Morphological representation of 2-D binary shapes using rectangular components

Abstract

The morphological skeleton transform is a shape representation scheme that decomposes a shape into a union of all maximal homothetics of a structuring element contained in the shape. In this paper, we develop an algorithm that generalizes the skeleton transform by allowing many different rectangles of different sizes and aspect ratios to be used as shape components. The shape components in our representations still have simple and well-defined mathematical characterizations. The representation is uniquely defined and the algorithm still is simple and efficient to implement. Experiments show that our representations use significantly less shape components than those produced by the skeleton transform. We also describe different ways to derive a new set of rectangular shape components with less overlapping from the original set of rectangles from our representation algorithm.

Introduction

Shape representation is an important problem in image processing and computer vision. In a vision system, shape representation is the basis for further shape-related processing and recognition. A good shape representation scheme makes it easier for a shape to be stored, transmitted, compared against, recognized, and learned. Certain properties of a shape representation scheme are desirable. A good shape representation should first provide an accurate and complete description of the given shape. The original shape should be allowed to be easily reconstructed. A good shape representation should also be compact. Efficient manipulation of the shape representation should be possible. More importantly, a good shape representation should be well defined. The representation should be generated according to simple, precise, and meaningful rules, instead of depending on some arbitrary decisions. A well-defined representation is more likely to capture the intrinsic characteristics of a given shape. We also want a representation to be easily computed. Computational efficiency is always a desirable feature in a computer imaging system.

A number of shape representation schemes have been developed over the years [1], [2], [3], [4]. One of the major schemes is structural shape description. In a structural shape description, a shape is described in terms of a number of shape components, or shape primitives, and the relationships among the components. Structural shape representation is suited for structure-based analysis and processing. Mathematical morphology is a shape-based approach to image processing [5], [6], [7]. It is no surprise that a number of morphological shape representation schemes have been developed. Most morphological shape representation schemes are region-based. That is, they produce structural descriptions for the given shapes. An important advantage of the morphological shape representation schemes is that basic morphological operations can be implemented very efficiently on many parallel image computers [8], [9], [10]. Another advantage of mathematical morphology is that it has a well-developed mathematical structure. Many properties of basic morphological operations have been developed that characterize the behavior of basic morphological operations. Therefore, it often is possible to give an intuitive interpretation to the effect of a morphological algorithm on a given image.

One of the leading morphological shape representation schemes is the morphological skeleton transform [11]. In the morphological skeleton transform, a given shape is decomposed into a union of all inscribable maximal “disks” of different sizes. A size n disk nB is a n-fold enlarged version of the basic structuring element B which in fact does not need to resemble a real disk. That is, the given shape is characterized in terms of the homothetics of the single basic structuring element B. One of the advantages of the skeleton transform is that each component has simple and well-defined mathematical characterizations. Each component is simply a maximal disk of a certain size. This definition is mathematically very simple and intuitively very easy to understand. The decomposition procedure is also simple, easy, and efficient to implement. A sequence of skeleton subsets can be obtained from a given shape using a series of basic morphological operations. Each skeleton subset contains centers of all maximal disks of a certain size. These skeleton subsets can be independently determined and they share some common operations. The morphological shape decomposition [12] is a scheme that decomposes a binary shape into a union of certain disks contained in the shape. The overlapping between disks of different sizes is eliminated. It has been shown [13] that the morphological skeleton transform typically uses fewer components than the morphological shape decomposition. In the morphological shape decomposition, shape components of different sizes are defined recursively and they are determined in a sequential order. Wang et al. [14] proposed a shape representation scheme using recursive morphological operations. Again, a given shape is decomposed into a union of certain disks contained in the shape. But the overlapping between the disks is completely eliminated. In this algorithm, each individual component is determined recursively. So it does not seem to be suitable for parallel implementations. According to their paper, this algorithm uses comparable numbers of components as the morphological shape decomposition. Therefore, this algorithm should typically use more components than the morphological skeleton transform.

One limitation of the morphological skeleton transform seems to be the fact that only one basic structuring element is used in the description of a given shape. If we could use more than one basic structuring elements, we might be able to construct a more efficient representation. Many efforts have been made by various researchers in this regard. Maragos [15] attempted to represent an image as a minimal union of translated and scaled patterns from a finite basic pattern class. It appears that the choice for the basic pattern set is not easy to make. Another difficulty seems to be the computational cost associated with determining the minimal union of maximal patterns. This basically is a search problem. Pitas and Venetsanopoulos [16] proposed an improvement over their earlier scheme by allowing the use of multiple structuring elements and minimal enclosing structuring elements. The time requirement of the algorithm increases as more structuring elements are allowed. The decision on how many and what kind of structuring elements should be used does not seem to be an easy one. A search-based shape representation scheme was proposed by Reinhardt and Higgins [17]. The basic representation scheme is similar to that of Pitas and Venetsanopoulos. But a search process is used to select the structuring elements used to form the shape components and the operations used to combine the shape components to rebuild the original shape. Xu [18] proposed a scheme that decomposes a shape into a collection of convex polygons that very often correspond quite well to the natural components of the shape. A search process is also used in this algorithm. Xu [19] also proposed a simple scheme that decomposes a shape into neck-free components that are roughly convex. A shape primitive generated by this scheme may not always be simple enough. Most of these schemes use various search procedures to select a representation. The shape components thus produced are more complex to characterize and the computational procedures are more time-consuming to implement. Another approach to handle the problem is to generalize the skeleton transform by first identifying all the maximal rectangles (or certain other elementary shapes with multiple parameters) contained in a given shape [20]. In general, there will be more redundant components than in the regular skeleton transform. A procedure can be developed to remove the redundant components by inspecting the rectangles in certain orders. However, when a different inspection order is used, a different minimal set of rectangles may be identified. Also, overlapping between the rectangles in a minimal set tends to be more severe than in the regular skeleton transform. The computational cost of this approach is also quite high. There are some other schemes [21], [22], [23], [24], [25]. They either are similar to one of the schemes that we have mentioned or emphasize other aspects of morphological shape representation.

In this paper, we develop a morphological shape representation scheme that decomposes a given shape into a union of rectangular shape components of different sizes and aspect ratios. That is, the given shape is characterized in terms of the shapes of many different rectangular structuring elements. Our rectangular shape components have simple and well-defined mathematical characterizations. They are maximal rectangles contained in a given shape under certain conditions. The decomposition procedure is simple, easy, and efficient to implement. No complicated search procedure is used in the algorithm. Our algorithm can be viewed as a generalized skeleton transform. It is a combination of a number of special and regular skeleton transforms using some special structuring elements. Compared to the regular skeleton transform, our algorithm uses smaller number of shape components to represent a given shape. Therefore, our algorithm produces more efficient representations, while maintaining most advantages of the regular skeleton transform. The paper is organized as follows. In Section 2, we present the representation algorithm that is formed by combining several skeleton transforms. Section 3 addresses the issue of reducing the overlapping between rectangular components produced in Section 2. Two overlapping reduction procedures are described. Section 4 contains examples of our algorithms applied to a number of shape images. Some comments are also made about the algorithms. Conclusions are given in Section 5.