k = 0; 7. While (X ) do the followig: a. M = 0; N = N 0 ; k = k + 1; b. While( NB > M & N 0) N = N 1 Let X e = X NB Choose z N X e such that ({z N } NB) X is maximal Let M N = ({z N } NB) X M = max =N0 N M Temporarily store z N c. Store z k ad N k : ({z k } N k B) X = M d. Empty the temporarily stored z N s e. Let X = X {z k } N k B 8. The SV morphological skeleto represetatio is the give by R(X) = k i=0 {z i}. 9. The recostructed image is X = k i=0 {z i} N i B. a sythetic image, the resultig SV morphological skeleto represetatio usig the optimal spatially-variat structurig elemets outlied i the algorithm is idetical to the SV erosio usig the same structurig elemet mappig. Figures 3(d) - 3(g) illustrate the process of selectig the structurig elemets for the house image usig a 3 3 square structurig elemet. The SV ad traslatio-ivariat morphological skeleto represetatios are show i Figs. 3(b) ad 3(c), respectively. The SV morphological skeleto represetatio has a compressio ratio which is more tha 3 times higher tha its traslatio-ivariat couterpart. We shall oce agai determie the umber of parameters eeded for represetatio of the skeleto ad igore the effects of the use of ecodig schemes usig ru-legths type techiques i computig the storage capacity requiremets. If the morphological skeleto represetatio is used for storage or commuicatio, the we eed = 648 parameters to trasmit the SV morphological skeleto represetatio. O the other had, the total umber of parameters required for trasmissio of the traslatio-ivariat morphological skeleto represetatio is equal to = Therefore, the SV morphological skeleto represetatio has a storage capacity gai that is 2.32 times higher tha the traslatio-ivariat morphological skeleto represetatio. cardiality of the morphological skeleto represetatio. The idea of the proposed algorithm is similar to the matchig pursuit algorithm [54]. The matchig pursuit algorithm adaptively decomposes a sigal ito waveforms that are the dilatios, traslatios ad modulatios of a sigle fuctio; thus providig a iterpretatio of the sigal structure. I the proposed algorithm, let X deote the origial image ad B a fixed structurig elemet. Table I describes a uiversal algorithm to costruct the optimal structurig elemet mappig for the spatially-variat morphological skeleto represetatio. The algorithm is a iterative process. At each iteratio, the algorithm selects the ceter of the dilated structurig elemet B that maximally itersects the image, for some iteger. The uio of these ceter poits costitutes the SV morphological skeleto represetatio. The exact recostructio of the origial image is guarateed give the set of ceter poits ad their correspodig iteger. It is easy to show that the resultig spatially-variat morphological skeleto represetatio usig the proposed algorithm is compact, i the sese that the set {(z i,n i ),i = 0,,k} is ot redudat; i.e., the recostructio based o ay partial subset of the resultig SV morphological skeleto represetatio would form a strict subset of the origial image. It is also iterestig to observe that the sets {{z i } N i B,i = 0,,k} are ot ecessarily disjoit. Thus, the optimal morphological skeleto represetatio is ot derived by a decompositio of the origial image ito o-overlappig shapes. Istead, overlappig shapes are exploited i order to reduce the umber of shapes required to cover the image. This approach allows for a substatial reductio i the cardiality of the morphological skeleto represetatio. Moreover, as we have see earlier i the example of the SV skeleto represetatio of V. CONCLUSIONS I this paper, we preseted a geeral theory of spatiallyvariat mathematical morphology (SVMM) ad showed its eormous potetial through two importat image processig applicatios. First, we itroduced SV alteratig sequetial filters ad SV media filters for SV morphological deoisig of degraded images. Simulatio results demostrated that, ot oly is the oise removal capability of the SV morphological filters dramatically higher tha their traslatio-ivariat couterparts, but also the topology ad geometrical structure of the origial image are preserved. Secod, we exteded the traslatio-ivariat morphological skeleto represetatio to the spatially-variat case. We have also developed a uiversal algorithm for optimal selectio of the spatially-variat structurig elemet mappig for skeletoizatio by miimizig the cardiality of the SV morphological skeleto represetatio. This approach has bee show to yield a substatial reductio i the cardiality of the spatially-variat morphological skeleto represetatio i compariso to its traslatio-ivariat couterpart. As a result of this ivestigatio, we have complemeted the elegat theory of spatially-variat mathematical morphology with powerful practical algorithms for image processig applicatios. The SV morphological framework preseted ca be applied to may classical image processig problems related to o-liear filterig. I the future, we pla to further explore the power of the SVMM framework by developig more sophisticated ad faster algorithms for o liear filterig i various image processig applicatios. We also pla to ivestigate the optimal spatially-variat structurig elemet mappig for morphological image restoratio applicatios by extedig the work of Schofeld [38] for traslatioivariat morphological filters. We further pla to explore

10 IEEE TRANSACTIONS ON IMAGE PROCESSING 10 Fig. 3. Morphological skeleto represetatio: (a) Origial image; (b) SV morphological skeleto represetatio (162 poits); (c) Traslatio-ivariat morphological skeleto represetatio (502 poits). the robustess of the spatially-variat morphological skeleto represetatio to oise degradatio by extedig the work of Schofeld ad Goutsias [45] for the traslatio-ivariat morphological skeleto represetatio. A formal aalysis of the optimal structurig elemet mappig ad ivestigatio of the robustess of the morphological structurig elemet require the use of radom set theory [6], [9], [51], [50]. APPENDIX A: PROOF OF PROPOSITIONS Proof: [Proof of Propositio 2] From subsectio II-A.4, we kow that Γ θ ad Φ θ are icreasig operators. Sice the class of icreasig operators is closed uder compositio, we coclude that the SVAF θ ad the SVASF N are icreasig. (a) Let us prove the idempotece of the SVAF. By the atiextesivity property of Γ θ, we have Γ θ [Φ θ (Γ θ (X))] Φ θ (Γ θ (X)), for all X P(ξ). Applyig Φ θ to the above iequality ad, sice Φ θ is icreasig, we obtai Φ θ {Γ θ [Φ θ (Γ θ (X))]} Φ θ [Φ θ (Γ θ (X))] = Φ θ (Γ θ (X)), where the last equality follows from the idempotece of Φ θ. Therefore, we have SVAF θ (SVAF θ ) SVAF θ. (42) By the extesivity property of Φ θ, we have, Φ θ (Γ θ (X)) Γ θ (X), X P(ξ). Applyig Γ θ to the above iequality ad sice Γ θ is icreasig, we obtai Γ θ [Φ θ (Γ θ (X))] Γ θ (Γ θ (X)) = Γ θ (X), (43) where the last equality follows from the idempotece of Γ θ. Applyig Φ θ to Eq. (43) ad sice Φ θ is icreasig, we obtai Φ θ [Γ θ (Φ θ (Γ θ (X)))] Φ θ (Γ θ (X)), Therefore, we have X P(ξ). SVAF θ (SVAF θ ) SVAF θ. (44) Usig Eqs. (42) ad (44), we establish the idempotece of the SVAF θ. (b) The idempotece of the SVASF N follows from the more geeral result established by Schofeld ad Goutsias i [8, Propositio 1]. All the ecessary coditios to apply Propositio [8, Propositio 1] are satisfied (see Sectio II- A.4). Proof: [Proof of Propositio 3] From Eq. (32) ad costrait (30), we have R (X) E θ (X) E θ+1 (X). (45) Assume that 1, 2 [0,N X ] are such that 1 < 2. From the defiitio of θ, we have E θ2 +1 (X) E θ 2 (X) E θ1 (X) E θ1 +1 (X). Therefore, (E θ2 (X) E θ2 +1 (X)) (E θ 1 (X) E θ1 +1 (X)) =. From Eq. (45), we obtai R 2 R 1 =. Eq. (37) is a direct cosequece of Eq. (32) ad the fact that E θ (X) X. This completes the proof. Proof: [Proof of Propositio 4] Let X P(ξ) such that E λ0 (X) =. From the defiitio of θ ad λ, we have λ 0 (z) θ 1 (z), for every z ξ. Therefore, E θ1 (X) E λ0 (X) =. From Eq. (32) ad the fact that ψ 0 ( ) =, we have R 0 (X) = E θ0 (X) ψ 0 (E θ1 (X)) = E θ0 (X) = X. From the costrait (30) ad sice E θ1 (X) =, we have E θ E θ1 =. Therefore, from Eq. (32), we have R (X) = for 1. Fially, we see that R(X) = {X}. (46) Now, we will show that R (X) satisfies E λ0 (R (X)) =. Let us take y R (X). From Eq. (32), we see that y E θ (X) ad y / ψ (E θ+1 (X)). From costrait (30), we see that y / E θ+1 (X) θ +1 (y) X. From the defiitio of θ +1, there exists z λ (y) such that θ (z) X z / E θ (X) = z / R (X). Sice z λ (y) ad by hypothesis λ (y) λ 0 (y) for [0,N X ], we have z λ 0 (y). Thus, λ 0 (y) R (X) for every y R (X). Cosider ow, y / R (X) ad cosider the set λ 0 (y). By defiitio of the mappigs λ, we have that y λ 0 (y). Sice y / R (X), we coclude that λ 0 (y) R (X) for every y / R (X). Thus, λ 0 (y) R (X) for every y ξ or E λ0 (R (X)) =. Fially, by usig Eq. (46) with X R (X), we prove Eq. (39). Proof: [Proof of Propositio 5] By the extesivity property of the SV dilatio, we have that E θ+1 (X) D λ (E θ +1 (X)) = ψ (E θ+1 (X)). By the extesivity property

11 IEEE TRANSACTIONS ON IMAGE PROCESSING 11 of SV closig, we have ψ (E θ+1 (X)) = D λ θ +1 (X)) Φ θ λ (E θ +1 (X))) = E θ (D θ λ (E θ +1 (X)))). (47) Usig Eq. (10), we have that D θ (D λ (X)) = D D λ (θ )(X). From the costructio of θ +1, we have that θ +1 (z) = t λ(z)θ (t) = D θ (λ (z)), for > 0 ad z ξ. Therefore θ +1(z) = {y ξ : z θ +1 (y)} = {y ξ : z D θ (λ (y))} = {y ξ : θ (z) λ (y) } = D λ (θ (z)). Therefore, D θ (D λ (X)) = D D λ (θ )(X) = D θ (X),X +1 P(ξ). Fially, Eq. (47) becomes ψ (E θ+1 (X)) E θ (D θ (E θ+1 (X))) = E θ (Γ θ+1 (X)). +1 Therefore, the sequece {ψ = D λ : [0,N X]} satisfies costrait (34) for every X P(ξ). Hece, by Theorem 2, R(X) is ivertible ad X = N X =0 D θ (R (X)) = NX =0 D θ (E θ (X) D λ (E θ +1 (X))). APPENDIX B: PROOF OF THEOREMS Proof: [Proof of Theorem 2] Sice costrait (30) must be satisfied for every X P(ξ), we ca choose X Γ θ+1 (X). Therefore, we obtai E θ+1 (Γ θ+1 (X)) ψ (E θ+1 (Γ θ+1 (X))) E θ (Γ θ+1 (X)), (48) for [0,N X ]. From Eq. (12), we observe that E θ+1 (Γ θ+1 (X)) = Φ θ (E θ+1 (X)). From the fact that +1 E θ+1 (X) is θ +1-closed, we have, usig Propositio 1, E θ+1 (Γ θ+1 (X)) = E θ+1 (X). (49) Substitutig the above equatio ito Eq. (48), we obtai Eq. (34). From Eq. (32), we have E θ (X) = R (X) ψ (E θ+1 (X)), (50) where ψ satisfies restrictio (30). Let Υ = Γ θ (X). Applyig D θ to Eq. (50) ad usig Eq. (34) ad the ati-extesivity of Γ θ, we obtai Υ = D θ (X)) D θ (E θ+1 (X))) D θ (X)) Γ θ (Γ θ+1 (X)) D θ (X)) Γ θ+1 (X) = D θ (X)) Υ +1. (51) Observe that Υ NX+1 = ad Υ NX = D θ (X)). By iteratig Eq. (51) for = k,k + 1,,N X, we obtai N X Υ k =k D θ (R (X))). (52) From Eq. (32) we have that R (X) E θ (X); thus, sice the SV dilatio is icreasig, D θ R (X) Γ θ (X), i.e., D θ (R (X)) Υ, (53) for = 0,1,,N X. We observe that θ +1 is θ -ope. Therefore, from Eq. (19), we have Γ θ+1 (X) Γ θ (X), which results i Υ 2 Υ 1, (54) for 1 2. From Eqs. (53) ad (54), we have N X =k Equatios (52) ad (55) prove that D θ (R (X)) Υ k. (55) N X Υ k = =k D θ (R (X)). (56) Observe that Υ 0 = Γ θ0 (X) = X. Therefore, we obtai Eq. (35). REFERENCES [1] E. R. Dougherty ad C. R. Giardia, Morphological Methods i Image Processig. Eglewood Cliffs: Pretice-Hall, Ic., [2] R. M. Haralick, S. R. Sterberg, ad X. Zhuag, Image aalysis usig mathematical morphology, IEEE Trasactios o Patter Aalysis ad Machie Itelligece, vol. 9, pp , [3] P. A. Maragos, A uified theory of traslatio-ivariat systems with applicatios to morphological aalysis ad codig of images, Ph.D. dissertatio, Georgia Istitute of Techology, July [4] P. A. Maragos ad R. W. Schafer, Morphological filters - part i: Their set-theoric aalysis ad relatios to liear shift-ivariat filters, IEEE Trasactios o Acoustics, Speech ad Sigal Processig, vol. 35, pp , [5], Morphological filters - part ii: Their relatio to media, orderstatistics ad stack filters, IEEE Trasactios o Acoustics, Speech ad Sigal Processig, vol. 35, pp , [6] G. Mathero, Radom Sets ad Itegral Geometry. New York: J.Wiley & Sos, [7] D. Schofeld, Optimal morphological represetatio ad restoratio of biary images: Theory ad applicatios, Ph.D. dissertatio, Joh Hopkis Uiversity, Baltimore, Maryl., [8] D. Schofeld ad J. Goutsias, Optimal morphological patter restoratio from oisy biary images, IEEE Trasactios o Patter Aalysis ad Machie Itelligece, vol. 13, pp , [9] J. Serra, Image Aalysis ad Mathematical Morphology. Sa Diego, Calif.: Academic Press, [10], Itroductio to mathematical morphology, Computer Visio, Graphics, ad Image Processig, vol. 35, pp , [11], Image Aalysis ad Mathematical Morphology. New York: Theoretical Advaces. Academic Press, 1988, vol. 2. [12] P. Soille ad M. Pesaresi, Advaces i mathematical morphology applied to geosciece ad remote sesig, IEEE Trasactios o Geosciece ad Remote Sesig, vol. 40, pp , Sept [13] A. D. Kishore ad S. Sriivasa, A distributed memory architecture for morphological image processig, i Iteratioal Coferece o Iformatio Techology: Codig ad Computig [Computers ad Commuicatios], April 2003, pp [14] M. H. Sedaaghi, Direct implemetatio of ope-closig i morphological filterig, Electroics Letters, vol. 33, o. 30, pp , Jauary [15] A. C. P. Loui, A. N. Veetsaopoulos, ad K. C. Smith, Flexible architectures for morphological image processig ad aalysis, IEEE Trasactios o Circuits ad Systems for Video Techology, vol. 2, o. 1, pp , March [16] D. Wag ad D. C. He, A fast implemetatio of 1-d grayscale morphological filters, IEEE Trasactios o Sigal Processig, vol. 42, o. 12, pp , December [17] H. J. A. M. Heijmas, Gray level morphology, Ceter for Mathematics ad Computer Sciece, Tech. Rep. AM-R9003, Jauary [18], From biary to gray level morphology, Ceter for Mathematics ad Computer Sciece, Tech. Rep. AM-R9009, May 1990.