Efficient morphological shape representation with overlapping disk components

1346 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 10, NO. 9, SEPTEMBER 2001

Efficient Morphological Shape Representation withOverlapping Disk Components

Jianning Xu

Abstract—This paper proposes a new morphological shaperepresentation algorithm, in which a two-dimensional (2-D)binary shape is represented as a union of certain disks containedin the given shape. The representative disks of different sizes mayoverlap. But excessive overlapping between them is avoided. Thealgorithm combines the advantages of the morphological skeletontransform (MST) and the morphological shape decomposition(MSD). The representative disks have simple and well-definedmathematical characterizations. The algorithm is simple andefficient to implement. The experimental results show that thenumber of representative disks used by our algorithm is significantlower than that used by the MSD. The overlapping level betweenthe representative disks is much lower than that of the MST.A simple procedure can be used to combine the representativedisks into more meaningful shape components. These shapecomponents seem to correspond better to the natural shape partsthan those generated by the MSD. It is also possible to build agood approximation for a given shape using only a small numberof major components.

Index Terms—Mathematical morphology, shape analysis, shapeapproximation, shape components, shape decomposition, shaperepresentation.

I. INTRODUCTION

SHAPE description and analysis is a fundamental problemin image processing and pattern recognition [1]–[4]. Good

shape representation or description schemes not only are impor-tant in developing shape analysis algorithms for shape matchingand recognition tasks [5], [6], they also are important in devel-oping efficient coding schemes for data compression purposes[7], [8] and developing video compression and image data re-trieval algorithms [9]–[11]. Certain properties of a shape rep-resentation scheme are desirable. A good shape representationshould have well defined mathematical characterizations. Therepresentation should be generated according to simple, precise,and meaningful rules, instead of depending on some arbitrarydecisions. A well-defined representation is more likely to cap-ture the intrinsic characteristics of a given shape explicitly. Agood shape representation should provide an accurate and com-plete description of the given shape. The original shape shouldbe allowed to be easily reconstructed or approximated. A goodshape representation should be compact. Efficient manipulationof the shape representation should be possible. A good shape

Manuscript received August 25, 2000; revised May 29, 2001. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Scott T. Acton.

The author is with the Computer Science Department, Rowan University,Glassboro, NJ 08028 USA (e-mail: [email protected]).

Publisher Item Identifier S 1057-7149(01)07463-2.

representation should be easily computed. Computational ef-ficiency is always a desirable feature in a computer imagingsystem.

A number of shape representation schemes have been devel-oped over the years. Structural shape description is one of them.In a structural shape description, a shape is first decomposedinto a number of shape components or primitives. The givenshape is represented in terms of these simpler componentsand the relationships among them. In recent years, a numberof morphological shape representation and decompositionalgorithms have been proposed [7], [18]–[32]. Mathematicalmorphology is a shape-based approach to image processing[12]–[14]. One advantage of mathematical morphology isthat basic morphological operations can be implementedvery efficiently on many parallel image computers [15]–[17].Another advantage of mathematical morphology is that it hasa well-developed mathematical structure, which provides afoundation for the analysis of morphological image processingalgorithms.

The morphological skeleton transform (MST) [18] is a shaperepresentation scheme where a given shape is represented as aunion of all the maximal disks contained in the shape. Thesemaximal disks in general overlap with each other. The max-imal disks of different sizes can be determined directly fromthe given shape. In this scheme, the shape components havevery simple and well-defined mathematical characterizations.The decomposition procedure is very simple and efficient to im-plement. The morphological shape decomposition (MSD) [19]is a scheme that decomposes a binary shape into a union of cer-tain disks contained in the shape. The overlapping between disksof different sizes is eliminated. The representative disks of dif-ferent sizes are defined recursively and therefore they are deter-mined in a sequential order. The characterization for the shapecomponents and the decomposition procedure are quite simple.One advantage of the MSD over the MST is that a structuraldescription of a given shape can be easily constructed basedon the shape components generated by the MSD. Wanget al.proposed a shape representation scheme using recursive mor-phological operations [21]. Again, a given shape is decomposedinto a union of certain disks contained in the shape. The overlap-ping between the representative disks is completely eliminated.The individual disks are defined recursively. The decompositionprocedure is still simple. But the recursive morphological oper-ations seem to be inherently serial and therefore not suitable forparallel implementations. Another problem seems to be that thedecomposition results depend on the order in which the imagepixels are examined.

1057–7149/01$10.00 © 2001 IEEE

XU: EFFICIENT MORPHOLOGICAL SHAPE REPRESENTATION 1347

Many efforts have been made by various researchers to im-prove representation efficiency. Maragos attempted to repre-sent an image as a minimal union of translated and scaled pat-terns from a finite basic pattern class [22]. It appears that thechoice for the basic pattern set is not easy to make. Anotherdifficulty seems to be the computational cost associated withdetermining the minimal union of maximal patterns. This basi-cally is a search problem. Pitas and Venetsanopoulos proposedan improvement over their earlier scheme by allowing the useof multiple structuring elements and minimal enclosing struc-turing elements [23]. The time requirement of the algorithm in-creases as more structuring elements are allowed. The decisionon how many and what kind of structuring elements should beused does not seem to be an easy one. A search-based shaperepresentation scheme was proposed by Reinhardt and Higgins[24]. The basic representation scheme is similar to that of Pitasand Venetsanopoulos. But a search process is used to select thestructuring elements used to form the shape components and theoperations used to combine the shape components to rebuild theoriginal shape. In an earlier work, we proposed a scheme thatdecomposes a shape into a collection of convex polygons whichvery often correspond quite well to the natural structures of thegiven shape [25]. A search process is also used in this algorithm.Most of these schemes use various search procedures to select arepresentation. The shape components thus produced are morecomplex to characterize and the computational procedures aremore time-consuming to implement. The representations gen-erated are in general less flexible to use.

There are some other schemes [7], [26]–[32]. They either aresimilar to some of the schemes that we have mentioned or em-phasize other aspects of morphological shape representation. Anumber of shape representation algorithms have been developedin our earlier works. In [26], we proposed a simple scheme thatdecomposes a shape into neck-free components that are roughlyconvex. A shape component generated by this scheme may notalways be simple enough. We also proposed a hierarchical rep-resentation scheme for two-dimensional (2-D) shapes [27]. Inthis scheme, a shape component in general is not a subimage ofthe given shape. In [28], we proposed a decomposition schemethat can be viewed as a variation to the MST. The heuristic al-gorithm proposed in [29] is based on the algorithm developed in[25]. By using a heuristic approach, the characterizations for theshape components become more complex and the relationshipbetween the given shape and the shape components become lessobvious. The computational cost is also higher than those of theMST and MSD.

Reinhardt and Higgins compared the two major search-freemorphological shape representation algorithms, the MST andMSD, both theoretically and empirically [20]. They concludedthat the MST usually uses fewer disks to represent a given shape.But the disks tend to be larger in size and there is a great deal ofoverlapping. The MSD generally uses larger number of disks.But many of them tend to be smaller in size. By removing over-lapping between representative disks of different sizes, the MSDgenerally lowers the level of redundancy and the reconstructioncost. But the number of disks used by the MSD and therefore thecoding cost typically becomes higher. This means that the rep-resentations generated by the MSD are in general not as con-

cise as the ones generated by the MST. Because of the heavyoverlapping in the MST, there is no simple and obvious wayof combining representative disks into more meaningful shapecomponents in the MST. Under the MSD, a given shape is de-composed into a number of components of different sizes anda structural description of the shape is built using these compo-nents. But these shape components are not always very intuitiveand some of them correspond poorly to the natural parts of thegiven shapes.

In this paper, we propose a morphological shape represen-tation scheme that combines the advantages of the MST andMSD. We want our new algorithm to be as simple and as ef-ficient as the MST and MSD. We also want our new algorithmto be able to generate structural shape descriptions so that thestructural shape analysis techniques can be applied. But we wantto achieve more concise representations and generate more nat-ural looking shape components than the MSD. In our scheme,a binary shape is represented as a union of disks of differentsizes. The representative disks of each size are selected to covera significant part of the given shape that is not covered by therepresentative disks of larger sizes. Overlapping between repre-sentative disks of different sizes is allowed. By insisting on cov-ering a significant part of the given shape by each new groupof disks, excessive overlapping is avoided. By allowing rep-resentative disks of different sizes to overlap, the number ofdisks used in our algorithm is significantly reduced comparedto that used in the MSD. Very often, it is lower than that usedin the MST. Under our algorithm, the representative disks aregrouped into a number of modestly overlapping shape compo-nents. These shape components seem to correspond better to thenatural shape components than those generated by the MSD. Agood approximation for a given shape can be constructed usingonly a small number of major components. The paper is orga-nized as follows. In Section II, we describe the representationalgorithm. It is also shown that a simple procedure can be usedto combine the representative disks together to form more mean-ingful shape components. The properties of the new algorithmare discussed in Section III. Section IV contains experimentalresults. Conclusions are given in Section V.

II. NEW ALGORITHM

In binary morphological image analysis, a 2-D image is de-fined as a subset of the 2-D Euclidean space or its dig-itized equivalent . In this paper, we deal only with dig-ital images that are defined as subsets of . For an image

and a point , the translation of byis defined

(1)

The two most fundamental morphological operations are dila-tion and erosion. They are defined as follows, respectively

(2)

(3)


Another pair of important morphological operations are openingand closing. They are defined in terms of dilation and erosion

(4)

(5)

In the MST, a binary shape is represented as a union of allmaximal disks contained in

(6)

where

(7)

and is the largest integer such that , andis a disk of size . In

fact, the unit disk does not need to resemble a real disk. Weassume that is convex and it contains the origin. The skeletonsubset contains the centers of all maximal inscribable disksof size . A maximal disk is not contained in a larger size inscrib-able disk. is the union of all maximal disks of sizein

. By avoiding redundant operations, all skeleton subsets canbe obtained using erosion and dilation operations with[18]. Maximal disks of different sizes may overlap. So, in gen-eral, we have

(8)

Another interpretation for these skeleton subsets is thatis theset of centers of all disks of sizein that are not containedin any representative (maximal) disks of larger sizes. We canrewrite (6) as

(9)

Therefore, dilations with will be needed to reconstructfrom all the skeleton subsets [18]. Note that some of the

skeleton subsets may be empty. So the number of union opera-tions needed can be less than.

In the MSD, a binary shape is represented as a union ofcertain disks contained in

(10)

where and

(11)

Again, is the largest integer such that . Notethat the sets of centers of representative disks of different sizes

must be determined in the order given andwe have

(12)

The set of centers of representative disks of size is de-termined by first removing all the representative disks of largersizes from the given shape, then finding all the centers of rep-resentative disks of sizein the remaining areas. These are thecenters of all disks of sizecontained in that do not inter-sect with any representative disks of larger sizes. Note that theoverlapping between disks of the same size still exists. To deter-mine all the center point setsdilation plus erosions with will be needed [19],assuming that all the center point sets are nonempty. Similar tothe MST, the original image can be reconstructed usingdilations with

(13)

Some of the ’s can be empty.Our new algorithm combines the ideas of both schemes. Sim-

ilar to both of them, we represent a given shape as a union of anumber of disks of different sizes

(14)

where and

(15)

Once again, is the largest integer such that .Similar to the MSD, the sets of centers of representative disksof different sizes must be determined se-quentially. However, similar to the MST, we have in general

(16)

The set of centers of disks of size is determined by firstidentifying all the centers of disks of sizecontained in whichis . Then we only use those centers that are inside theparts of the original shape that have not been represented bylarger representative disks. In other words, these are disks ofsize whose centers are not in any representative disks of largersizes. The implementation of this algorithm is quite straight-forward. We first use erosions with to get the eroded

. Then we go through thefollowing steps to get all the center point sets:

Step 1) ;Step 2) ;Step 3) ;Step 4) ; if , then go to step 2.

In step 3, dilations with are used. Therefore,dilations are used in all iterations. The total number of basicmorphological operations used is

. Here, we are assuming that all center point sets arenonempty. The reason for that we use less erosion operationsthan the MSD is that in our algorithm the erosions are performedon the original , while in the MSD the erosions are performedon the residue images from the set difference operations. TheMSD requires the same set of basic erosion operations used byour algorithm plus many additional erosion operations. Similar


to the other two algorithms, the original imagecan be recon-structed using dilations with

(17)

In fact, some of the s can be empty.Our algorithm can be viewed as a compromise between the

MST and MSD. In the MST, a disk is selected as a represen-tative disk if it is not completely contained in a representativedisk of a larger size. Therefore, there can be a great deal of over-lapping between representative disks of different sizes and theredundancy level can be high. In the MSD, a disk is selected ifit does not intersect with a representative disk of a larger size.The overlapping between representative disks of different sizesis completely eliminated. But the previous work [20] and ourexperiments show that the MSD typically uses more represen-tative disks than the MST. One explanation for this is that thenonoverlapping condition often forces us to use a larger numberof smaller disks to represent the same area which otherwise canbe represented by a smaller number of larger overlapping disks.In our algorithm, we allow some overlapping between repre-sentative disks of different sizes. A disk is selected if its centeris not in a representative disk of a larger size. This is a simpleand straightforward way to avoid severe overlapping betweendisks of different sizes and this scheme can be easily and ef-ficiently implemented. Compared to the MST, the overlappinglevel is much lower. Therefore the redundancy level is muchlower. Compared to the MSD, the moderate overlapping reducesthe number of representative disks needed.

Reinhardt and Higgins’s work [20] also concluded that thereconstruction cost of the MST is typically higher than that ofthe MSD. However, the cost measures used in their work arebased on the direct implementations of (6) and (10). By usingformulae (9), (13), and (17), all three algorithms will need exact

dilation operations. Only the number of union operations willdepend on the number of nonempty skeleton or center point sets.Therefore, the difference between the reconstruction costs of thethree algorithms is not very significant.

Consider the example in Fig. 1. We use the squarecentered at the origin as our unit “disk.” Under the MST, fourskeleton points are obtained as shown in Fig. 1(a). We use aninteger to represent a center of a representative disk of size.Such a point is said to be a representative point of order. Underthe MSD, ten representative points are used. They are shown inFig. 1(b). Under our new algorithm, five representative pointsare used. They are in Fig. 1(c). In this example, the MST usesthe fewest points and the MSD uses the most. Our algorithm is inthe middle. Compared to the MST, we do not use the disk of sizetwo, which overlaps heavily with other disks. As a result, twoadditional disks of size zero are used. Compared to the MSD, weuse a disk of size one that overlaps with the disk of size three.This disk of size one is responsible for reducing the number ofdisks of size zero from nine to three.

Another example is given in Fig. 2. For this image, the MSTuses five representative points; the MSD uses fifteen points; andour new algorithm uses four points. For this example, our algo-rithm uses the fewest points and the MSD uses the most. Com-pared to the MST, we use one less disk of size one, which hap-

Fig. 1. Representation example: (a) MST, (b) MSD, and (c) new algorithm.

Fig. 2. Representation example: (a) MST, (b) MSD, and (c) new algorithm.

pens to be redundant. Compared to the MSD, the use of the sizetwo disk eliminates the need for those size zero disks and onedisk of size one. But, one additional disk of size one, which isalso redundant, is used in a different area.

In these two examples, we have seen that the MST and ouralgorithm may use redundant representative disks. In fact, theMSD may also use redundant representative disks. For the MST,a minimal skeleton has been defined as a minimal set of repre-sentative points that can still completely represent the originalshape [18]. Similar minimal representations can be defined forthe MSD and our new algorithm. Clearly, a minimal representa-tion uses fewer representative points and has lower overlappinglevel. However, a minimal representation is not unique. There-fore, they in general are not appropriate for shape matching andother similar shape analysis applications.

One advantage of the MSD is that not only can a givenshape be represented as a union of certain disks, there is alsoan easy and natural way to divide these disks into a numberof groups, each of which represents a more natural shape part[19]. A similar decomposition procedure can also be developedfor our new representation algorithm. For an image, ourrepresentation algorithm produces a sequence of center pointsets: . For each center point set , weidentify all the connected components: .Each such connected component corresponds to a shapecomponent in . The union of all such componentsis .

Consider the shape in Fig. 3. The representative points pro-duced by our representation algorithm are shown in Fig. 3(a).After combining connected representative points of different or-ders together and dilating each connected component by a diskof the corresponding size, five shape components are generated.

is divided into and . is divided into and .Fig. 3(b) has and . Fig. 3(c) has ,

, and . These shape components are more natural thanthe individual disks. For this example, the decomposition resultbased on our representation algorithm is in fact very similar tothe one based on the MSD. If we apply the same procedure tothe representative points produced by the MSD, which are given


Fig. 3. Example of combining representative disks into larger shapecomponents.

in Fig. 3(d), five very similar shape components are generated.They are shown in Fig. 3(e)–(f).

Let us go back to the two earlier examples. If we apply theprocedure to the set of representative points produced by ourrepresentation algorithm in Fig. 1(c), five shape componentsare identified, one for each representative point. If we applythe procedure to the set of representative points producedby the MSD in Fig. 1(b), only two shape components areidentified. All the points of order zero are combined together.The latter seems to be a better interpretation of the originalshape. For the image in Fig. 2, the decomposition based on ourrepresentation algorithm has three shape components. Theycorrespond quite well to the natural structures of the shape. Thedecomposition based on the MSD has four shape components.Here, some shape components do not correspond very wellto the natural shape parts.

The shape components produced by the decomposition pro-cedure can be organized into a graph structure. Each componentis represented by a node in the graph. An edge is used to con-nect two nodes if the corresponding components are connectedto each other in the shape image. Other information about theshape and shape components can also be incorporated into therepresentation. Shape matching algorithms based on such struc-tural representations are under investigations.

III. PROPERTIES OF THEALGORITHM

Our algorithm is defined in a discrete space. We only use diskstructuring elements of discrete sizes. Clearly, under our algo-rithm, a shape representation is uniquely defined. Each centerpoint set is uniquely defined in terms of the given image and thehigher order center point sets. The corresponding representativedisks cover a portion of the given shape not covered by largerrepresentative disks. The size zero disks are selected to cover allthe points in the shape that are not covered by larger represen-tative disks. Therefore, the union of all the representative disksis the given shape. It is clear that our algorithm can be easilygeneralized to three-dimensional (3-D) spaces.

Our shape representation is translation invariant, providedthat the translation factor is an integer. However, our shape rep-resentation is in general not scaling and rotation invariant. In adiscrete space, a scaling operation by an arbitrary factor and arotation operation by an arbitrary degree are not well defined.Before our algorithm is applied to a shape image, both scalingand rotation normalization transforms should be performed. Itseems that if our algorithm is generalized to the continuous

Fig. 4. Triangle shape and four of its disk components.

space , then the resulting transform should be scaling androtation invariant.

We now describe some simple properties of our algorithm. Itis clear that all the center point sets are contained in the givenimage .

Proposition 1:

(18)

This follows from the assumption that contains the origin.The next proposition says that the center point sets of differentorders are not adjacent to each other. In fact, they are at leastcertain distances away from each other.

Proposition 2:

(19)

This follows directly from the definition. The next propositionsays that each center point set consists of line segments of zerowidth in the discrete sense.

Proposition 3:

(20)

Proof: If for some , then we have a point. Therefore, . Now,

. Since is not containedin any representative disks of size or larger and thereforenot contained in any representative disks of size or larger.Thus, should be selected asa representative disk of size . This contradicts the earlierconclusion that . Q.E.D.

In the MSD, small shape components do not always rep-resent small shape details defined by boundary lines. For ex-ample, consider the triangle shape and its four disk componentsin Fig. 4. The smallest component shown does not touch theboarder of the shape. In our new algorithm, all representative


Fig. 5. Two basic structuring elements and some discrete disks formed by them: (a)B ; (b)B ; (c) 2B; (d) 3B; and (e)4B.

disks contain boundary points of the given shape. For a repre-sentative point is a representative diskof size . The representative disk contains boundarypoints means that containsbackground points. So we have the following.

Proposition 4: If , then .Proof: If , then is the center of

a disk of size in . Since , is not containedin any representative disks of size or larger and thereforenot contained in any representative disks of size or larger.Therefore, should be selected as a representativedisk of size . This contradicts the assumption that

. Q.E.D.In the following, we describe a number of simple properties

of the new algorithm in relations to the MST and MSD. Thecomparison is motivated by the work of Reinhardt and Higgins[20]. Some definitions and results are similar to the ones in [20].We first compare our new algorithm with the MST. In order todescribe a condition for the two algorithms to be equivalent, weneed some definitions. For a skeleton subset, we define theoverlapping subset of

and (21)

The disjoint subset of is defined

(22)

Note that

(23)

This is the union of all maximal disks of size or larger in ,which is same as the union of all disks of size or larger in

. We know that contains the centers of all maximal disks ofsize in . Then, has those points of that are contained insome larger size disks in; while has those points of thatare not contained in any larger size disks in. For the image inFig. 2(a), and .

Proposition 5:

1) (24)

2) When (25)

3) (26)

Proof: 1) This follows directly from the definitions. 2)Since contains the centers of all disks of size,

point iff is a center of a disk of size andis not in a disk of size . Equivalently, is a center of a max-

imal disk of size and is not in a disk of size . This issaying that . 3) A point implies that is not inany disks of size one or larger. Since contains all the pointsof that are not in some disks of size one or larger, we musthave . Q.E.D.

This proposition says that the MST and our algorithm use thesame number of representative disks of size. The MST mayuse more representative disks of size . And our algorithmmay use more disks of size 0. For the image in Fig. 1, we have

.Proposition 6: For , if for ,

then

for (27)

and, when

(28)

Proof: We know that . Let us assume thatfor . A point iff is a

center of a disk of size and is not in any representative disksof size or larger for our algorithm. By assumption, theseare all the maximal disks of size or larger. Equivalently,

is a center of a maximal disk of sizeand is not in anymaximal disks of size or larger. By definition, we have

. Since , we have . The prooffor is similar. Q.E.D.


Fig. 6. Nine images used in the experiments: (a) teapot; (b) lamp; (c) telephone; (d) puzzle; (e) letters; (f) digits; (g) Lenna; (h) house; and (i) tree.

If Proposition 6 is true for , then we have the condi-tion for the MST and our algorithm to be equivalent. In gen-eral, this proposition provides a condition for the first few highorder center point sets from our algorithm to be same as the onesfrom the MST. When this happens, for the first unequal pair, theskeleton subset may contain more points than our center pointset. For the image in Fig. 2, we have and for

. We also have . This proposition indicatesthat the MST may use more certain large size disks than our al-gorithm under certain conditions.

Proposition 7: For , if

for all (29)

then

for all (30)

Proof: By definition, . From the given condition,for . By Proposition 6,

for . Q.E.D.

This proposition says that if a skeleton representation doesnot use certain large size disks, then our algorithm will not usedisks of those sizes either. For the image in Fig. 2, we have

. But if we have for all where, we can still have for some .

For the image in Fig. 1, we have and .We now compare our algorithm with the MSD. For a center

point set , we define the overlapping subset of

and

(31)

The disjoint subset of is defined

(32)

contains the center points of those representative disks ofsize that intersect with some representative disks of larger


sizes. contains the center points of those representativedisks of size that do not intersect with any representative disksof larger sizes. For the image in Fig. 2(c),and .

Proposition 8:

1) (33)

2) When (34)

3) (35)

The proof follows directly from the definitions and is omitted.This proposition indicates that the MSD and our algorithm usethe same number of representative disks of size. But our al-gorithm may use more disks of size . This seems to implythat our algorithm may use less small size disks. The experi-ments will show that our algorithm typically uses fewer repre-sentative disks than the MSD.

Proposition 9: For , if for ,then

for (36)

and, when

(37)

The proof is similar to that for Proposition 6 and is omitted.If Proposition 9 is true for , then we have the conditionfor the MSD and our algorithm to be equivalent. In general, thisproposition indicates that our algorithm may use more certainlarge size disks than the MSD under certain conditions. For theimage in Fig. 1, we have and for ,and .

Proposition 10: For , if

for all (38)

then

for all (39)

The proof is similar to that for Proposition 7 and is omitted.This proposition says that if our algorithm does not use certainlarge size disks, then the MSD will not use disks of those sizeseither. For the image in Fig. 2, we have . But ifwe have for all , where , wecan still have for some . For the image inFig. 1, we have , while .

IV. EXPERIMENTS

Our representation algorithm was tested with nine binaryshape images. To achieve better approximations to real disksand keep the size of unit disk down, we used two basic struc-turing elements and as given in Fig. 5 to define disksof different sizes. In our implementation, the unit disk isdefined to be , the size two disk is defined to be ,

TABLE INUMBER OF REPRESENTATIVEPOINTS USED BY THREE DIFFERENT

ALGORITHMS

TABLE IINUMBER OF BASIC MORPHOLOGICAL OPERATIONSUSED BY THREE

DIFFERENTALGORITHMS

and the size three disk is defined to be . Ingeneral, a disk of sizeis defined

if is odd,if is even.

(40)

Some of these disks are in Fig. 5. Note that with this definition,the formula for calculating skeleton subsets becomes

if is even,if is odd.

(41)

In fact, this is a generalized-step skeleton transform as definedin [32]. The formula (9) now becomes

if is odd

if is even(42)

This generalization can be easily applied to the MSD and ournew algorithm. Formulae in (13) and (17) can be generalizedsimilarly. Formulae in (11) and (15) need not be changed. Wejust need to note that

if is oddif is even

(43)


Fig. 7. Decomposition examples.

where is either or . Proposition 3 in Section III becomes

if is even and ,if is odd and .

(44)

All other propositions in Section III still hold without modifica-tions.

The nine images used in the experiments are given in Fig. 6.Three algorithms were implemented. They are the MST, theMSD, and our algorithm. For each algorithm, we calculatedthe number of representative points used for each shape image.These numbers are given in Table I. Comparing the MST withthe MSD, we can see that the number of representative points

used is always lower with the MST except in one case. TheMST allows representative disks of different sizes to overlap.Overlapping often allows a larger disk to cover an area of theoriginal shape that would otherwise require many smaller disks.Now compare our algorithm with the MST. In our algorithm, weonly use those disks whose centers are in the areas of the orig-inal shape that are not covered by the larger representative disks.This way, the overlapping level is significantly reduced. It is in-teresting that the number of representative points used by ouralgorithm is either very close to or lower than that used by theMST. By reducing overlapping, we very often lower the numberof representative disks needed, because excessive overlappingcan also lead to less efficient use of representative disks. Now


Fig. 8. Approximation examples: (a)–(c) MSD and (d)–(f) our new algorithm.

we compare our algorithm with the MSD. The number of repre-sentative points used by our algorithm is always lower than thatused by the MSD. Very often, they are significantly lower. Inthis case, the moderate overlapping between disks of differentsizes leads to more efficient use of representative disks. This isa major advantage of our algorithm.

We also calculated the number of basic morphological opera-tions used for each shape image. They are given in Table II. TheMST always uses the least number of basic morphological op-erations. The numbers used by our algorithm are slightly lowerthan those used by the MSD. The actual computational costs ofthese algorithms will depend on the underlying computer archi-tecture, the algorithms used to implement the basic morpholog-ical operations, and the types of shape images being processed.

Three more examples of using the decomposition proceduredescribed at the end of Section II are shown in Fig. 7. For eachshape, we show the representative points produced by our repre-sentation algorithm as well as the shape components formed bycombining connected representative points and performing dila-tion operations. The airplane shape is decomposed into 24 shapecomponents; the truck shape is decomposed into 27 compo-nents; the bird shape is decomposed into 22 components. Theseshape components seem to correspond to the natural structuresof the given shapes quite well. Severe overlapping between com-ponents is avoided and each shape component represents a sig-nificant part of the given shape that is not covered by other shapecomponents. The moderate overlapping between componentsof different size orders provides the flexibility that allows theoriginal shape to be accurately approximated using only a smallnumber of large components. The ability to generate flexibleand more natural shape components is another major advantageof our algorithm.

In Fig. 8, we have the approximations of the first three shapeimages from Fig. 6 using shape components generated by boththe MSD and our algorithm. The MST is not known as a struc-tural shape decomposition algorithm and there is no simple and

TABLE IIICOMPARISON BETWEEN THE MSD AND OUR ALGORITHM FOR

SHAPE APPROXIMATION

obvious way of combining representative disks into more mean-ingful shape components in the MST. Therefore, this experimentdoes not include the MST. For the teapot shape, we first use allthe shape components of the form , where isa connected component of the center point set. There are 14such components. The approximation is in Fig. 8(a). In Fig. 8(d),we have the approximation using all the components of the form

, where is a connected component of thecenter point set . There are 15 such components. The rep-resentation error for the MSD is 6.96%; while it is 5.58% forour algorithm. The representation error is calculated as the ratiobetween the number of points not represented and the numberof points in the original shape. We can see that our approxima-tion has much better visual quality than that of the MSD. Theapproximation by the MSD contains holes that do not exist inthe original shape. It also has rougher boundary curves. Our ap-proximation uses one more component. But even after removinga small component, our approximation should still have betterquality than the one produced by the MSD. We also have theapproximations for the lamp and phone shapes in Fig. 8. Thenumbers of components used and the values of representationerrors are given in Table III. They all indicate that our represen-tation scheme allows good approximations of the given shapesusing small numbers of shape components. With about the samenumbers of shape components, our approximations have betterquality than those produced by the MSD in both the visual andnumerical sense.


V. CONCLUSIONS

In this paper, we have developed a new morphological shaperepresentation algorithm that does not require searching. In ouralgorithm, a given shape is represented as a union of a numberof disks contained in the given shape. Overlapping is allowedbetween representative disks of different sizes. But, excessiveoverlapping between disks of different sizes is avoided. Theexperimental results have shown that this algorithm combinesthe advantages of the two major search-free morphologicalshape representation schemes. The number of representativedisks used by our algorithm for a given image is typically eithervery close to or lower than that used by the MST. It is muchlower than that used by the MSD. Compared to the MST, theoverlapping level between the representative disks of differentsizes in our algorithm is significantly reduced. A given shapeis decomposed into a number of modestly overlapping shapecomponents. These shape components seem to correspondbetter to the natural shape parts than those generated bythe MSD. A good approximation for a given shape can beconstructed using only a small number of major components.Compared to the MSD, our algorithm seems to produce betterquality approximations with similar number of components.Similar to both the MST and MSD, our algorithm is simpleand straightforward. It is also easy and efficient to implement.The representative disks selected have simple and well-definedmathematical characterizations. This algorithm can be usedto generate shape representations for structure-oriented shapeanalysis and processing applications. The algorithm can begeneralized to use nondisk shape structuring elements. It canalso be easily generalized to represent 3-D binary shapes.

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Jianning Xu received the B.S. degree in computerengineering from Harbin Institute of Technology,Harbin, China, in 1982, and the M.S. and Ph.D.degrees in computer science from Stevens Instituteof Technology, Hoboken, NJ, in 1984 and 1988,respectively.

In 1988, he joined the faculty of the ComputerScience Department, Rowan University (formerlyGlassboro State College), Glassboro, NJ, where hecurrently is an Associate Professor. His researchinterests include image processing, pattern recogni-

tion, and mathematical morphology.

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Efficient morphological shape representation with overlapping disk components