A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object...

1798 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 10, OCTOBER 2000

A Generalized Fuzzy Mathematical Morphologyand its Application in Robust 2-D and 3-D Object

RepresentationVassilios Chatzis, Associate Member, IEEE,and Ioannis Pitas, Senior Member, IEEE

Abstract—In this paper, the generalized fuzzy mathematicalmorphology (GFMM) is proposed, based on a novel definitionof the fuzzy inclusion indicator (FII). FII is a fuzzy set used asa measure of the inclusion of a fuzzy set into another, that isproposed to be a fuzzy set. It is proven that the FII obeys a set ofaxioms, which are proposed to be extensions of the known axiomsthat any inclusion indicator should obey, and which correspondto the desirable properties of any mathematical morphologyoperation. The GFMM provides a very powerful and flexibletool for morphological operations. The binary and grayscalemathematical morphologies can be considered as special cases ofthe proposed GFMM. An application for robust skeletonizationand shape decomposition of two-dimensional (2-D) and three-di-mensional (3-D) objects is presented. Simulation examples showthat the object reconstruction from their skeletal subsets that canbe achieved by using the GFMM is better than by using the binarymathematical morphology in most cases. Furthermore, the use ofthe GFMM for skeletonization and shape decomposition preservesthe shape and the location of the skeletal subsets and spines.

Index Terms—Fuzzy morphology, fuzzy sets, morphologicalskeletons, object representation, shape decomposition.

I. INTRODUCTION

M ATHEMATICAL morphology is a very rich and pow-erful tool used for the representation and analysis of bi-

nary and grayscale images [1]–[6]. The morphological imagerepresentation has been used for the description of the geomet-rical characteristics of image objects as well as for binary imagecompression. The morphological skeleton and the morpholog-ical shape decomposition are the very popular methods for mor-phological shape representation [7], [8]. Although several com-parative studies and properties analysis can be found in the liter-ature [7]–[10] the main disadvantage of both methods is the lackof robustness especially in impulsive noise, something that canbe considered as a general characteristic of all the morpholog-ical operations [11]. Several efforts have been done to reducethe sensitivity of morphological operations in impulses. Softmorphological operations have been proposed in [12], whichcan restrict the problem in some cases. The properties of softmathematical morphology have been investigated in [13], [14].Fuzzy morphological operations have also been proposed and

Manuscript received April 7, 1999; revised March 24, 2000. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Scott T. Acton.

The authors are with the Department of Informatics, Aristotle Universityof Thessaloniki, Thessaloniki, Greece (e-mail: [email protected];[email protected]).

Publisher Item Identifier S 1057-7149(00)06913-X.

investigated rather recently [15]–[20], but still the fuzzy and softmorphologies are not efficient in binary object representation,since skeletonization and shape decomposition have strong de-pendency on the presence of outliers as in the classical binarymathematical morphology.

In this paper, a novel generalized fuzzy mathematical mor-phology (GFMM) is proposed. The GFMM is based on a fuzzyinclusion indicator (FII), a fuzzy set used as a measure of theinclusion of a fuzzy set into another. It is proven that the FIIobeys a set of axioms, which are extensions of the known ax-ioms that any inclusion indicator should obey, and correspondto desirable characteristics that any mathematical morphologyoperation should have. The binary and grayscale mathematicalmorphologies can be considered as special cases of the proposedGFMM. The GFMM provides a very powerful and flexible tool.In this paper an application for robust skeletonization and shapedecomposition of two-dimensional (2-D) and three-dimensional(3-D) objects is presented.

The paper has the following structure. In Section II, thefuzzy sets relations and operations are presented in brief. InSection III, the FII is defined grounded on an axiomatic basis.The FII is also compared with other inclusion indices. InSection IV, the GFMM is defined. The basic morphologicaloperations are presented and the compatibility with binary andgrayscale mathematical morphology is shown. The flexibilityand robustness of the GFMM is presented by using an appli-cation example. Simulation results on the use of the GFMMfor robust skeletonization and shape decomposition of imagesand volumes are presented in Section V. Conclusions aredrawn in Section VI. The Appendix includes the proofs of thecompliance of the FII on the axioms proposed in Section III.

II. RELATIONS BETWEEN FUZZY SETS

The fuzzy set [21], is usually described by its membershipfunction that maps the space to the interval

(1)

The value is called the degree of membership of the pointto the set . A more effective representation of a fuzzy setis

the stack of its -cuts , which are defined as the crisp sets ofthe points . The fuzzy set can be reconstructedfrom its -cuts using a procedure symbolized as

(2)

1057–7149/00$10.00 © 2000 IEEE

CHATZIS AND PITAS: GENERALIZED FUZZY MATHEMATICAL MORPHOLOGY 1799

The classical crisp set relations (subset) and operations(union and intersection ) have been extended to fuzzy sets.Although several definitions can been found in the literature, themost frequently used are the classical definitions that follow. Let

be two fuzzy sets with membership values . Theirunion and intersection are the fuzzy sets and withmembership functions and given by

(3)

(4)

A fuzzy set is considered to be subset of the fuzzy setifand only if the membership degree of is less than orequal to the membership degree of for all

(5)

or equivalently by using the-cuts

(6)

The complement of a fuzzy set is usually defined as thefuzzy set with membership function

(7)

The translation of a fuzzy set by a crisp vector is the fuzzyset with membership function

(8)

Similarly, the centrally symmetric fuzzy set has member-ship function . Finally, a fuzzy set is calledconvex if , .

III. FUZZY INCLUSION INDICATOR

A. Axiomatic Basis of the Fuzzy Inclusion Indicator

An inclusion indicatoror inclusion grade operator mea-sures the belief in the proposition “is a subset of ,” where

are fuzzy sets. It has been defined in the literature as atwo argument function that maps any pair of fuzzy sets

into the interval [0, 1], and satisfies a number of axioms(properties) [15]–[17].

A1) if and only if;

A2) ;A3) ;A4) ,

;A5) ;A6) ;A7) ;A8) .Another axiom has been proposed [16] as follows:A9) if and only if

which has been found too restrictive, since it is not satisfied bymost of the inclusion indicators proposed in the literature [15].

The aforementioned axioms are not independent, since, for ex-ample, A3) can be derived by applying A5) to A2) and A7) canbe derived by applying A5) to A6). Equivalences between thesenine axioms and the desired properties of a fuzzy mathemat-ical morphology are extensively investigated in [17]. It is justmentioned here that A4) is strongly related to the translation in-variance property, A5) to the principle of duality, A1) and A9)lead to the compatibility of the constructed fuzzy mathematicalmorphology with the binary and grayscale one, A2) and A3) arerelated to the increasing (decreasing) property of erosion withrespect to the reference (structuring element) set, and A6), A7),and A8) are related to the compatibility with union and inter-section in classical morphology.

In the following, a GFMM will be constructed based on thenovel definition of an inclusion indicator as a fuzzy set ratherthan a number, defined in the interval . Large membershipvalues strengthen the belief that “is a subset of .” The use ofa fuzzy inclusion indicator in the construction of a fuzzy math-ematical morphology guarantees that fuzzy mathematical oper-ations on fuzzy signals result to fuzzy signals.

Definition 1: A fuzzy inclusion indicatoris any two argu-ment function , that maps a pair of fuzzy sets into an-other fuzzy set with domain of definition the interval , andsatisfies the following axioms.

B1) if and only if;

B2) ;

B3) If is convex and ;

B4) ;

B5) In the general case: . Ifis convex then ;

B6) ;

B7) ;

where is the zero fuzzy set with membership functiondefined on its interval of confidence. These seven

axioms, B1)–B7), are extensions of the previously reportedaxioms A1)–A4) and A6)–A8) that are fundamental for theconstruction of any fuzzy mathematical morphology. Thus, theGFMM that will be constructed based on these seven axiomswill have the desired properties of any mathematical mor-phology operation: translation invariance [B4)], compatibilitywith the binary and grayscale mathematical morphology [B1)],increasing (or decreasing) property of erosion with respect tothe reference (structuring element) set [B2), B3)], compatibilitywith union and intersection in classical morphology [B5), B6),B7)]. The principle of duality [A5)] will be taken into accountin the construction of the GFMM, by defining the dilation asthe dual operation of the erosion.

B. Fuzzy Inclusion Indicator Definition

Theextension principle (EP)is often used in fuzzy theory toextend mathematical laws of crisp numbers to fuzzy sets [22],[23]. It provides the theoretical warranty that the fuzzificationof the parameters or arguments of a function results in com-putable fuzzy sets. Let be fuzzy sets and

be a crisp function. Then, the extension


principle transfers the fuzziness of throughthe function to a fuzzy set where

if

if(9)

An EP-like operation that has been proposed in [24], [25] todefine the compatibility index will help us to define the fuzzyinclusion index of a fuzzy set with respect to another fuzzyset . The compatibility index has been defined as:

(10)

However, since the compatibility index applies a( ) operator, it cannot be used for the definition of an erosionoperation. We can reach the same conclusion, by observing thatthe compatibility index does not satisfy the axioms B5) and B6).

In this paper, a novel fuzzy inclusion indicator is proposed,by applying a ( ) operator on the EP-like operation(10) as follows. Let be two fuzzy sets. A fuzzy inclusionindicator is defined as a fuzzy set with membershipfunction

(11)

If there is no such that , then the inclusion indexis not defined for the certain . However, the fuzzy set cor-responds to the structuring element of the mathematical mor-phology and can be selected by the user. Thus, it can be selectedto define a fuzzy inclusion indicator for all . The pro-posed fuzzy inclusion indicator (11) satisfies the axioms B1–B7.The proofs for 1–D fuzzy sets are presented in the Appendix. Al-though most of the proofs can be easily extended to-dimen-sional fuzzy sets, the extensions are not straigthforward whenthe convexity property is required and the-dimensional fuzzysets should have specific properties in order to satisfy some ofthe axioms. Details are given in the Appendix.

The main difference between the proposed fuzzy inclusionindicator and other known indicators is that the proposedindicator is generally a fuzzy set defined in instead of acrisp number in . Although this characteristic providesfuzziness propagation through the inclusion operation, a crispvalue could be desirable in some cases. Then, a defuzzificationprocess should be followed, symbolized as . The resultingcrisp value of inclusion should vary from 0 to 1:

(12)

A variety of defuzzification processes have been proposed in theliterature [23], [25], [26]. The most usual ones are the center ofgravity method, the center of maxima method and the mean ofmaxima method as well as generalizations or weighted versionsof them.

Some examples of fuzzy inclusion indicators between fuzzysets, crisp sets and crisp numbers are given in Fig. 1. We canobserve in Fig. 1(a) that the domain of definition of the inclu-sion operation is assigned by the included set(structuring

Fig. 1. Examples of fuzzy inclusion indicators of (a) two fuzzy sets and of (b)a crisp and a fuzzy set.

element). Subsets of, defined from its membership functionthrough the equation , provide the subdomainswhere the operator is applied on the values of the member-ship function of the reference set (signal or image).The correspondence of the valuesto the valuesconstructs the fuzzy set that is the fuzzy inclusion indicator

. It is also shown in Fig. 1(b) that the fuzzy inclusionof a crisp set into a fuzzy set is the minimum membership valueof in the crisp set, corresponding to . It is also easy toshow that the fuzzy inclusion of a crisp set or a crisp value intoa crisp set are extensions of classical set theory relations.

C. Other Inclusion Indices

In [17] a fuzzy mathematical morphology is defined based onthe indicator

(13)

The indicator (13) is similar to the proposed fuzzy inclusion in-dicator (11), since it applies an operator on values of themembership function of the fuzzy set that belong in subsetsof the support of . The integral can be considered as a specialcase of the defuzzification process of the fuzzy indicator. Themain difference of the two indicators is that the subsets in (13)are the -cuts of the fuzzy set , while the subsets in (11) aredefined as the subsets of the support ofwhere ,a process that is similar to the extension principle one. In gen-eral, the indicator (13) satisfies modified axioms A6) and A7)that contain inequalities instead of equalities. Thus, the associ-ated erosion definition is not an erosion according to the generaldefinition of [27].

Several other inclusion indicators have been proposed in theliterature in the last decade and have been successively used forthe definition of fuzzy mathematical morphologies. A familyof functions, which satisfy special characteristics, has beenproposed in [16] to define a family of inclusion indicators as

(14)


This indicator satisfies the eight axioms A1)–A8) and is studiedin detail in [16], [17]. Another inclusion indicator that leads tofuzzy morphological operations proposed in [28] is

(15)

which has been used especially in the study of fuzzy convexity.Other inclusion indicator indices, properties and comparativestudies can been found in [15]–[17], [19], [29].

IV. GENERALIZED FUZZY MATHEMATICAL MORPHOLOGY

A. Definition of the Basic Morphological Operations

Based on the previously defined fuzzy inclusion indicator(11), the GFMM framework will be constructed. In the fol-lowing, the four basic operations of any morphology, erosion,dilation, opening and closing will be defined. Let be twofuzzy sets. The erosion of the fuzzy setby the fuzzy struc-turing element , is defined as the fuzzy set havingmembership function given by

(16)

where denotes the fuzzy inclusion indicator andis a de-fuzzification procedure. The dilation is then definedto obey to the duality principle

(17)

where denotes the centrally symmetric of the fuzzy struc-turing element and the complement of the fuzzy setasin (7). Opening and closing are defined in terms of erosion anddilation

(18)

(19)

It can be observed in (16) that the membership value of theeroded set is the value of the fuzzy inclusion indicator mem-bership function at the point resulting from the defuzzificationprocedure of the fuzzy inclusion indicator. Since the fuzzy in-clusion indicator membership function takes its values from themembership function of the reference set, the eroded set takesalso values coming from the reference set. The neighbor of thereference set that takes part in the erosion procedure is definedby the fuzzy structuring element. The above mentioned charac-teristics are similar with the classical morphological ones. Thefuzziness of the structuring element forms sets with equal mem-bership, which are responsible for the topological characteristicsand special features of the corresponding morphological opera-tions.

B. Compatibility with Classical Mathematical Morphologies

In the following, it will be shown that the classical binarymathematical morphology (BMM) can be considered as a spe-cial case of the proposed GFMM. Let be two classicalsets defined in a subset of and any correspond toa value in through a function . The above setsare considered as a reference and a structuring element set, re-spectively, of the classical BMM. In the GFMM framework thetwo valued reference set is equivalent to a fuzzy set with a

bi-valued membership function . Let alsobe a fuzzy set defined in the same domain as the classical set

, having a membership value , in the domainof definition. For this special case of , the fuzzy inclusionindicator membership function (11) is simplified as follows:

ifotherwise

(20)

i.e., it is defined only in the set . Then, any defuzzificationprocess in (16) leads to and the corresponding fuzzyerosion operation is simplified to

ifotherwise

(21)

which is equivalent to the definition of the classical binary ero-sion.

The grayscale mathematical morphology (GMM) assumesthat the reference set takes values through a function ,in a certain convex set. By normalizing the values of the convexset in , any membership function of a certain fuzzy setcan substitute the function. The fuzzy structuring element isdefined as in the previous case. For this special case of ,the fuzzy inclusion indicator membership function (11) is sim-plified as follows:

(22)

i.e., it is defined only in the set . Then, any defuzzificationprocess in (16) leads to and the corresponding fuzzyerosion operation is simplified to

(23)

which is equivalent to the definition of the classical grayscaleerosion. It can be concluded from (21) and (23) that BMMand GMM can be considered as special cases of the proposedGFMM.

C. How to Use the GFMM—Robustness

It was presented in the previous section that, by using a crispset as a fuzzy structuring element, the GFMM can include theBMM and the GMM as special cases and inherit the proper-ties of the corresponding morphological operations. In the fol-lowing, it will be shown that the GFMM characteristics can behandled by the membership function of the fuzzy structuringelement. The role of the structuring elements membership func-tion is important during the calculation of the fuzzy inclusionindicator. The membership function of the fuzzy structuring el-ement defines classical sets of points with equal degree ofmembership in . The meaning of the membership equality isthat it defines a locus of topological similarity. Then, theop-eration of the inclusion indicator takes place among the pointsthat belong to each classical set of equal membership separately,and the infimum values are used to construct the fuzzy inclusionindicator set. Similarly with the definition of a fuzzy structuringelement in other fuzzy mathematical morphologies, the fuzzystructuring element can be separated in two parts, the core wherethe membership function equals unit, and the fuzziness, which isthe rest of the element. The core is responsible for the properties


Fig. 2. Example of a fuzzy erosion operation: (a)10 � 10 binary image; (b)5 � 5 fuzzy structuring element with pyramidal fuzziness; (c) and (d) fuzzyinclusion indicators of the mentioned translated structuring element for varioustranslations; and (e) the eroded image using the maximum value to defuzzifythe fuzzy inclusion indicators.

which come from the classical morphological operation. Thefuzziness of the membership function of the fuzzy structuringelement can cause different behavior of the GFMM operations,and its suitable selection, can lead to desirable characteristics.An example of a selection that leads to a robust mathematicalmorphology will be presented in the following.

Let be a binary image with a rectangular object,corrupted by an impulse of zero value in pixel , shown inFig. 2(a). Let be a fuzzy structuring element with pyramidalmembership function defined on a classical set, shownin Fig. 2(b). In such cases, when the core set ( ) of thestructuring element has support area less than a pixel, the cor-responding classical erosion (with structuring element less thana pixel) would leave the image unchanged. The fuzzy inclusionindicators of the translated structuring element in theimage for various translationsare shown in Fig. 2(c) and (d).

The result of the erosion depends on the defuzzification method.In this example, each pixel of the eroded image that isshown in Fig. 2(f) has been substituted by the maximum valueof the fuzzy inclusion indicator membership function. This ex-ample emphasizes that a robust morphology in the presence ofimpulsive noise can be constructed by using a pyramidal fuzzystructuring element. When the core set has area greater than apixel area, the robustness is combined with the properties ofthe classical erosion operation, that are not presented in this ex-ample.

V. APPLICATION IN ROBUSTOBJECTSHAPE REPRESENTATION

The morphological skeleton and the morphological shape de-composition are popular methods for shape representation. Eachmethod represents an objectas a number of components. Al-gebraic combinations of the components reconstruct the object.The representation of using the morphological skeleton is [7]

(24)

where the sets known asskeletal subsetsare

(25)

where and are morphological erosion and dilation,is the morphological opening of by , denotes setdifference, is the th-order homothetic of and is thelargest integer such that .

The morphological shape decomposition ofis describedby [8]

(26)

where the sets known asspinesare

(27)where is again the largest integer such that .

Skeletonization and shape decomposition based on theGFMM morphology can be applied by substituting the clas-sical erosion, dilation and opening operations that are usedduring their processes, with the corresponding operations ofthe GFMM defined in (16)–(18). Since is a binary image,the simplified erosion equation given in (21) can be used.Simplified versions of dilation and opening are derived easily.The fuzzy structuring element was chosen to have a size of

pixels, a core set of pixels where , andpyramidal fuzziness. The lack of property B3 in the 2-D caseis employed in our experiments in order to have robust fuzzyrepresentations. Extensive simulations were applied by usingtwo reference images. The first one contains an orthogonalspiral and it is shown in Figs. 3(a) and 5(a). The second onecontains a puzzle piece and it is shown in Figs. 4(a) and 6(a).


Fig. 3. BMM and GFMM skeletons. (a) Originalspiral image; (b) BMM skeleton; (c) reconstructed object from the BMM skeleton; (d) GFMM skeleton; (e)reconstructed object from the GFMM skeleton; (f)spiral image contaminated with impulsive noise (1%); (g) BMM skeleton; (h) reconstructed object from theBMM skeleton; (i) GFMM skeleton; (j) reconstructed object from the GFMM skeleton; (k)spiral image contaminated with impulsive noise (5%); (l) BMMskeleton; (m) reconstructed object from the BMM skeleton; (n) GFMM skeleton; and (o) reconstructed object from the GFMM skeleton.

Fig. 4. BMM and GFMM skeletons. (a) Originalpuzzleimage; (b) BMM skeleton; (c) reconstructed object from BMM skeleton; (d) GFMM skeleton; (e)reconstructed object from GFMM skeleton; (f)puzzleimage contaminated with impulsive noise (1%); (g) BMM skeleton; (h) reconstructed object from the BMMskeleton; (i) GFMM skeleton; (j) reconstructed object from the GFMM skeleton; (k)puzzleimage contaminated with impulsive noise (5%); (l) BMM skeleton;(m) reconstructed object from the BMM skeleton; (n) GFMM skeleton; and (o) reconstructed object from the GFMM skeleton.


Fig. 5. BMM and GFMM shape decomposition. (a) Originalspiral image; (b) BMM spines; (c) reconstructed object from the BMM spines; (d) GFMM spines;(e) reconstructed object from the GFMM spines; (f)spiral image contaminated with impulsive noise (1%); (g) BMM spines; (h) reconstructed object from theBMM spines; (i) GFMM spines; (j) reconstructed object from the GFMM spines; (k)spiral image contaminated with impulsive noise (5%); (l) BMM spines; (m)reconstructed object from the BMM spines; (n) GFMM spines; and (o) reconstructed object from the GFMM spines.

Fig. 6. BMM and GFMM shape decomposition. (a) Originalpuzzleimage; (b) BMM spines; (c) reconstructed object from the BMM spines; (d) GFMM spines;(e) reconstructed object from the GFMM spines; (f)puzzleimage contaminated with impulsive noise (1%); (g) BMM spines; (h) reconstructed object from theBMM spines; (i) GFMM spines; (j) reconstructed object from the GFMM spines; (k)puzzleimage contaminated with impulsive noise (5%); (l) BMM spines; (m)reconstructed object from the BMM spines; (n) GFMM spines; and (o) reconstructed object from the GFMM spines.


TABLE IRESULTS FROM THE SKELETONIZATION OF THE ORIGINAL AND NOISY SPIRAL IMAGES

USING THE CLASSICAL BMM AND GFMM

TABLE IIRESULTS FROM THE SKELETONIZATION OF THE ORIGINAL AND NOISY PUZZLE IMAGES

USING THE CLASSICAL BMM AND THE GFMM

Each image was contaminated by impulsive (salt and pepper)noise. The noise probability varied from 0.1% to 20%. Sincethe location of the impulses with respect to the object stronglyaffects the resulting skeletal subsets and spines, several experi-ments were performed for the same noise probability.

The skeletonization results are presented in Tables I and IIfor the spiral and thepuzzleimage, respectively. The numberof the skeletal subsets, the area of the noisy objectas a per-centage of the original, the total area of the skeletal subsets as apercentage of the original, and the area of the reconstructed ob-

ject from the skeletal subsets as a percentage of the originalshape area are presented in these tables. The representationerror is also presented as the sum of the positive error definedas the number of pixels that belong toand do not belong to

, and the negative error defined as the number of pixels thatbelong to and do not belong to . It should be noted that,in order to avoid reconstructing the white impulses on the back-ground, the last step of the reconstruction was omitted. It can beobserved that the representation error is always less by using theproposed GFMM than by using BMM. The reconstructed area


TABLE IIIRESULTS FROM THE SHAPE DECOMPOSITION OF THEORIGINAL AND NOISY SPIRAL IMAGES


TABLE IVRESULTS FROM THE SHAPE DECOMPOSITION OF THEORIGINAL AND NOISY PUZZLE IMAGES


is also greater in the GFMM case. Generally, object reconstruc-tion from their skeletal subsets, that can be achieved by usingthe GFMM is better than obtained by using the BMM. How-ever, the most important characteristic of the use of GFMM isthe preservation of the shape and the location of the skeletalsubsets. An indication of this property is the number of skeletalsubsets, presented in Tables I and II. Figs. 3 and 4 illustrate theabove mentioned property. The original spiral image, the BMMskeleton, the reconstructed from the BMM skeleton object, theGFMM skeleton and the GFMM reconstructed object are shown

in Fig. 3(a)–(e). In Fig. 3(f)–(o) the skeletonization process isapplied on the noisy spiral images shown, and the correspondingBMM and GFMM skeletons and reconstructed object are alsopresented. The corresponding results when skeletonization isapplied on the original and two noisy puzzle images are pre-sented in Fig. 4.

The shape decomposition results are presented in Tables IIIand IV for thespiral and thepuzzleimage, respectively. Thenumber of the spines, the area of the noisy object, thearea of the spines and the area of the reconstructed object


Fig. 7. (a) Originalbodyvolume; (b) and (c) visualizations of the BMM and GFMM skeleton; (d) noisybodyvolume; (e) and (f) visualizations of the BMMskeleton; and (g) and (h) visualizations of the GFMM skeleton.

Fig. 8. (a) Originalbodyvolume; (b) and (c) visualizations of the BMM and GFMM spines; (d) the noisybodyvolume; (e) and (f) visualizations of the BMMspines; and (g) and (h) visualizations of the GFMM spines.

as percentages of the original, and the representation errorare presented in these tables. The representation error and thereconstructed area are more or less similar by using the GFMMand the BMM. The shape and the location of the spines arepreserved better in the case of GFMM. The number of spinesis, in most cases, closer to the number of the original objectsspines by using the GFMM shape decomposition. Figs. 5 and6 illustrate the above mentioned property. The original spiralimage, the BMM spines, the reconstructed object from theBMM spines, the GFMM spines and the GFMM reconstructedobject are shown in Figs. 5(a)–(e). In Figs. 5(f)–(o) the shapedecomposition process is applied on the noisy spiral imagesshown, and the corresponding BMM and GFMM skeletonsand reconstructed object are also presented. The corresponding

results when skeletonization is applied on the original and twonoisy puzzle images are presented in Fig. 6.

Simulations were also performed on 3-D data. The BMM andGFMM skeletonization and shape decomposition were appliedon the binary volume of a human body, consisted of 187 frames,each frame of pixels, shown in Figs. 7(a) and 8(a)[30], [31]. The visualization of the morphological skeleton ofthe body by using either BMM or GFMM are shown in Fig. 7(b)and (c). Both skeleton and shape decomposition processes werealso applied on a noisy body volume contaminated with impul-sive noise (0.1%) shown in Figs. 7(d) and 8(d). The visualizationof the morphological skeleton of the noisy body by using BMMare shown in Fig. 7(e) and (f). The same projections by usingGFMM are shown in Fig. 7(g) and (h). Two projections of the


Fig. 9. (a) and (b) Originaltooth volume; (c) the noisytooth volume; (d) ten frames of the original volume BMM skeleton; (e) the same frames of the noisyvolume BMM skeleton; and (f) the same frames of the noisy volume GFMM skeleton.

spines of the body by using either BMM or GFMM are shown inFig. 8(a) and (b). Two projections of the spines of the noisy bodyby using BMM are shown in Fig. 8(c) and (d). The same projec-tions by using GFMM are shown in Fig. 8(e) and (f). It can beobserved that, as in the 2-D case, the location and the shape ofthe morphological skeleton and spines are better preserved byusing the GFMM. The BMM and GFMM skeletonization andshape decomposition were also applied on the nonconvex bi-

nary volume of a tooth, consisted of 170 frames, each frame ofpixels, shown in Fig. 9(a) and (b). The volume was

corrupted by impulsive noise (0.1%) shown in Fig. 9(c). Tenframes of the result of classical BMM skeletonization when ap-plied on the original volume are shown in Fig. 9(d). The sameten frames of the skeleton when BMM and GFMM skeletoniza-tion are applied on the noisy volume are shown in Fig. 9(e) and(f), respectively. It can also be observed that the location and


the shape of the morphological skeleton are better preserved byusing GFMM.

VI. CONCLUSIONS

In this paper, the GFMM was proposed. It was based on anovel definition of an FII, which is a fuzzy set defined as ameasure of the inclusion of a fuzzy set into another. It wasproven that the FII obeys a set of axioms, which are exten-sions of the known inclusion indicator axioms, and which cor-respond to the desirable properties of any mathematical mor-phology operations. The proposed GFMM was compared withthe classical Binary and Grayscale mathematical morphologiesand it was shown that they can be considered as special cases ofthe GFMM. The GFMM provides a very powerful and flexibletool for morphological operations. An application for robust 2-Dand 3-D object representation using skeletonization and shapedecomposition was investigated. Extensive simulations showedthat the reconstruction of noisy objects from their skeletal sub-sets, that can be achieved by using the GFMM is better than byusing the classical BMM in most cases. Besides, the use of theGFMM for skeletonization and shape decomposition preservesthe shape and the location of the skeletal subsets and spines, andtherefore, can be efficiently used for object representation espe-cially in cases of impulsive noise.

APPENDIX

In the following, it will be proven that the fuzzy inclusion in-dicator proposed in (11) satisfies the axioms B1)–B7). It is as-sumed that are 1-D fuzzy sets defined in a specificsubset of the universe. Specific comments are given regardingthe extension of the proofs to-dimensional fuzzy sets.

B1): if and only if.

Proof:

(A1)

This proof can be easily extended to-dimensional fuzzy sets.B2): .

Proof: From (5) implies , .This implies that for any subset ofsuch that

(A2)

This proof can be easily extended to-dimensional fuzzy sets.B3): In the 1-D case, if the fuzzy set is convex then

.

Proof: From (6) implies , .Let us symbolize as and the lower and upper limits of theclassical set , and similarly the limits and of the clas-sical set . If the fuzzy sets are not convex, the -cuts

can be considered as unions ofand , respectively,convex sets. Then, the limits of these sets are symbolized as,

, , . Since the following relations are valid:

(A3)

The fuzzy set should be convex in the common domain of def-inition of all the fuzzy sets . This means that the fuzzyset should have one increasing and one decreasing part in thisdomain, although one of then can be missed out. Then, the mem-bership value of on either or will al-ways be less than or equal to the membership value of all the,

points. Thus,and consequently . The extension of thisproof to -dimensional fuzzy sets is valid only in special casesof .

B4): .Proof: Equality holds since the following equalities of the

membership functions are valid:

(A4)

This proof can be easily extended to-dimensional fuzzy sets.B5): In the general case: .If is convex then: .

Proof: The following relation holds for the crisp sets thatare of interest:

(A5)

Furthermore, the points that belong in the setand do not belong in the set are locatedbetween the minimum and maximum of the set

(A6)

The inequality in the above relation can be substituded byequality if is convex, since the minimum and maximumvalue of both the crisp sets in (A5) are the same. It can be easilyproven that the superset relation holds also for-dimensionalfuzzy numbers. Equality holds only for-dimensional convexfuzzy sets with special membership functions.

B6): .


Proof: The following equalities between the membershipfunctions are valid:

(A7)

This proof can be easily extended to-dimensional fuzzy sets.B7): .

Proof: It should be proven that

(A8)

where

(A9)

and

(A10)

By using (A9) and (A10) and classical arithmetic it can beproven that the inequality of (A8) is valid. This proof can beeasily extended to-dimensional fuzzy sets.

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Vassilios Chatzis(S’96–A’99) was born in Kavala,Greece, in 1968. He received the Diploma degree inelectrical engineering in 1991 and the Ph.D. degreein informatics in 1999, both from the Aristotle Uni-versity of Thessaloniki, Greece. His research inter-ests include nonlinear image and signal processingand analysis, fuzzy logic and pattern recognition.

He is currently a Technical Manager with Net-INVEST Ltd., Thessaloniki, Greece.

Dr. Chatzis is a member of the Technical Chamberof Greece.

Ioannis Pitas (S’83–M’84–SM’94) received theDiploma degree in electrical engineering in 1980 andthe Ph.D. degree in electrical engineering in 1985,both from the University of Thessaloniki, Greece.

Since 1994, he has been Professor with the De-partment of Informatics, University of Thessaloniki.From 1980 to 1993, he was Scientific Assistant,Lecturer, Assistant Professor, and Associate Pro-fessor in the Department of Electrical and ComputerEngineering at the same university. His currentinterests are in the areas digital image processing,

multidimensional signal processing and computer vision. He has publishedover 250 papers and contributed in eight books in his area of interest. He isco-editor ofMultidimensional Systems and Signal Processing.

Dr. Pitas has been member of the European Community ESPRIT Parallel Ac-tion Committee. He has also been an Invited Speaker and/or member of the pro-gram committee of several scientific conferences and workshops. He has beenAssociate Editor of IEEE TRANSACTIONS ONCIRCUITS AND SYSTEMS. He iscurrently Associate Editor of IEEE TRANSACTIONS ON NEURAL NETWORKS.He was Chair of the 1995 IEEE Workshop on Nonlinear Signal and Image Pro-cessing (NSIP’95) and he will be General Chair of the 2001 IEEE InternationalConference on Image Processing (ICIP’01).

A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object...

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