Application of mathematical morphology to problems related to Image Segmentation

Transcription

1 Application of mathematical morphology to problems related to Image Segmentation Bala S Divakaruni and Sree T. Sunkara Department of Computer Science, Northern Illinois University DeKalb IL mrdivakaruni [at] gmail dot com, mail.teja [at] gmail dot com. Abstract - Mathematical morphology (MM) which is based on the algebra of nonlinear operators, operands on object shape and in many respects supersedes the linear algebraic system of convolution. And it has become a powerful tool in Digital Image Processing (DIP). It allows processing images to enhance fuzzy areas, segment objects, detect edges and analyze structures. Fuzzy systems are capable of representing diverse, non-exact, uncertain and inaccurate knowledge or information. They use qualifiers that are very close to human way of expressing knowledge, such as bright, medium dark, dark etc. They are based on fuzzy logic, which represents a powerful approach to decision making. Our approach is to apply mathematical morphological principals to some application of Image analysis like Segmentation. And also to apply Fuzzy mathematical morphological principles and compare the results with basic mathematical morphology to show how the fuzzy sets specifically utilized in MM have turned into a functional tool in DIP. Keywords: Mathematical morphology, Segmentation, Fuzzy Logic, fuzzy mathematical morphology. 1 INTRODUCTION Mathematical Morphology, created to characterize the physical and structural properties of diverse materials, relies on geometric, algebraic and topologic concepts as well as on the theory of sets. The main idea lying behind MM is assessing images geometric structures by overlapping small patterns, called structuring elements, in different parts of the same, [1]-[3]. The basic operations of MM are erosion and dilation. The remaining morphological operators are defined based on the combination of these two. MM performs many tasks- Pre- Processing, segmentation using object shape, and object quantification- better and more quickly than the standard approaches. Morphological tools are implemented in most advanced image analysis packages, and can be applied easily in a qualified way. Morphology is very often used in applications where shape of the objects and speed is an issue-for example, analysis of microscopic images, industrial inspection, etc. A primary difference between Traditional Mathematical Morphology (TMM) and Fuzzy Mathematical Morphology (FMM) is the structuring element. While TMM considers it as an image, FMM does so as a fuzzy set when evaluating morphological operators. 2 MORPHOLOGICAL METHODS 2.1 Morphological Operations In order to apply mathematical morphological transformations on an image we have to first understand the basic Mathematical Morphological Operations. Any morphological transformation is based on these operations. There are 4 basic or fundamental operations in Morphology. The 4 basic morphological operations are: Dilation, Erosion, Opening and Closing. These operations are considered on both types of images Binary and Gray- Scale images Morphological Operations on Binary Images Dilation on Binary Images Dilation adds pixels to the boundaries of objects in an image. The number of pixels added to the objects in an image depends on the size and shape of the structuring element used to process the image The sets of black and white pixels constitute a description of a binary image. Assume that only black pixels are considered, and the others are treated as white background. Now any of the operations Dilation, Erosion, Opening, Closing can be performed on this binary image. Dilation is represented by the symbol. Dilation combines two sets using vector addition. Also called as the Minkowski set addition.it is like (a,b)+(c,d)=(a+c,b+d). The dilation X B is the point set of all possible vector additions of pairs of elements, one from each of the sets X and B. X B = {p I : p=x + b, x X and b B } (2.7) The following sets illustrate dilation.

2 X = {(1,0), (1,1), (1,2), (2,2), (0,3), (0,4)} B = {(0,0), (1,0)} X B ={(1,0), (1,1), (1,2), (2,2), (0,3), (0,4), (2,0), (2,1), (2,2), (3,2), (1,3), (1,4)} In the above, X denotes the set of pixels that constitute an image (a binary image) B represents the structuring element. X B means that dilation of X with respect to the structuring element B. Figure 1 Binary dilation using point sets: (a) A binary image; (b) Structuring element; (c) Dilation; Above figure represents an image formed by the point set X.. A pixel is plotted at the positions represented by the point set X. Similarly the Structuring element B can be represented as shown below. Now when we apply dilation on X using the Structuring element B, i.e. X B the resulting image is as shown below: Thus X B is the point set representation of the image above. Erosion on Binary Images Erosion removes pixels from the boundaries of objects in an image. The number of pixels removed from the objects in an image depends on the size and shape of the structuring element used to process the image Erosion is represented by the symbol. Erosion combines two sets using vector subtraction. Erosion is also called as the Minkowski set subtraction. It is like (a,b)- (c,d)=(a-c,b-d). The erosion X B is the point set of all possible vector subtractions of pairs of elements, one from each of the sets X and B. X B = {x I : p=x - b, p X for every b B }(2.8) The following sets illustrate erosion. X ={(1,0),(1,1),(1,2),(0,3),(2,3),(1,3),(1,4),(3,3)} B = {(0,0),(1,0)} X B ={(0,3),(1,3),(2,3)} In the above, X B means that dilation of X with respect to the structuring element B. Figure 2: Binary erosion using point sets; (a) a binary image; (b) Structuring element; (c) Erosion; Above figure represents an image formed by the point set X. A pixel is plotted at the positions represented by the point set X. Similarly the Structuring element B can be represented as shown above. Now when we apply erosion on X using the Structuring element B, the resulting image is as shown. Thus X B is the point set representation of the image above. Note that the resulting pixels should lie in X. The erosion explained above corresponds to a 5Χ4 matrix Opening on a Binary Image Dilation and Erosion are not inverse transformations If an image is eroded and then dilated, the original image is not re obtained. Instead, the result is a simplified and less detailed version of the original image. Erosion followed by dilation creates an important morphological transformation called opening. Opening is represented by the symbol ο. The opening of an image X by the structuring element B is given as X ο B = (X B) B (2.9) Now let us look at the point set and the 1-D input signal working of opening. As in dilation and erosion let X denote the image point set and B is the structuring element. The following are the points on the plane they represent, these points when represented as pixels form a binary image. X ={(1,0),(1,1),(1,2),(0,3),(2,3),(1,3),(1,4),(3,3)} B = {(0,0),(1,0)} X B ={(0,3),(1,3),(2,3)} (X B) B ={(0,3),(1,3),(2,3),(3,3)} Hence X ο B = (X B) B ={(0,3),(1,3),(2,3),(3,3)} Figure 3 Binary opening using point sets: (a) A binary image; (b) Structuring element; (c) Opened Image; Closing on a Binary Image As already stated, Dilation and Erosion are not inverse transformations If an image is eroded and then dilated, the original image is not re obtained. Instead, the result is a simplified and less detailed version of the original image. Dilation followed by Erosion is closing. Closing of an image is represented by. The closing of an image X by the structuring element B is given as X B = (X B) B (2.10)

3 As in dilation and erosion let X denote the image point set and B is the structuring element. The following are the points on the plane they represent, these points when represented as pixels form a binary image. X = {(1,0), (1,1), (1,2), (2,2), (0,3), (0,4)} B = {(0,0), (1,0)} X B ={(1,0), (1,1), (1,2), (2,2), (0,3), (0,4), (2,0), (2,1), (2,2), (3,2), (1,3), (1,4)} (X B) B = {(1,2),(2,0),(2,1),(2,2), (3,2), (1,3), (1,4)} Hence X B = (X B) B ={(0,3),(1,3),(2,3),(3,3)} We look at the internal working of dilation on the Gray- Scale images. That is the array representation. Here every pixel or an element of an array has its own intensity values. If its intensity values are restricted to 0 and 1 then it turns out to be a Binary image. Figure 5 Gray-Scale dilation using a signal Figure 4 Binary opening using point sets: (a) A binary image; (b) Structuring element; (c) closed Image; The structuring element is moved across the image as before. The maximum value inside the structuring element is then set as the output Morphological Operations on Gray-Scale Images Dilation on Gray-Scale Images In Gray-Scale Dilation we deal with digital image functions of the form f(x,y) and b(x,y) where f(x,y) is the input image and b(x,y) is the structuring element,itself a sub image function. Here we assume that these functions are discrete. That is if Z denotes the set of real integers, the assumption is that (x, y) are integers from ZxZ and that f and b are functions that assign a gray-scale value to each distinct pair of coordinates (x,y). If the gray levels are also integers, then Z replaces R. Gray-Scale dilation of f by b, denoted as f b is defined as f b)(s,t) = max {f(s-x, t-y)+b(x,y) (s-x),(t-y) Df ; (x,y) Db} (2.11) Where Df, Db are the domains of f and b. Note that f and b are functions not sets. The condition that (s-x) and (t-y) have to be in the domain of f, and x and y have to be in the domain of b, is analogous to the condition in binary definition of Dilation, where 2 sets have to overlap by at least one element. The above equation can also be implemented by a simple 1-D function using only 1 variable. That is given as (f b)(s) = max {f (s-x)+b(x) (s-x) Df ; (x) Db} (2.12) Until now we have seen the definitions of dilation on binary image and the theoretical concepts. But for digital implementation a change in the formulae is given as (f b)(x) = f(x-y) (2.13) Figure 6: Explanation of Gray-Scale dilation using a structuring element Erosion on Gray-Scale Images (f b)(s,t) = min{f(s+x,t+y)-b(x,y) (s+x),(t+y) Df ; (x,y) Db} (2.14) Where Df, Db are the domains of f and b. Note that f and b are functions not sets. The condition that (s+x) and (t+y) have to be in the domain of f, and x and y have to be in the domain of b, is analogous to the condition in binary definition of Erosion,

4 where the structuring element has to be completely contained by the set being eroded. The above equation can also be implemented by a simple 1-D function using only 1 variable. That is given as (f b)(s) = min{f(s+x)-b(x) (s+x) Df ; (x) Db} (2.15) Now let us first look at the internal working of erosion on Gray-Scale images. Figure 10: Gray-Scale Opening using a signal Closing a Gray-Scale Image Closing a gray-scale image is similar to opening a grayscale image. Let us look at the internal working of closing on Gray-Scale images. That is the array representation. Figure 7: Gray-Scale erosion using a signal Having discussed 1-D dilation, we now study the effect of the 2-d dilation where both the image and structuring element are 2-Dimensional. Figure 11: Gray-Scale Closing using a signal The above is a 1-D matrix. However in real implementations the size of the image will be 256X 256. Figure 8: different structuring elements used in 2-D The output can be as below Input Image Output Image Figure 9: Explanation of Gray-Scale erosion using a structuring element Opening a Gray-Scale Image Opening a gray-scale image is eroding a gray-scale image and then dilating it. Let us look at the internal working of opening on Gray-Scale images. That is the array representation. Here every pixel or an element of an array has its own intensity values. 3 IMAGE SEGMENTATION Image Segmentation is one of the applications of Digital Image Processing which refers to the process of extracting the required features from the image. 3.1 Definition Segmentation sub-divides an image into its constituent regions or objects. The level to which segmentation is to be carried out depends on the problem being solved. 3.2 Standard Techniques There are many standard techniques which are available for segmentation. Thresholding, Histogram Based Segmentation, Clustering. 3.3 Morphological Segmentation Morphological Segmentation uses only simple mathematical morphological operators to segment the image. The results produced by morphological segmentation are as satisfactory as the standard techniques used for segmentation. 4 FUZZY SYSTEMS Fuzzy systems are capable of representing diverse, nonexact, uncertain and inaccurate knowledge or information. They use qualifiers that are very close to human way of expressing knowledge, such as bright, medium dark, dark etc. Fuzzy systems can represent knowledge from complex

5 knowledge and even knowledge from contradictory sources. They are based on fuzzy logic, which represents a powerful approach to decision making. 4.1 FUZZY logic Fuzzy logic has been developed to overcome the obvious limitations of numerical or crisp representation of information. Consider the use of knowledge represented by production rule for recognition of balls; using the production rule, the knowledge about balls may be represented as if circular then ball. If the object in a two-dimensional image is considered circular then it may represent a ball. Our experience with balls, however, says that they are usually close to, but not perfectly, circular. Thus, it is necessary to define some circularity threshold so that all reasonably circular objects from our set of objects are labeled as balls. Here is the fundamental problem of crisp descriptions: how circular must an object be to be considered circular? If humans represent such knowledge, the rule for ball circularity may look like if circularity is HIGH then object is a ball with HIGH confidence. Clearly, high circularity is a preferred property of balls. Such knowledge representation is very close to common sense representation of knowledge, with no need for exact specification of the circularity/non-circularity threshold. Fuzzy rules are of the form if X is A then Y is B, Where X and Y represent some properties and A and B are linguistic variables. Fuzzy logic can be used to solve object recognition and other decision-making tasks, among others. 4.2 FUZZY Reasoning Fuzzy reasoning is performed in the context of a fuzzy system model that consists of control, solution, and working data variables; fuzzy sets; hedges; fuzzy rules; and a control mechanism. In fuzzy reasoning, information carried in individual fuzzy sets is combined to make a decision. The functional relationship determining the degree of membership in related fuzzy regions is called the method of composition and results in definition of a fuzzy solution space. To arrive at the decision, de-fuzzification is performed. Processes of composition and de-fuzzification form the basis of fuzzy reasoning. 4.3 FUZZY sets and FUZZY membership functions A fuzzy set S in a fuzzy space X is a set of ordered pairs (2.30) Where represents the grade of membership of x in S. The range of the membership function is a subset of non-negative real numbers whose supremum is finite. For convenience, a unit supremum is widely used (2.31) 4.4 Fuzzy Mathematical Morphology Unlike binary images, when working with gray level images, Boolean logic cannot be applied. As a consequence, it is necessary to consider fuzzy logic. Every level of gray is associated with a value between zero and one (See Table 4). To this end, a continuous and strictly decreasing function :{0,1,2,...,255} [0,1], called fuzzy function, is defined, which, when composed with the image function, results in a fuzzification of the image, in other words: f : [0,1]. Table 1 Categories Category Truth value Absolutely Black 0 Almost Black 0.1 Quite Black 0.2 Somewhat Black 0.3 More Black than White 0.4 As White as Black 0.5 More White than Black 0.6 Somewhat White 0.7 Quite White 0.8 Almost White 0.9 Absolutely White 1 Traditional morphology relies on the theory of sets; fuzzy morphological operators can be defined by means of fuzzy logic. Let A and B belong to the set of images parts, then: (2.32) Where I denotes the binary implication. The fuzzy erosion of an image f can be defined by a structuring element B, in a point x like: (2.33) In a similar way, being A and B part of f set of parts, it is known that:

6 (2.34) Where C is the binary conjunction. Then, the fuzzy dilation of an image f can be defined by a structuring element B, in a point x like: (2.35) Following the steps of the morphological theory, fuzzy opening is described as: and fuzzy closing as: Fuzzy Top-Hat by opening as: (2.36) (2.37) (2.38) Lastly, fuzzy Top-Hat by closing is defined as: (2.39) Structuring element The main difference between Traditional Mathematical Morphology and Fuzzy Mathematical Morphology is the way in which the structuring element is regarded. TMM considers it with an image; i.e., the structuring element is a B: Dom B {0,1,2,...,255}function. Conversely, FMM considers it as a fuzzy set in which the rules mentioned in Table 1 are met. A quick way to obtain fuzzy structuring elements is to compose function B with the fuzzy function elements becomes an easy task. 4.5 Proposed method. By so doing, building fuzzy structuring The proposed method follows these steps: Step 1: Image Fuzzification. Step 2: Image fuzzy dilation. Step 3: Calculation of fuzzy Top-Hat transform. Step 4: Image Defuzzification. Step 5: Visualization. Step 1: To be able to operate with fuzzy logic in gray level images, the first step is to fuzzify the image. This consists in taking the values of each pixel from the original image to values between 0 and 1. There are several techniques that allow so. The one employed in this work was the sigmoid function: (2.41) If any value is below zero, it is assigned a zero value. Step 3: The Fuzzy Top-Hat transform was calculated. The objective is to extract the locally brilliant elements from the image. To do so, fuzzy Top-Hat by opening was used. The purpose is to obtain the graph peak representing the object to be segmented (brilliant object). The strategy is to create a new image with the irrelevant information, that is to say, to eliminate the peak applying an opening to the original image. Then, by subtracting it from the original image, a new one is obtained built from the information of interest. Step 4: To achieve defuzzification, the inverse function of the sigmoid function was used. Step 5: Visualization of the segmented structures. 5 RESULTS AND DISCUSSION Medical imaging constitutes an ideal field to apply fuzzy morphology techniques. The segmentation of tree structure images such as retinal angiographic images is not simple if TMM traditional methods are employed; the reason being that lines are fuzzy and their contours not sharp. The application of the above described algorithm to this type of images enables lines enhancement and optimum results. As an example of its potential applications, we propose the use of fuzzy morphological operators to segment branching images. The algorithm was applied to 50 retinal angiographic images. This type of images accounts for a great deal of noise which leads to notorious difficulty when it comes to applying traditional segmentation techniques. When processing these images with the proposed algorithm, the lines of interest were well segmented. Below, several images to which traditional operations for binary, gray-scale images and fuzzy operators were applied using Kleene-Dienes conjunction are shown. The efficiency of the proposed method can be clearly appreciated. (2.40) Step 2: The image was dilated by means of Kleene- Dienes conjunction. The structuring element used is 5 5 in size and cone shaped. Its values range from 0 to 1, and it is obtained from the following formula:

7 (d) (e) Figure 12: (a) Original Images; (b) binary dilation; (c) binary erosion; (d) binary opening; (e) binary closing; (d) (e) (f) Figure 13; (a) Original Images; (b) Gray-scale dilation; (c) gray-scale erosion; (d) gray-scale opening; (e) gray-scale closing; (f) gray-scale segmentation; (d) Figure 14; (a) Original Images; (b) Fuzzy Top-Hat; (c) Binarization; (d) Final Image; 6 CONCLUSIONS Non-linear techniques have proved to be more power full. One such technique is used by Mathematical Morphology. MM deals with only non-linear techniques. We have successfully implemented these non-linear techniques. Results obtained by Basic Mathematical Morphology were impressive (We could detect the microscopic growth/ calcium deposition around the metallic rod which has been implanted in the ankle region). Fuzzy Mathematical Morphology constitutes a valid tool to segment tree-structure images such as angiographic images, in which the edges of the blood vessels are not clearly cut. These operators constitute a considerable contribution to Digital Image Processing. Even though the mathematical bases for these techniques are complex, their implementation is simple, quick and easier on the user. We present an algorithm providing a far more efficient way of recognizing ramifications in angiographic retinal images as compared with the segmentation achieved by traditional DIP techniques, Lines segmentation in this kind of images is complex due to the fuzzy areas which hinder object extraction. The method herein proposed offers an efficient segmentation applicable to similar images, thereby facilitating professionals work by reducing analysis subjectivity. This method appears to be a robust approach and a novel contribution to medical images quantification. 7 Acknowledgements We thank Vimala Gayathri Divakaruni and Dr. Rama Murthy Suri (GRIET) for their support and contribution. 8 REFERENCES [1] J. Serra, Image Analysis and Mathematical Morphology, London Ed. Academic Press, [2] Binary Image Processing Ritterand Wilson, [3] Extensions to Gray Scale Images from: Papers by Jonesand Svalbe, 1994, Park and Chin1995, Aneli, [4] Current work in the theory and Applications of Morphological Image Processing by Rosenfel, [5] T. Deng and H. Heijmans, Grey-scale Morphology Based on Fuzzy Logic, Journal of Mathematical Imaging and Vision, Springer Netherlands, vol. 16, no. 2, pp , [6] R. Gonzalez and R. Woods, Digital Image Processing, Editorial Adison Wesley, [7] E. Dougherty, An Introduction to Morphologic Image Processing, SPIE, Washington, [8] Overview of Binary and Gray Scale Morphology by Basart and Gonzalez, [9] F. Zana and J.C. Klein, Segmentation of Vessel- Like Patterns using Mathematical Morphology and Curvature Evaluation, IEEE Transactions on Image Processing, vol.10, no.7, pp , [10] SDC Morphology Toolbox for MATLAB 5. User s Guide. SDC Information Systems, 2001.