Binary Shape Characterization using Morphological Boundary Class Distribution Functions

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1 Binary Shape Characterization using Morphological Boundary Class Distribution Functions Marcin Iwanowski Institute of Control and Industrial Electronics, Warsaw University of Technology, ul.koszykowa 75, Warszawa POLAND; Summary. In the paper a new method for binary shape characterization is proposed. It is based on the analysis of binary image pixels belonging to the internal boundary of an object performed by means of the mathematical morphology. By using morphological operators, internal boundary pixels are classified into three groups. This classification is performed for increasing sizes of structuring elements used. The changes within the class assignments are described by the boundary class distribution functions. These functions may be used as features in the pattern recognition process. 1 Introduction In the process of binary shape recognition the shape features are widely used. In this paper new way of describing the binary shape is proposed. It is based on the mathematical morphology [7, 2]. By means of morphological operators, in the proposed method, pixels belonging to the internal object 1 boundary are classified into one of three spectral classes. The internal boundary is obtained here as the difference between image and its erosion. Every pixel belonging to the boundary is classified into one of the following classes: 1. Core boundary pixels - all pixels which have within their neighborhood at least one pixel of eroded object (referred to as core region). For regular shapes most of (or even all) boundary pixels belong to this group. For more complex shapes growing number of boundary pixels belong to the next two classes. 2. Corridors - pixels which are not core boundary pixels and which connect two core regions. As a result of the erosion a single object can be split into two or more objects (imagine e.g. eroded shape of a digit 8 with filled holes) The boundary pixels lying between these spitted areas are classified as corridors. 1 Object is defined as a single connected component of pixels of value 1.

2 2 Marcin Iwanowski 3. Branches - all the other boundary pixels. This class refers to all regions which do not connect multiple objects after erosion, but are too thin and too long to be classified as core boundary. All these classes are shown on Fig. 1. The internal boundary in this example consists of all pixels marked as a, b and c and was computed using the structuring element containing all 8-neighbors and the pixel itself aaaaa...aaaaabb a111a...bbbbba111a a111a...a111aab a111acccccccca1111ab a111acccccccca111aa a111a...a111a a111a...bbbba111aab a111a...bbbba1111a aaaaa...aaaaaab Fig. 1. Test shape (left) and four types of boundary pixels (right): a - core boundary, b - branches, c - corridors. The proposed method classifies each internal boundary pixel to the appropriate class. The only parameter used in the method used is the structuring element of the morphological operators. Depending on this element, borders of various shapes and thicknesses can be obtained. To characterize the shape, the percentage of pixels belonging to a given class among all the pixels is computed. The proposed method is based on the assumption that the assignment of pixels to one of three classes depends also on the form of the structuring element used. For structuring elements of increasing sizes (i.e. covering pixel s neighborhood of increasing radius) the classification of boundary pixels changes differently for different shapes. Thus, the distribution of number of pixels belonging to each class for increasing sizes of the structuring element offers detailed shape description. The functions introduced in the paper are called boundary class distribution functions. They may be used as feature vector for shape description and recognition. The paper is organized as follows. Section 2 contains the basic notions. Section 3 describes the pixel classification method. Section 4 introduces the boundary pixels distribution functions and discusses the choice of the structuring element. Section 5 presents some results. Finally, section 6 concludes the paper. 2 Basics 2.1 Internal boundary of binary object The internal boundary is defined as the set of pixels belonging to the object which have within their neighborhoods at least one pixel which does not belong

3 Shape Characterization using Morphological Distribution Functions 3 to the object. In the proposed method this boundary is computed using the morphological gradient which is one of the most widely known morphological operators [7, 2]. In case of the binary images the word gradient is usually replaced by contour. In the proposed method, the morphological gradient (contour) by erosion has been applied. It is defined as: ρ B (X) = X \ (X B) (1) were X B stands for the erosion of binary image X with structuring element B. The contour line obtained by using the Eq. 1 is the internal contour of the object X. Usually the gradient is computed with the elementary or directional structuring element. In the first case the closest neighborhood in a given grid is considered. It results in either 4- or 8-pixel neighborhood. In case of the directional structuring element, the neighbor(s) located only in a given direction are taken into account. Apart from these traditional structuring elements also the thick gradient has also been defined in the literature [2]. In this case wider neighborhood is considered. The thickness and shape of these boundaries is defined by the size and shape of the structuring element. This feature of thick gradient makes it useful for shape analysis in the proposed method. 2.2 Morphological reconstruction Morphological reconstruction is the operation which allows extracting from the binary image only given objects (connected components). These objects are indicated using the supplementary image called marker image. Morphological reconstruction is defined as a series of geodesic (or conditional) dilations which are performed until idempotence of dilated image is reached. The morphological reconstruction (by dilation) is defined as: R Y (X) = (...(((X B) Y ) B) Y )... Y ) (2) }{{} k times where X B stands for the dilation of X with structuring element B and k is the lowest number of iterations after which the resulting image stops to change. Image X used is referred to as the marker image because it contains markers which indicated object on the second image - Y. The latter is called the mask image. The marker image Xis reconstructed according to the content of the mask Y, which remains unchanged. The morphological reconstruction can be computed either using the above definition or using specialized algorithms, which provides us with much faster computation of this operator [3]. 2.3 Anchored skeletonization Skeletonization is used in the proposed methodology to find the corridor areas i.e. these areas that connect two core regions. The way of performing skele-

4 4 Marcin Iwanowski tonization is based on the notion of simple (or deletable) pixel [6] i.e. such a pixel that its removal does not change the homotopy of the binary image. A foreground pixel p belonging to the image X is simple if and only if it satisfies the following three conditions: 1. N G (p) X, 2. N G (p) X C, 3. S CC G (N 8 (p) X) such that N G (p) X S, where X C stands for the complement of image X. N G and N G represents the closest neighborhood of foreground and background pixels respectively. Due to connectivity paradox, different connectivity should be used for the foreground and for the background. So either N G = N 8 represents 8 closest neighbors (horizontal, vertical and diagonal one) and N G = N 4 represents 4 closest neighbors (without diagonal one), or inversely. Function CC G used in the above equation returns the set of G-connected components of its argument. The fast computation of simpleness can be achieved using the look-up table for the neighborhood configurations. By the successive removal of simple pixels the image is thinned. Image obtained by this thinning performed till idempotence is a skeleton which is also referred to as homotopic marking. In order to get symmetrical thinning, usually the operation is performed in two stage iterative process. In the stage phase the simple pixels are detected within the whole image (but not yet removed). The removal is performed in the second stage. These two stages are performed iteratively until idempotence is reached. In order to have the possibility to control the thinning process, the notion of anchor pixels has been introduced [5]. These pixels are defined separately and - by definition - cannot be removed during the thinning process even if they are simple. The anchored skeletonization requires thus two input images - the image to be thinned and the image containing anchor pixels referred to as the anchor image. 3 Boundary pixel classification The pixels belonging to the internal contour defined by the Eq. 1 are classified into three spatial classes [1]: core boundary, isolated region, corridor and branch. First, by using the morphological erosion with the given structuring element, the image object is shrinking. The shrunk areas of object will be referred to as core regions. The difference between the initial object and core area is the internal boundary of the object. Boundary pixels are further classified into three groups. The first of them is the locus of all pixels that are neighbors of the core region. Such pixels are classified as core boundary pixels. In fact core boundary pixels can be detected as the difference of input image opening (i.e. erosion followed by dilation) and erosion. It can be defined as the following function: f cb (X,B) = ( (X B) B T) \ (X B) (3)

5 Shape Characterization using Morphological Distribution Functions 5 where X stands for initial image, and B T = { p : p B} for the transposition of structuring element B. f cb (X,B) stands for the function which transforms binary image X to the image containing core boundary pixels using the structuring element B. By subtracting core boundary from from the internal contour, the remaining part of the contour (rest of the boundary) is obtained: f rob (X,B) = X \ ((X B) f cb (X,B)) (4) The pixels belonging to this remaining part are further classified into two classes. To the first class belongs all regions that are stretched between at least two core areas. They are corridors joining them. Pixels that are not corridors are simply branches of the core regions. In order to find corridors, the anchored skeletonization is applied. The motivation for applying this transform was the fact that, when performing the anchored skeletonization by removal of simple pixels of the input image with the core areas used as anchors, the parts of input image that connects core region are detected. Since however, the result of skeletonization is one-pixel thick line, it does not cover the whole corridor areas. In order to the complete areas, the reconstruction is used. The whole operation can be expressed by the following expression: f co (X,B) = R frob (X,B)(Skel(X,(X B)) f rob (X,B)) (5) where Skel(X 1,X 2 ) stands for the anchored skeletonization of the image X 1 with anchor image X 2, and R represents the reconstruction operator according to the Eq. 2. Using the equation 5, the second class - corridors - is detected. The remaining part of internal contour are the branches - the third class: f br (X,B) = X \ ((X B) f cb (X,B) f co (X,B)) (6) 4 Boundary pixel classes distribution functions The number of pixels belonging to each class depends, first of all, on the shape of the object being analyzed. But is also strongly depends on the structuring element B used - on its shape and size. Since the method is based on the classification of the internal boundary pixels, the way of obtaining this boundary (contour) is of primary importance. Usual way of computing the boundary makes use of 4- or 8-connected elementary structuring element, which consists of a central pixel plus 4 (horizontal and vertical) or 8 (horizontal, vertical and diagonal) closest neighbors respectively. To get more information on the shape the thick gradient approach [2] is more convenient in this case. Structuring element in this approach consist of neighboring pixels covering wider pixel neighborhood, usually a neighborhood of a given radius. Let then

6 6 Marcin Iwanowski B (n) be the neighborhood of a radius n i.e. set of the all the closest (to the central point) pixels which are not farther than n. The structuring element B (n) can be defined in various ways. The simplest (and fastest) is based on superposition by successive dilations of n elementary structuring elements B (n) = B B... B where B stands for an elementary structuring element. Since B can be either 4- or 8-connected, the final shape of B (n) can vary depending on it. The distance formulation which allows defining the radius is different for both cases. In the case of 4-connected B, the B (n) contains the pixels which are within the circle of radius n defined on the basis of the city-block distance. In the case of 8-connected B the distance is implied by the max-norm. In order to get the neighborhood of radius n according to the Euclidean distance the superposition by dilations cannot be used and the structuring element B (n) have to be computed individually for every n. This yields to slower computation but, on the other hand, gives the neighborhood closer to usual way of considering (which is based on Euclidean distance). The classification of boundary pixels using the series of structuring elements B (n) for increasing n allows obtaining the boundary class distribution functions for each boundary class. They are defined as: D(s) = f(x,b (s) ) (7) where. stands for the number of pixels of the argument. There are three functions defined according to the Eq. 7: core boundary distribution D cb for f f cb, where f cb is computed using the Eq. 3; corridors distribution D co for f f co (Eq. 5) and branches distribution D br for f f br (Eq. 6). Alternatively the distribution functions can be defined using the relative number of pixels of each class, i.e. the ratio of the pixels of given class to the total number of pixels of the internal boundary: D (s) = f(x,b(s) ) ρ B (s) (X) (8) where ρ stand for the internal boundary according to the Eq Results Figures 2 and 3 show two test figures: spades and club and their class distribution functions computed according to the Eq. 7. The classification was computed using the structuring element which approximated a disc of a given radius defined by Euclidean distance. The radii from 1 to 34 was taken into account. The distribution functions, especially branch and corridor ones are visibly different for both shapes. Thanks to that, they may be used for a shape classification. For some shapes only one function is sufficient to differentiate the shapes. In Fig. 4 the branch class distribution functions of another four test shapes

7 Shape Characterization using Morphological Distribution Functions 7 (a) numer of pixels belonging to each class branches core boundary corridors size of the structuring element (b) Fig. 2. Test shape spades (a) and its distribution functions (b). number of pixels belonging to each class branches core boundary corridors (a) size of the structuring element (b) Fig. 3. Test shape club (a) and its distribution functions (b). are shown. Contrary to the previous example, they was computed according to the Eq. 8 and presents the ratio of the pixels belonging to the branch class to the number of all internal boundary pixels. Similiarily to the previous example the structuring covering the neighborhood of a given radius according the the Euclidean distance was applied. Distribution functions for each shape varies from one shape to the other, what makes them a good feature for shape recognition. branch pixel coefficient hand rabbit cat hawk 0.1 (a) size of the structuring element (b) Fig. 4. Four test shapes (a) and their branch class distribution functions (b).

8 8 Marcin Iwanowski 6 Conclusions In the paper the morphological method for binary shape description was proposed. It is based on the spatial classification of the boundary pixels into three groups: core boundary, corridors and branches. The classification method is based on the morphological operators of erosion, dilation and reconstruction as well as the anchored skeletonization. The parameter of the classifier is the structuring element used. Depending on the choice of this element various classification results are obtained. By computing the number of pixels belonging to each class as a function of the size of a structuring element, the boundary pixel class distribution functions are obtained. These functions characterize the shape and allow differentiating various shapes, what was also shown in the paper on some examples. Owing to that class distribution functions may be used as features for shape representation and pattern recognition. References 1. M.Iwanowski, Morphological Boundary Pixel Classification, Prof. of IEEE EU- ROCON Conference, Warsaw P.Soille, Morphological image analysis, Springer Verlag, 1999, L.Vincent, Morphological grayscale reconstruction in image analysis: applications and efficient algorithms, IEEE Trans. on Image Processing, Vol.2, No.2, April J.Serra, L.Vincent, An overview of morphological filtering, Circuit systems Signal Processing, 11(1), L. Vincent, Efficient computation of various types of skeletons, In: Loew, M., ed. Medical Imaging V: Image Processing. Volume SPIE-1445 (1991) pp T.Kong, A.Rosenfeld, Digital topology: Introduction and survey,computer Vision, Graphics, and Image Processing, 1989, vol.48, pp J.Serra, Image analysis and mathematical morphology, vol.1, Academic Press, 1983.

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