Image thresholding by minimizing the measures of fuzzines

-This paper introduces a new image thresholding method based on minimizing the measures of fuzziness of an input image. The membership function in the thresholding method is used to denote the characteristic relationship between a pixel and its belonging region (the object or the background). In addition, based on the measure of fuzziness, a fuzzy range is defined to find the adequate threshold value within this range. The principle of the method is easy to understand and it can be directly extended to multilevel thresholding. The effectiveness of the new method is illustrated by using the test images of having various types of histograms. The experimental results indicate that the proposed method has demonstrated good performance in bilevel and trilevel thresholding. Image thresholding Measure of fuzziness Fuzzy membership function I. I N T R O D U C T I O N Image thresholding which extracts the object from the background in an input image is one of the most common applications in image analysis. For example, in automatic recognition of machine printed or handwritten texts, in shape recognition of objects, and in image enhancement, thresholding is a necessary step for image preprocessing. Among the image thresholding methods, bilevel thresholding separates the pixels of an image into two regions (i.e. the object and the background); one region contains pixels with gray values smaller than the threshold value and the other contains pixels with gray values larger than the threshold value. Further, if the pixels of an image are divided into more than two regions, this is called multilevel thresholding. In general, the threshold is located at the obvious and deep valley of the histogram. However, when the valley is not so obvious, it is very difficult to determine the threshold. During the past decade, many research studies have been devoted to the problem of selecting the appropriate threshold value. The survey of these papers can be seen in the literature31-3) Fuzzy set theory has been applied to image thresholding to partition the image space into meaningful regions by minimizing the measure of fuzziness of the image. The measurement can be expressed by terms such as entropy, {4) index of fuzziness, ~5) and index of nonfuzziness36) The "ent ropy" involves using Shannon's function to measure the fuzziness of an image so that the threshold can be determined by minimizing the entropy measure. It is very different from the classical entropy measure which measures t Author to whom correspondence should be addressed. probabil ist ic information. The index of fuzziness represents the average amount of fuzziness in an image by measuring the distance between the gray-level image and its near crisp (binary) version. The index of nonfuzziness indicates the average amount of nonfuzziness (crispness) in an image by taking the absolute difference between the crisp version and its complement. In addition, Pal and Rosenfeld ~7) developed an algorithm based on minimizing the compactness of fuzziness to obtain the fuzzy and nonfuzzy versions of an ill-defined image such that the appropriate nonfuzzy threshold can be chosen. They used some fuzzy geometric properties, i.e. the area and the perimeter of an fuzzy image, to obtain the measure of compactness. The effectiveness of the method has been illustrated by using two input images of bimodal and unimodal histograms. Another measurement, which is called the index of area converge (IOAC), ts) has been applied to select the threshold by finding the local minima of the IOAC. Since both the measures of compactness and the IOAC involve the spatial information of an image, they need a long time to compute the perimeter of the fuzzy plane. In this paper, based on the concept of fuzzy set, an effective thresholding method is proposed. Given a certain threshold value, the membership function of a pixel is defined by the absolute difference between the gray level and the average gray level of its belonging region (i.e. the object or the background). The larger the absolute difference is, the smaller the membership value becomes. It is expected that the membership value of each pixel in the input image is as large as possible. In addition, two measures of fuzziness are proposed to indicate the fuzziness of an image. The optimal threshold can then be effectively determined by minimizing the measure of fuzziness of an image. The performance of the proposed approach is compared