Efficient morphological shape representation by varying overlapping levels among representative disks

Pattern Recognition 36 (2003) 429–437www.elsevier.com/locate/patcog

E�cient morphological shape representation by varyingoverlapping levels among representative disks

Jianning Xu ∗

Computer Science Department, Rowan University, Glassboro, NJ 08028, USA

Received 27 July 2001; accepted 20 March 2002

Abstract

The morphological skeleton transform, the morphological shape decomposition, and the overlapped morphological shapedecomposition are three basic morphological shape representation schemes. In this paper, we propose a new way of generalizingthese basic representation algorithms to improve representational e�ciency. In all three basic algorithms, a 0xed overlappingpolicy is used to control the overlapping relationships among representative disks of di2erent sizes. In our new algorithm,di2erent overlapping policies are used to generate shape components that have di2erent overlapping relationships amongthemselves. The overlapping policy is selected dynamically according to local shape features. Experiments show that comparedto the three basic algorithms, our algorithm produces more e�cient representations with lower numbers of representativepoints. ? 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Mathematical morphology; Shape analysis; Shape representation; Structural shape representation; Shape decomposition; Shapecomponents; Overlapping policy

1. Introduction

Shape representation and description is a very impor-tant problem in computer vision and image processing[1–4]. Shape representation provides the foundation fordeveloping various shape-related processing and analysisalgorithms. They include algorithms for image coding [5,6],image recognition [7–9], video compression [10,11], andimage data retrieval [9,12]. Many di2erent shape represen-tation and description schemes have been developed. Oneof the major schemes is structural shape representation.In a structural shape representation, a shape is describedin terms of a number of simpler shape parts and the rela-tionships among the shape parts. To construct a structuralshape representation, a given shape is 0rst decomposed intosimpler shape parts or components.

Certain properties of a structural shape representation al-gorithm are desirable. The shape components generated by

∗ Corresponding author. Tel.: +1-856-256-4500; fax: +1-856-256-4915.

E-mail address: [email protected] (J. Xu).

the algorithm should be mathematically simple and wellde0ned. Such shape components allow us to easily see therelationships between the components and the original shapeand therefore can often help us to develop e2ective shapeanalysis algorithms. The shape components should alsobe natural and intuitively meaningful. Such componentsallow us to use intuition and heuristics in developing andinterpreting shape-related processing and operations. Wealso want the representation to be compact. E�cient repre-sentation leads to e�cient storage, transmission, and pro-cessing. A good shape representation should be easily ande�ciently computed. Computational e�ciency is alwaysdesirable. There are some other desirable properties [20].Some of the above-mentioned properties are in fact in con-Bict. For example, if a representation scheme allows a shapeto be represented in many di2erent ways, then by usingan exhaustive search we can always 0nd the most e�cientrepresentation with the lowest coding cost. However suchan approach is hardly ever used.

In recent years, a number of morphological shaperepresentation algorithms have been proposed [19–37].Mathematical morphology is a shape-based approach to

0031-3203/02/$22.00 ? 2002 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.PII: S0031 -3203(02)00075 -4

430 J. Xu / Pattern Recognition 36 (2003) 429–437

image processing [13–15]. One advantage of mathemati-cal morphology is that basic morphological operations canbe implemented very e�ciently on many parallel imagecomputers [16–18]. Another advantage of mathematicalmorphology is that it has a well-developed mathematicalstructure, which provides a foundation for the analysis ofmorphological image processing algorithms.

One of the leading morphological shape representationalgorithms is the morphological skeleton transform (MST)[19]. In the MST, a given shape is represented as a unionof all maximal disks contained in the shape. These maxi-mal disks in general overlap heavily with one another. An-other important morphological shape representation schemeis the morphological shape decomposition (MSD) [20]. Inthe MSD, a given shape is represented as a union of cer-tain disks contained in the shape. The overlapping amongrepresentative disks of di2erent sizes is eliminated. A newmorphological shape representation algorithm that can beviewed as a compromise between the MST and the MSDwas recently proposed [21,22]. In this scheme, overlap-ping among representative disks of di2erent sizes is al-lowed. But severe overlapping is avoided. We can call thisalgorithm overlapped morphological shape decomposition(OMSD).

Many e2orts have been made by various researchers toimprove the representational e�ciencies of these basic algo-rithms. Many di2erent approaches have been used. One ap-proach is to allow more than one basic structuring element tobe used in the description of a given shape [23,24,26,29,35].Another approach is to allow di2erent types of operationsto be used to combine shape components together to formthe original shape [24,25,28]. Some algorithms use limitedsearching to achieve representational e�ciency [25,26]. Re-dundancy removal procedures can also be used to determinea minimal representation from a representation generated bya basic scheme [19,31].

In this paper we consider a new approach of generaliz-ing the basic representation schemes to improve represen-tational e�ciency. The three basic algorithms di2er primar-ily in the overlapping relationships among representativedisks of di2erent sizes. Each algorithm employs a di2erentstrategy in selecting a new group of representative disks sothat they will have certain overlapping relationships withlarger-size representative disks. In this paper, we will com-bine the strategies used by these three algorithms togetherinto a single algorithm. In our algorithm, each group ofrepresentative disks are selected to best describe a localshape area by applying the best-0tting overlapping policy.Our algorithm is intended to provide a new way of general-izing the basic representation algorithms and it can be com-bined with some other e�ciency improvement schemes thathave been proposed by other researchers.

The paper is organized as follows. In Section 2, we de-scribe the new algorithm. The experiments and some dis-cussions are included in Section 3. Section 4 contains con-clusions.

2. The new algorithm

In binary morphological image analysis, a 2-D image isde0ned as a subset of the 2-D Euclidean space R× R or itsdigitized equivalent Z × Z . In this paper, we deal only withdigital images that are de0ned as subsets of Z × Z . For animage A ⊆ Z × Z and a point u∈ Z × Z , the translation ofA by u is de0ned

(A)u = {a+ u | a∈A}: (1)

The two most fundamental morphological operations are di-lation and erosion. They are de0ned as follows, respectively,

A⊕ B =⋃b∈B

(A)b; (2)

A B =⋂b∈B

(A)−b: (3)

Another pair of important morphological operations areopening and closing. They are de0ned in terms of dilationand erosion

A ◦ B = (A B)⊕ B; (4)

A • B = (A⊕ B) B: (5)

In the MST, a binary shape X is represented as a unionof all maximal disks contained in X

X =N⋃i=0

Si ⊕ iB; (6)

where

Si = (X iB) \ ((X iB) ◦ B); (7)

and N is the largest integer such that X NB =�, andiB=B⊕B⊕· · ·⊕B (i times) is a disk of size i (0B={(0; 0)}).In fact, the unit disk B does not need to resemble a real disk.We assume that B is convex and symmetric about the x- andy-axes. The skeleton subset Si contains the centers of allmaximal inscribable disks of size i. Si ⊕ iB is the union ofall maximal disks of size i in X . Maximal disks of di2erentsizes may overlap. So in general we have

(Si ⊕ iB) ∩ (Sj ⊕ jB) =�; i = j: (8)

Another interpretation for these skeleton subsets is that Siis the set of centers of all disks of size i in X that are notcontained in any representative (maximal) disks of largersizes.

In the MSD, a binary shape X is represented as a unionof certain disks contained in X

X =N⋃i=0

Li ⊕ iB; (9)

where LN = X NB and

Li =

(X \

(N⋃

j=i+1

Lj ⊕ jB))

iB; 06 i ¡N: (10)

J. Xu / Pattern Recognition 36 (2003) 429–437 431

Again, N is the largest integer such that X NB =�. Notethat the sets of centers of representative disks of di2erentsizes LN ; LN−1; : : : ; L0 must be determined in the order givenand we have

(Li ⊕ iB) ∩ (Lj ⊕ jB) = �; i = j: (11)

The set of centers of representative disks of size i; Li, isdetermined by 0rst removing all the representative disksof larger sizes from the given shape, then 0nding all thecenters of disks of size i in the remaining areas. These arethe centers of all disks of size i contained in X that do notintersect with any representative disks of larger sizes. Notethat the overlapping among representative disks of the samesize still exists.

In the OMSD, we also represent a given shape X as aunion of certain disks contained in X

X =N⋃i=0

Ci ⊕ iB; (12)

where CN = X NB and

Ci = (X iB) \(

N⋃j=i+1

Cj ⊕ jB); 06 i ¡N: (13)

Once again, N is the largest integer such that X NB =�.Similar to the MSD, the sets of centers of representativedisks of di2erent sizes CN ; CN−1; : : : ; C0 must be determinedsequentially. However, similar to the MST, we have in gen-eral

(Ci ⊕ iB) ∩ (Cj ⊕ jB) =�; i = j: (14)

The set of centers of representative disks of size i; Ci, isdetermined by 0rst identifying all the centers of disks of sizei contained in X which is X iB. Then we only use thosecenters that are inside the parts of the original shape thathave not been represented by larger representative disks. Inother words, these are disks of size i whose centers are notin any representative disks of larger sizes.

For the MSD and OMSD, there is an easy and natu-ral way to divide the representative disks into a number ofgroups, each of which represents a more natural shape part[20,22]. We consider the case of the OMSD. For an im-age X , the OMSD algorithm produces a sequence of centerpoint sets: CN ; CN−1; : : : ; C0. For each center point set Ci, weidentify all the connected components: Ci1; Ci2; : : : ; Cij ; : : : .Each such connected component Cij corresponds to a shapecomponent Cij ⊕ iB in X . The union of all such shapecomponents is X . A similar procedure can be applied to theMSD. Because of the heavy overlapping among maximaldisks, the MST is not normally considered a shape decom-position scheme.

To show the relative advantages and disadvantages of thedi2erent strategies employed by the three basic algorithms todeal with overlapping among representative disks, we lookat a number of simple examples. In Fig. 1, we have a sim-ple shape image and its three representations generated by

Fig. 1. Representations generated by di2erent algorithms: (a) MST;(b) MSD; (c) OMSD.



the three basic algorithms. The representative disks are indi-cated by their center points and we call them representativepoints. In this and following examples, we use the 3 × 3square centered at the origin as the unit “disk”. In this ex-ample, the most e�cient representation with only two repre-sentative points is generated by the MST. In Fig. 2 we havea similar shape and its three representations. This time, theMSD generates the most e�cient representation containingtwo representative points. In Fig. 3, we show another similarshape and its di2erent representations. In this example, themost e�cient representation with two representative pointsis generated by the OMSD.

The three shapes used in these examples can all be viewedas comprising two disks of di2erent sizes. For the 0rst shape,there is much overlapping between the two disks. There-fore, the MST is the most e�cient scheme. The two disksin the second example do not overlap. The MSD is the bestapproach for the situation. In the third example, there ismoderate overlapping between the two disks. Therefore, theOMSD produces the most e�cient representation. Some-times the three algorithms will generate the same results asthe example in Fig. 4 shows.

In each of the preceding examples, after the size-2 diskis extracted from the given shape, only one connected com-ponent remains and the best scheme to use to representthe remaining shape part is always easy to determine. Butin general things are not always this simple. Consider the


Fig. 4. Representation example: the three basic algorithms generatethe same result.

Fig. 5. Representation example: applying di2erent overlapping poli-cies to the same shape.

next example in Fig. 5(a). After subtracting the size-2 diskfrom the given shape, we see three connected componentsas shown in Fig. 5(b). Not a single scheme will handlethe three connected components equally well. From earlierexamples we can see that the MSD is the best approach forrepresenting the component to the left of the size-2 disk; thebest strategy for the component below the size-2 disk is theMST; and the last component is best characterized using theOMSD scheme. The 0nal representation generated by us-ing di2erent overlapping strategies for di2erent connectedcomponents is shown in Fig. 5(c).

For a more complicated shape image, this process of se-lecting the best representational strategy for a connectedcomponent in the residue image needs to be repeated manytimes in a hierarchical fashion. In general, we 0rst identifythe largest disks contained in the given shape. After sub-tracting these disks from the given shape, the best represen-tational strategy is selected to generate representative disksof a certain size for one of the connected components in thedi2erence image. The representative disks thus selected maynot cover the connected component completely. The newgroup of representative disks are also subtracted from thegiven image and the same process is repeated over and over.

Now we present the details of the algorithm. For a givenshape image X , let N be the largest integer such thatX NB =�. X NB is the 0rst set of representative points.For a point p in X NB; p ⊕ NB is a representativedisk of size N . The 0rst group of representative disks are(X NB)⊕NB. Let C1; C2; : : : be the connected componentsof XNB. There could be only one such component. Similarto the MSD and OMSD, the 0rst group of shape componentsgenerated by our algorithm are C1 ⊕ NB; C2 ⊕ NB; : : : .

Let P = C1 ⊕ NB ∪ C2 ⊕ NB ∪ · · · = (X NB) ⊕ NB.Thus, P represents the parts of the given shape that havebeen covered by the shape components that we have identi-0ed so far. Let R=X \P be the parts of the given shape thatare not covered by the current components yet. We de0ne

the scale of an image Y to be the largest integer n such thatY nB =�. Let R1; R2; : : : be the connected components ofR. There could be only one such component. Without loss ofgenerality, we assume that R1 is a connected component ofthe largest scale. Intuitively, R1 represents one of the mostsigni0cant remaining parts. The next group of representativedisks is selected to represent R1 e�ciently. When determin-ing the relationships between the new group of representa-tive disks for R1 and the existing representative disks, wecan choose from three representational strategies: maximaloverlapping (MST), no overlapping (MSD), and moderateoverlapping (OMSD).

The process of selecting R1 and an e�cient strategy forrepresenting R1 will be repeated. For each such R1, threecenter point sets D1; D2, and D3 are determined. D1 is theset of centers of all the largest disks in X that overlap withR1. In other words, all the points in D1 represent disks ofthe same size and each such disk covers at least one point inR1. Such a disk is not completely contained in an existingrepresentative disk. In fact, such a disk is not containedin the union of all existing representative disks. But heavyoverlapping with existing representative disks is possible.We can say that D1 corresponds to the MST strategy. D1 iscalculated by selecting the largest integer n so that

(X nB) ∩ (R1 ⊕ nB) = D1 =�: (15)

Note that for a point p in R1 ⊕ nB; p⊕ nB contains at leastone point in R1. If p is also in X nB, then p ⊕ nB is asize-n disk in X containing at least one point in R1.D2 is the set of centers of all the largest disks in R1. Such

disks do not overlap with current representative disks. ThusD2 corresponds to the MSD strategy. D2 is calculated byselecting the largest integer n so that

R1 nB = D2 =�: (16)

In order for this formula to have the same format as the onefor D1 in Eq. (15), we can rewrite it as

(X nB) ∩ (R1 nB) = D2 =�: (17)

D3 is the set of centers in R1 of the largest disks in X . D3

is calculated by selecting the largest integer n so that

(X nB) ∩ R1 = D3 =�: (18)

Clearly, it is possible for this set of disks to overlap withexisting representative disks. But severe overlapping isavoided since their centers are outside the current represen-tative disks. Clearly, D3 corresponds to the OMSD scheme.Note that the di2erence between the formula in (15), (17),and (18) is the way in which R1 is modi0ed.Let si be the size of the disks represented byDi for i=1; 2,

and 3. The representation e�ciency factor ei forDi is de0ned

ei =|(Di ⊕ siB) ∩ R1|

|Di| ; i = 1; 2; and 3: (19)


Fig. 6. Representation example using the new algorithm.

The e�ciency factor ei is the ratio between the number ofpoints in R1 covered by the maximal disks under consid-eration and the number of representative points used. TheDi with the highest e�ciency factor is selected for R1. Ifthe choice is not unique, we can either introduce additionalcriteria or break the tie arbitrarily. The Di thus selected isour new set of representative points. The new group of rep-resentative disks are Di ⊕ siB. Let Di1 ; Di2 ; : : : be the con-nected components of Di. Again, there could be only onesuch component. The new group of shape components forthe given shape are Di1 ⊕ siB; Di2 ⊕ siB; : : : .

We now update both P and R. Note that P is the unionof all the shape components that have been identi0ed so far.Therefore P = P ∪ Di1 ⊕ siB ∪ Di2 ⊕ siB ∪ · · ·= P ∪ Di ⊕siB · R = X \ P represents the areas of the given shape yetto be covered. The same process is repeated. That is, oneof the largest-scale connected components of R is identi0edand the most e�cient covering strategy is determined. Theprocess continues until we 0nally have R= �.

Let’s consider another example. For the image inFig. 6(a), we have N=2 and X 2B has only one connectedcomponent—X 2B itself. The 0rst shape component is(X 2B) ⊕ 2B = P. The center point set X 2B andthe shape component represented are given in Fig. 6(b).

R=X \P is in Fig. 6(c) and R1 is the connected componentto the left of the 0rst shape component. For this R1, threecenter point sets D1; D2, and D3 are determined. They arein Fig. 6(d)–(f). The corresponding e�ciency factors aree1 =14=6; e2 =12=2, and e3 =14=4. Clearly e2 has the high-est value. Since D2 has only one connected component—D2

itself, D2 ⊕ B is selected as the next shape component.The updated P is in Fig. 6(g) and the updated R is in

Fig. 6(h). The same process is repeated. This time, let R1

be the largest connected component with six points in Fig.6(h). Again, we need to determine three new center pointsets D1; D2, and D3 for this new R1. They are in Fig. 6(i)–(k). From these center point sets, we have e1=4=2, e2=6=6,and e3 = 6=6. Clearly, e1 has the highest value. Therefore,D1⊕B is our next shape component. This process is repeatedover and over. The 0nal representation is given in Fig. 6(l).

3. Experiments and discussions

In our experiments, our new algorithm was implementedalong with the three basic algorithms. To achieve betterapproximations to real disks and keep the size of unit disksmall, we used two basic structuring elements B1 and B2


Fig. 7. Two basic structuring elements and some discrete disksformed by them: (a) B1; (b) B2; (c) 2B; (d) 3B; (e) 4B.

as given in Fig. 7 to de0ne disks of di2erent sizes. In ourimplementation, the unit disk B is de0ned to be B1, thesize-two disk 2B is de0ned to be B1⊕B2, and the size-threedisk 3B is de0ned to be B1 ⊕ B2 ⊕ B1. In general, a disk of

Fig. 8. Shape images used in the experiments: (a) teapot; (b) lamp; (c) telephone; (d) butterBy; (e) dog; (f) 0sh; (g) puzzle piece;(h) letters; (i) digits.

size i is de0ned

iB =

{(i − 1)B⊕ B1 if i is odd;

(i − 1)B⊕ B2 if i is even:(20)

Some of these disks are in Fig. 7. Note that with this de0ni-tion, the formula for calculating skeleton subsets becomes

Si =

{(X iB) \ ((X iB) ◦ B1) if i is even;

(X iB) \ ((X iB) ◦ B2) if i is odd:(21)

In fact, this is a generalized-step skeleton transform as de-0ned in [36]. This generalization can also be easily ap-plied to the MSD, OMSD, and our new algorithm. For thesealgorithms, we just need to note that

X ∗iB =

{(X ∗(i − 1)B)∗B1 if i is odd;

(X ∗(i − 1)B)∗B2 if i is even;(22)

where ∗ is either ⊕ or .We applied the new algorithm to nine binary shape

images given in Fig. 8. Table 1 shows the numbers of repre-sentative points used by four di2erent algorithms: the MST,MSD, OMSD, and our new algorithm. From these numbers,we can see that our new algorithm always uses the lowest


Fig. 9. Reconstruction using major components.

Table 1Numbers of representative points used by four di2erent algorithms

MST MSD OMSD New algorithm

Teapot 188 349 217 141Lamp 238 439 248 178Telephone 300 587 305 208ButterBy 346 514 328 238Dog 357 481 316 269Fish 328 526 317 247Puzzle piece 286 450 281 172Letters 567 585 460 436Digits 731 717 613 560

number of representative points. That is, our algorithm pro-duces the most e�cient representations. The improvementin representational e�ciency is achieved through employ-ing a more complicated representation scheme. Our exper-iments show that our scheme can actually take advantageof the variations in the relationships among di2erent naturalshape parts. In Fig. 9, we show the partial reconstruction forthe 0rst three shapes from Fig. 8 using only a small numbermajor components for each shape. The teapot shape uses 13components, the lamp uses 11, and the telephone also uses11. We can see di2erent overlapping relationships amongthe components.

Among the three basic algorithms, either the MST or theOMSD produces the lowest number of representative points.The reduction in the number of representative points fromthe previous best achieved by our new algorithm ranges from5% (letters) to 39% (puzzle piece). In fact, the reductionsfor most of the nine shape images are quite signi0cant. Thelowest levels of improvements are achieved on the lettersand digits images. These two shape objects can be viewed asconsisting mainly of elongated parts of similar scales. Ournew algorithm achieves high representation e�ciency bye2ectively representing di2erent overlapping relationshipsamong di2erent object components of di2erent shapes andscales. In the letters and digits images, the shapes and scalesof the object components are too uniform and they provideless opportunities for our scheme to be e2ective.

In the three basic algorithms, all representative disks of thesame size are determined at the same time. In our new algo-

rithm, representative disks are generated in groups, each ofwhich is selected for a connected component in the residueimage after all current groups of representative disks aresubtracted from the original image. However, all represen-tative disks of the same size can also be collected together.Let Ei be the union of all center point sets of size-i rep-resentative disks, i = 0; 1; : : : ; N . Then similar to the basicalgorithms, we have

X =N⋃i=0

Ei ⊕ iB: (23)

Therefore, under our new scheme a given shape is still rep-resented as a union of certain disks contained in the givenshape. But instead of determining all representative disks ac-cording to a 0xed global policy, we select our representativedisks according to local shape features. The criteria used areonly locally optimal. No global optimization is attempted.

In our new algorithm, the three basic representationschemes are combined into a uni0ed and more generalscheme. This new scheme is also more complicated. In thebasic algorithms, all representative disks of a certain sizeare determined at the same time for the whole shape image.In our new algorithm, representative disks are determinedfor local shape components. A local shape component isidenti0ed as a connected component in the di2erence imagebetween the original image and all current representativedisks. To generate a group of representative disks, we exam-ine all three overlapping policies and select the best-0ttingone. All these require extra-operational steps and compu-tational time. The computational costs of the three basicalgorithms have been discussed in [19,20,22]. In fact, theyare quite similar with the MST being the most e�cient one.The computational cost of our algorithm is much higher thanthose of the basic algorithms. For example, computationtime for the pot image using the MST algorithm is 0:67 sand it is 41:96 s for our new algorithm on a SGI Origin 200.The computational e�ciency can be improved signi0cantlywith specialized hardware for morphological operations.

In determining the overlapping levels among di2erentgroups of representative disks, we choose from three over-lapping policies: maximal overlapping (MST), no overlap-ping (MSD), and moderate overlapping (OMSD). In fact,there are many other possible policies that lie between the


two extremes—MST andMSD. However, to include all pos-sible overlapping relationships in our algorithm will signif-icantly increase the complexity of the algorithm. The threepolicies used come from the three basic algorithms andthey represent three important overlapping relationships. Allother situations can be approximated using the policies thatwe have adopted.

In our algorithm, di2erent strategies are used to handle theoverlapping relationships between a new group of represen-tative disks, all of which have the same size, and all existingrepresentative disks. The overlapping among disks in thesame group is not controlled. Similar to the basic algorithms,we use basic morphological operations to generate the cen-ters of such disks and therefore, severe overlapping existsamong these disks. To control the overlapping levels amongrepresentative disks in the same group, we will have to se-lect each representative disk individually. However, such analgorithm may not be considered a real morphological one.One of the major advantages of morphological algorithmsis that they are suitable for parallel implementations. Select-ing representative disks individually will imply sequentialimplementations at the representative-point level. Selectingrepresentative disks individually will also signi0cantly in-crease the complexity of the algorithm. One way to reducesuch overlapping is to use a redundancy removal procedure[19,31]. In fact, this problem is not unique to our algorithm.It is shared by all three basic algorithms.

In our algorithm, the representative disks and shape com-ponents still have relatively simple mathematical character-izations. The representative disks are maximal disks underthree di2erent conditions. These conditions are derived fromthe basic algorithms. The shape components are formed bycombining the representative disks with connecting centerpoints. This is similar to the approach used in the MSD andOMSD. The union of all such shape components is the givenimage. At the same time, these disks and the shape compo-nents are still quite natural and intuitive. They are generatedusing the same basic representation schemes used in the ba-sic algorithms. They correspond to shape parts of di2erentsize scales. By combining di2erent schemes together, theserepresentative disks and shape components have di2erenttypes of overlapping relationships among themselves. Thismixing of overlapping relationships allows our algorithmto generate more e�cient or more precise representations.Very often, more precise representations correspond betterto the structures of the given shapes and therefore can beconsidered more natural.

4. Conclusions

In this paper, we have proposed a new scheme for gener-alizing the three basic morphological shape representationalgorithms to improve representational e�ciency. The re-sulting algorithm combines the di2erent overlapping policiesemployed by these basic algorithms to generate shape com-

ponents that have di2erent overlapping relationships amongthemselves. Experiments show that the new algorithm pro-duces more e�cient representations than the ones producedby the basic algorithms. The better representational e�-ciency is achieved by making representational decisions ac-cording to local shape features. The overall representationalstructure of the new algorithm is very similar to the onesof the basic algorithms. Therefore, the shape componentsgenerated by the new algorithm still have relatively simplemathematical characterizations. The more e�cient or moreprecise representations generated by the new algorithm canoften be considered more natural as well.

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About the Author—JIANNING XU was born in Nanjing, China, in 1959. He received the B.S. degree in Computer Engineering from HarbinInstitute of Technology, Harbin, China, in 1982 and the M.S. and Ph.D. degree in Computer Science from Stevens Institute of Technology,Hoboken, NJ, USA, in 1984 and 1988, respectively. Currently, he is an Associate Professor at the Department of Computer Science, RowanUniversity, Glassboro, NJ, USA. His research interests include image processing, pattern recognition, and mathematical morphology.

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Efficient morphological shape representation by varying overlapping levels among representative disks