Granulometry and size distribution

Mathematical Morphology and Its Application – Akira Asano ISSN 1858-1633 2005 ICTS 5 Fig. 5. Subwindows of [ n 2 + 1] pixels.

3.3.2. Average filter: The simplest average filter

operation, that is, the average of two pixel values x and y, is expressed by the minimum and the supremum, as follows: or as shown in Fig. 7. Fig. 6. Median expressed by the maximum and minimum. Fig. 7. Average expressed by the maximum and minimum.

4. GRANULOMETRY AND TEXTURE ANALYSIS

Texture is an image composed by repetitive appearance of small structures, for example surfaces of textiles, microscopic images of ores, etc. Texture analysis is a fundamental application of mathematical morphology, since it was developed for the analysis of minerals in ores. In this section, the concept of size in the sense of mathematical morphology and the idea of granulometry for measuring granularity of image objects are explained. An example of texture analysis applying granulometry by the author is also presented.

4.1. Granulometry and size distribution

Opening of image X with respect to structuring element B means residue of X obtained by removing smaller structures than B. It indicates that opening works as a filter to distinguish object structures by their sizes. Let 2B, 3B, . . . , be homothetic magnifications of the basic structuring element B. We then perform opening of X with respect to the homothetic structuring elements, and obtain the image sequence XB, X2B, X3B, . . . . In this sequence, XB is obtained by removing the regions smaller than B, X2B is obtained by removing the regions smaller than X2B, X3B is obtained by removing the regions smaller than 3B, . . . . If B is convex, it holds that X XB X2B X3B . . . . The size of rB is defined as r, and this sequence of opening is called granulometry [10]. We then calculate the ratio of the area for binary case or the sum of pixel values for gray scale case of XrB to that of the original X at each r. The area of an image is defined by the area occupied by an image object, i. e. the number of pixels composing an image object in the case of discrete images. The function from a size r to the corresponding ratio is monotonically decreasing, and unity when the size is zero. This function is called size distribution function. The size distribution function of size r indicates the area ratio of the regions whose sizes are greater than or equal to r. Information and Communication Technology Seminar, Vol. 1 No. 1, August 2005 ISSN 1858-1633 2005 ICTS 6 Fig. 8. Granulometry and size density function The r-times magnification of B, denoted rB, is usually defined in the context of mathematical morphology as follows: where {0} denotes a single dot at the origin. Let us consider a differentiation of the size distribution function. In the case of discrete sizes, it is equivalent to the area differences of the image pairs corresponding to adjacent sizes in X B , X 2B , X 3B , .... For example, the area difference between X 2B and X 3B corresponds to the part included inX 2B but excluded fromX 3B , that is, the part whose size is exactly 2. The sequence of the areas corresponding to each size exactly, derived as the above, is called pattern spectrum [13], and the sequence of the areas relative to the area of the original object is called size density function [14]. An example of granulometry and size density function is illustrated in Fig. 8. Size distribution function and size density function have similar properties to probability distribution function and probability density function, respectively, so that such names are given to these functions. Similarly to probability distributions, the average and the variance of size of objects in an image can be considered. Higher moments of a size distribution can be also defined, which are called granulometric moments, and image objects can be characterized using these moments [14–16] .

4.2. Application to texture analysis