A novel fuzzy entropy approach to image enhancement and thresholding
Introduction
In the field of image processing, people have to deal with many ambiguous situations. Ambiguity caused by projecting a 3-D object into a 2-D image or digitizing analog pictures into digital images, and the uncertainty related to boundaries and nonhomogeneous regions are very common. Some of definitions, such as edges, contrast, enhancement, etc., are fuzzy as well. Fuzzy set theory is a useful mathematical tool for handling the ambiguity or uncertainty.
However, selecting the fuzzy region of membership function is a fundamental and important task. In order to apply the fuzzy theory, most researchers use a predetermined approach. For instance, multi-level segmentation is determined by many local maxima of fuzzy entropy with a predetermined window (fuzzy region) which is shifted along the histogram 8, 9, 11, 12. Basically, the predetermined window approach has some inherent problems. First, there is no theoretical evidence to explain why the fuzzy region should be set like that. Second, when the range of intensity is large, it cannot effectively select the fuzzy region. The selection of fuzzy region should depend on the nature of the images. That is, different images with different characteristics need to use fuzzy regions with different sizes.
The purpose of this work is to automatically determine the fuzzy region based on the maximum fuzzy entropy principle and apply it to find the membership function. The search of the best combination of a, b and c of the membership function is conducted using a genetic algorithm. Then, we process the fuzzified images for enhancement and thresholding.
Section snippets
Fuzzy theory and maximum entropy principle
In this section, we give brief descriptions about the fuzzy set theory and entropy which is used as a measurement of fuzziness of a fuzzy set. A new fuzzy entropy definition used in our experiments will be introduced as well.
Genetic algorithm
Genetic algorithm (GA) maintains a set of possible solutions which are encoded as chromosomes. Like physiological reproduction, the algorithm generates next generations by crossovers and mutations. The crossover strategy involves the elitist model which significantly improves the performance on the unimodal surface problems.
For a genetic algorithm, several parameters need to be defined 2, 10. They are described below.
- 1.
Coding method. Coding method represents the problem parameters as a finite
Determine the parameters for S-function
In this section, we will explain how to use genetic algorithm to find a combination of (a,b,c) such that H(A,a,b,c) has the maximum value.
Experimental results
In this section, we test the performance of the proposed approach among different images. The original images are shown in panels (a) of Fig. 1(a)Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7(a). The corresponding histograms are shown in panels (b) of Fig. 1(b)–Fig. 7(b). Each image has 256 gray levels from 0 (the darkest) to 255 (the brightest). The first subsection describes the experimental results of N=3 and N=5. The second subsection describes the effect on an image by different N values.
Discussions
There are many formulas for computing the entropy of a fuzzy set as described in Section 2.2such as , , . Here we want to explain why we need to propose a new method. Notice that the membership function used in our experiments is an asymmetric S-function with three parameters (a,b,c) and our goal is to keep the entropy as large as possible. Besides, the solution of (a,b,c) should be related to the histogram distribution.
In Zadeh's formula Eq. (5), P(xi) is from the intensity distribution of an
Conclusions
An automatically selecting fuzzy region approach with a new definition of fuzzy entropy is developed. The new entropy definition overcomes the drawbacks of the existing entropy definitions. The fuzzy region is found by a genetic algorithm based on the maximum fuzzy entropy principle. The image can keep as much information as possible when the image is transformed from the intensity domain to the fuzzy domain. In most of the existing methods, the cross-over point of S-function is set as the
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