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Pergamon Pattern Recoonition, Vol. 28, No. 11, pp. 1783 1792, 1995

Elsevier Science Ltd Copyright 1995 Pattern Recognition Society

Printed in Great Britain. All rights reserved 0031-3203/95 $9.50+.00

0031-3203(95)00036-4

SHAPE DECOMPOSITION AND REPRESENTATION USING A RECURSIVE MORPHOLOGICAL OPERATION

DEMIN WANG,t VERONIQUE HAESE-COAT, and JOSEPH RONSIN:~ "t" IRISA/INRIA, Campus Universitaire de Beaulieu, 35042 Rennes Cedex, France

$ Laboratoire d'Automatique, Equipe Image, Institut National des Sciences Appliqures 35043 Rennes Cedex, France

(Received 10 May 1994; in revised forra 22 February 1995; received for publication 17 March 1995)

Abstract--This paper presents a recursive morphological operation developed in order to perform efficient shape representation. This operation uses a structuring element as a geometrical primitive to evaluate the shape of an object. It results in a set of loci of the translated structuring elements that are included in the object but which do not overlap. The analysis of its computational complexity shows that it is usually less time-consuming than morphological erosion. By using this operation, an object decomposition algorithm is then developed for shape representation. It decomposes an object into a union of simple and non- overlapping object components. The object is represented by the sizes and loci of its object components. This representation is information preserving, shift and scale invariant, and non-redundant. It has been compared with skeletons, morphological decomposition, chain codes and quatrees in terms of compression ability and image processing facility. Experimental results shows that it is very compact, especially if information loss is allowed. Because of the non-overlap between object components, many image processing tasks can be easily performed by directly using this shape representation.

Shape representation Object description Image decomposition Mathematical morphology Recursive algorithm Data compression Image coding

l. INTRODUCTION

Shape representation is a very important issue in digital image processing and computer vision. It pro- vides descriptions of binary image objects in a com- pressed form which can be used for the design of automatic image analysis, pattern recognition and computer vision systems, as well as for image coding techniques.

A good shape representation scheme should have the following properties: (i) information preserving: it should contain all the information concerning the image under consideration, and exact reconstruction of the original object should be possible. (ii) Math- ematically tractable: it should be efficient and easy to use for various image analysis and computer vision applications. This usually requires the representation to be invariant under translation and scaling. More- over, a hierarchical representation is often desired. (iii) Compact: it should be non-redundant and provide high data compression. Non-redundant representa- tion is defined as allowing a reconstruction of the original object, however, removal of any one of its elements would violate this reconstruction.

In recent years, a multitude of shape representation schemes based on different theories have been develop- ed. The schemes related to mathematical morphol- ogy t1'2"3~ include granulometric size distribution, skeletonization and morphological decomposition. Granulometric size distribution was conceived by Matheron ~I) as a descriptor of granularity or texture

within an image. Its derivative is often referred to as pattern spectrum. 141 Granulometric size distribution and pattern spectrum give good quantitative descrip- tions of shapes. They have been employed to study shape-size complexity 14) and to classify texture im- ages. tS) However, original objects cannot be recon- structed from these representations. Skeletonization was initially called medial axis transform where the skeletons were defined as the locus of maximal disks that can be inscribed inside objects. ~6) It has been proved that skeletons can be obtained by morphologi- cal operations. Iz) Skeletons give accurate representa- tions of shapes. Pitas and Venetsanopoulos ~71 have proposed a morphological decomposition algorithm for shape representation. This algorithm decomposes a binary object into a union of simple subsets by using a family of structuring elements of different sizes. The extension of this decomposition to multilevel images has been presented in references (8 and 9) for texture classification and segmentation, and in reference (10) for range image analysis. Another shape decomposi- tion algorithm based on mathematical morphology, has been proposed by Ronse and Macq." 11 Its prin- ciple is similar to that in reference (7). These shape decompositions are information preserving, invariant under translation and scaling.

In morphological skeletonization and shape decom- positions, erosions are used to extract representative points and dilations are used to reconstruct original images from these representative points. The points extracted by erosion are usually connected to each

1783

1784 D. WANG et al.

other, instead of being isolated. In the reconstruction procedure using dilation, these connected points are then replaced by structuring elements located at these points. Due to the resulting overlap between the struc- turing elements these representations have two draw- backs: firstly it is difficult to perform various image analysis tasks by directly using these representations, and secondly there is redundancy in these representa- tions. The globally minimal skeletonization proposed by Maragos and Schafer t12~ is a non-redundant repre- sentation. However, it eliminates only a part of the overlap. To remove all of the overlap, a new mor- phological operation must be developed to replace the erosions and dilations previously used.

This paper presents a recursive morphological oper- ation. A non-redundant shape representation, based on this operation, is then proposed. The rest of the paper is organized as follows. In Section 2, we first describe the recursive morphological operation. Then we study its properties and its computational complex- ity. In Section 3, the non-redundant shape representa- tion based on the new operation is presented. Its compactness, i.e. data compression capability, is studied and compared with other schemes. The ease with which this representation can be directly used in image processing is demonstrated. Finally, Section 4 summarizes the results of our research.

2. RECURSIVE MORPHOLOGICAL OPERATION

To obtain non-redundant shape representation, we have developed a recursive morphological operation. In this section, we first present the definition of the operation. Then, we analyse its properties and discuss its computational complexity.

2.1. Definition of the operation

In mathematical morphology, a discrete binary image is considered as a point set A defined on a dis- crete two-dimensional (2-D) rectangle region con- sisting of an M x N regularly spaced lattice. Let B denote the structuring element, which is usually a set of simple shape, and A\B denote the set differ- ence between A and B. We use A + x, instead of A x, to denote the translation of A by point x because the notation A + x will avoid confusion with the variables having subscripts which will be used in the paper. This notation has been used by Maragos 14~ and Dougherty~5,13~ to denote translated sets. The erosion and dilation of A by B are respectively denoted by A e B and A ~) B, and defined as: TM

AB= NA-x={x lB+x~_A} (1) xeB

AB= UA+x= UB+x, (2) xeB xeA

where x is the point in 2-D space Z 2. As indicated by equations (1) and (2), erosion of A by

B can be implemented by moving B over all points in

llIIIIIII llllllllTI Il l lITTI

\ A

A @ B

B

iiilllll- IIllllll l l l I l l J l

TIITTTII A Q B O B

Fig. 1. Erosion and dilation.

Z 2. At each point x, we examine whether B + x is included or not in A. The set of points corresponding to positive answers forms the erosion result. Dilation of A by B can be implemented by replacing each point xeA by the translated structuring element B + x and considering their union.

If representative points in a shape representation are extracted by erosion, and original images are recon- structed by dilation of the representative points with the same structuring element [as in the morphological skeletons and morphological decomposition pres- ented in references (12) and (7)], then the representative points may be adjacent to one another. In the recon- structed image, adjacent representative points are re- placed by overlapping structuring elements, as shown in Fig. 1. The overlap between the structuring elements leads to representation redundancy, that is, the re- moval of some representaive points may not alter the reconstruction results. For example, if we keep only the leftmost and rightmost points of A e B in Fig. 1 as the representative points, the dilation of these points is a union of two non-overlapping structuring elements and remains equal to (A e B) ~3 B.

We expect to represent an object by a few isolated representative points so that the reconstruction of the object is made by a union of non-overlapping structur- ing elements. Such representative points can be ob- tained by the following recursive procedure. Let structuring element B be translated point by a point in Z 2. At the first point (i, j), we examine whether B + (i, j) is included or not in object A. If not, B is translated to the next point and we repeat the examination. Other- wise, point (i,j) is taken as one of the representative points. Then B + (i,j) is subtracted from A, B is trans- lated to the n