Abstract-This paper presents an automated algorithm for image enhancement. A novel parametric indices of fuzziness (PIF) is introduced, which serves as the optimization criterion of the contrast enhancement procedure. The proposed PIF comprises the Sugeno class of involutive fuzzy complements and the first order fuzzy moment of the image. The PIF as the measure of fuzziness should be maximized, and the maximum of PIF is tuned based on the first-order fuzzy moment of the image. The parameters of the transformation function are found by the genetic algorithm aiming to maximize the PIF. Finally, several experiments are made to demonstrate the efficiency of the proposed method.

Keywords: Image enhancement, Measure of fuzziness, Fuzzy complements, Parametric indices of fuzziness, Genetic algorithm.

I. INTRODUCTION

igital image processing is the study of theories, models and algorithms for the manipulation of the images [15]. It spans a wide variety of topics

such as digitization, filtering, segmentation, enhancement, etc. Image enhancement is one of the most important tasks in the digital image processing and it almost always means replacing the gray-level value of every pixel in an image with a new value depending on some type of local information, aiming to improve the appearance of images in terms of human brightness perception [15]. The principal objective of enhancement is to process an image so that the result is more suitable that the original image for a specific application [14]. Many algorithms for image enhancement have been proposed that encompass a variety of operations such as noise removal, debluring, gray level dynamic range modification. Contrast enhancement is among them and is often part of image processing systems in the preprocessing and/or post processing stage.

The fuzzy set theory [9] has been successfully applied to many image processing and pattern recognition problems. Since fuzzy logic can easily incorporate heuristic knowledge about a specific application in the form of rules, it is ideally suited for building an image enhancement system [15]. This has led to the development of a variety of image enhancement methods based on fuzzy logic. Despite the fact that contrast enhancement is a usual task in digital image processing, the concept of image contrast lacks of a precise definition that can serve as an optimization criterion.

The concepts of fuzziness and fuzzy entropy have been extensively used in the field of image processing as an optimization criterion but characterized by the limitation of their maximum being committed to the fixed point 0.5; i.e. the midpoint of the unit interval. This can be considered as a drawback when such measures are to be applied in real-world images, as argued in [11], since this point with inherent maximum fuzziness is a fixed point and thus it is not application-dependent. [12] presented an automated algorithm based on a class of parametric indices of fuzziness having the ability of adjusting the location of the maximum in any point of the (0,1) interval and thus modeling the ambiguity and vagueness of images in a more adaptive way Motivated by [8, 12], we use a parameter-dependent complement, the S-function used in [12] as the transformation function and the first-order fuzzy moment of image to maximize the proposed parametric index of fuzziness (PIF). In contrast with selecting the symmetric of the first-order fuzzy momentwith respect to the midpoint of the unit interval as the maximum of PIF, we set the maximum of PIF corresponding to the location of fuzzy moment. Finding the optimal parameters of transformation function is performed by maximizing the PIF, however an exhaustive searching the state space of the S-function parameters is needed which is computationally expensive. An optimization method can be used to significantly reduce the computational cost of the algorithm, in order to be applicable in real-time applications. Genetic algorithm (GA) is a global optimal search tool, which is used to find the optimal parameter set of the S-function.

This paper is organized as follows: first, we briefly describe the image in fuzzy domain. After representation of the linear and parametric indices of fuzziness, we introduce the proposed PIF in section 5. Then we tune the PIF by exploiting the statistics of the image in the fuzzy domain in the same section. In section 6, we use the S-function as the transformation function to modify the membership values. Finally, in section 7 the genetic algorithm as an optimization search tool is used to find the optimal parameter set of the S-function. Comprehensive experiments are made to demonstrate the efficiency of the proposed algorithm in section 8. Discussion and conclusions are made in section 9.

An automated GA-based fuzzy image enhancement method

Omid Khayat, Javad Razjouyan, Mina Aghvami, Hamid Reza Shahdoosti, and Babak Loni.

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978-1-4244-2760-4/09/$25.00 ©2009 IEEE

II. IMAGE REPRESENTATION IN THE FUZZY DOMAIN

Let us consider an image A of size pixels, having gray levels ranging from to . The image A can be viewed as an array of fuzzy singletones [5-7]. Each element of the array is the membership value of

the gray level , corresponding to the th pixle,

regarding to an image property such as brightness, edginess, homogeneity, etc. For the task of image enhancement we consider the property "brightness" of the gray levels. Using the fuzzy sets notation image A can be represented as:

III. PARAMETRIC INDICES OF FUZZINESS

Contrast enhancement is a common task in digital image processing systems. Several methods have been proposed for measuring the grayness ambiguity of an image and therefore providing a criterion for the enhancement procedure. One of the most common measures of fuzziness is the "linear index of fuzziness", which for a set A is defined as

Where denotes the frequency of occurrence of the gray level g. The linear index of fuzziness of equation (2) considers the intersection between fuzzy sets and their complements, using a t-norm such as min operator, that here the algebraic product similar to [8] is selected as the intersection operator.

IV. INVOLUTIVE FUZZY COMPLEMENTS

The complement of membership function was defined at first by Zadeh [9] as (3). Sugeno and

Yager also introduced a class of involutive fuzzy complements as (4) and (5) respectively.

In the proposed method, the Sugeno class of fuzzy complements is considered.

Fig. 1. The equilibrium of the sugeno fuzzy complement as a function of .

V. THE NOVEL PIF

In the proposed method, we use a class of "parametric indices of fuzziness", based on the Sugeno class of involutive fuzzy complements and algebraic product t-norm operator

Where is the first-order fuzzy moment of image A. And is the Sugeno class of fuzzy complements, in

which , is a constant. The first-order fuzzy moment of image A is defined as

The Sugeno class of parametric indices of fuzziness has the property of assigning different weights as the membership moves away from the two values 0 and 1, since it has the advantage of centering the maximum of the PIF at any point of the (0,1) interval. This property of the proposed parametric indices of fuzziness overcomes the intrinsic limitation that the linear index of fuzziness imposes, where the location of the maximum is committed to the fixed point . Selecting different parameters results

in varying the equilibrium of the Sugeno fuzzy complement and consequently the point where PIF attains a maximum. The equilibrium as a function of is given by

Fig. 1 illustrates the curve of the equilibrium of the sugeno fuzzy complement as a function of . In the proposed algorithm, the location of the maximum of the PIF is calculated based on the first-order fuzzy moment of the image. The "optimal" equilibrium point is

selected as

Where and are the minimum and maximum gray levels of the image A respectively. Evaluating (8) for

yields

For any we ensure that the constraint of is satisfied. Fig. 2 illustrates as a

function of the first-order fuzzy moment .

Fig. 2. The curve of the optimal parameter as a function of the first order fuzzy moment .

VI. IMPLEMENTATION

Fuzzy image processing consists of three sequential stages: (a) fuzzification, (b) suitable membership values modification, and (c) defuzzification.

A. Fuzzification

In the fuzzification stage of the proposed algorithm, the membership function is initialized according to

Where and are the minimum and maximum gray levels of the image respectively. Using (11) for

initializing the membership values has the advantage of stretching out the membership function over the unit interval.

B. Membership values modification

In the second stage, an S-function is used to transform the membership values of its gray levels. The S-function with a parameter set , which determines the shape of the S-function, is used as [12]. The parameters satisfy the condition of , and can be any point in the interval

For a specific combination of the S-functions parameters, which is considered optimal, the PIF of the generated image is maximized. Images possessing maximal grayness ambiguity are expected to be more suitable in terms of human brightness perception [8]. Therefore, the modification of the membership values for performing contrast enhancement is carried out according to

Where denotes the new membership value corresponding to the gray level . The PIF of the resulting image can be regarded as a function of the parameter set P, and is maximized for

Where is the number of gray levels, is the first order fuzzy moment of the image A, P is the parameter set of the S-function and is a constant to prevent the denominator to become zero. It should be mentioned that searching exhaustively the state space of the S-function's parameters is computationally expensive, since the search space is about . Therefore, genetic algorithm as an optimization method can be used to significantly reduce the computational cost of the algorithm, in order to be applicable in real-time applications.

C. Defuzzification

Finally, in the defuzzification stage of the proposed algorithm and after having obtained the optimal transformation parameters , the contrast enhanced image is derived by scaling the membership values up to the range of using the following formula

Where and are the original and the transformed gray levels, respectively, and L the number of intensity values.

The proposed algorithm can be formulated as follows

Step 1. Initialize the membership values by (11).

Step 2. Calculate the image histogram .Step 3. Calculate the first-order fuzzy moment

of the image by (7).Step 4. Calculate the optimal ( ) by (10).Step 5. Find the optimal parameter set

by GA. Step 6. Perform the transformation by the

obtained S-function (modification of the membership values).

Step 7. Generate new gray levels by (15).

VII. GENETIC ALGORITHM

Genetic algorithm (GA) can handle any kind of objective functions and any kind of constraints without much mathematical requirements about the optimization problems [16, 17 & 18]. GA has been touted as a class of general-purpose search strategies for optimization problems.[19] In GA, variables of a problem are represented as genes in a chromosome, and the chromosomes are evaluated according to their fitness values. GA starts with a set of randomly selected chromosomes as the initial population that encodes a set of possible solutions. Through natural selection and genetic operators, mutation and crossover, chromosomes with better fitness are found. The genetic operators alter the composition of genes to create new chromosomes called offspring. The selection operator is an artificial version of natural selection, a Darwinian survival of the fittest among populations, to create populations from generation to generation, and chromosomes with better fitness have higher probabilities of being selected in the next generation.

VIII. EXPERIMENTAL RESULTS

The proposed algorithm has been tested using various images of different types. In order to evaluate the performance of the proposed method, we compared our results to those derived from the maximum entropy principle of fuzzy events (MEP-FE) [1], the -enhancement method based on optimization of image fuzziness [8], and the automated image enhancement method based on PIF [12]. For our simulations we used gray-scale images of size pixels with 8 bit per pixel gray-tone resolution. For obtaining the optimal transformation function the parameters of S-function should be found by GA. The size of the initial population for GA is 20. The crossover probability , mutation probability and the maximum number of generation . The fitness function for GA is calculated by maximizing parametric indices of fuzziness as

Fig. 3. Comparison of the proposed method with different contrast enhancement methods. (a) original low-contrasted image. Images processed by (b) MEP-FE, (c) -enhancement, (d) automated image enhancement method of [12], and (e) proposed method.

Comparing the images of cameraman shown in Fig. 3, one can observe the efficiency of the proposed method in enhancing the highly low-contrasted images. Fig. 3(a) illustrates a highly low contrasted image with a majority of its pixels concentrated at the lower intensity levels as can be seen from Fig. 4(a), which shows the histogram of the initial image. The images of Figs. 3(b), 3(c) and 3(d) are the enhanced images using the MEP-FE, -enhancement method and the automated image enhancement method of [12] respectively. As it can be seen from Fig. 3(b), the image, although enhanced, possesses a large number of pixels in the lower and higher bins of the histogram. This over-enhancement results in image that appear somewhat saturated. Also the images of Figs. 3(c) and (d) do not reveal the hidden details and they are still dark. In contrast,

in the image of Fig. 3(e), enhanced by the proposed method, details have been revealed elaborately and the image appears neither dark nor bright.

Fig. 4. Histograms of (a) the initial low contrasted image of cameraman and (b) the image of cameraman processed by the proposed algorithm. (c) membership modification function with

.

The histogram of the enhanced image of cameraman, depicted in Fig. 4(b), is more uniformly distributed over the total gray-level range. The corresponding transformation function, S-function, is shown in Fig. 4(c). The maximum of PIF has been achieved with parameter set .

Fig. 5. Comparison of the proposed method with different contrast enhancement methods. (a) original low-contrasted image. Images processed by (b) -enhancement, (c) automated image enhancement method of [12], and (d) proposed method.

Another demonstration of the proposed method is shown in Fig. 5. The images obtained using the -enhancement method and the automated image enhancement method of [12] are shown in Figs. 5(b) and (c) respectively. As it can be seen from the Fig. 5(b), the -enhancement method improves the image of Lena

negligibly, hence, the histograms of Figs. 6(a) and (b) are almost the same.

Fig. 6. Histograms of (a) the initial low contrasted image of Lena , the image of Lena processed by (b) the -enhancement method, (c) the automated image enhancement method of [12] and (d) the proposed algorithm .

The result of the automated image enhancement method of [12], shown in Fig. 5(c), is somewhat bright. Fig. 5(d), derived using the proposed PIF and proposed , reveals more details such as high frequency edges and has more distributed histogram over the total gray-level range.

Fig. 7. Comparison of the proposed method with different contrast enhancement methods. (a) original low-contrasted image. Images processed by (b) -enhancement, (c) automated image enhancement method of [12], and (d) proposed method.

Finally, in order to further justify the efficiency of the proposed PIF and , the image of peppers are enhanced by the -enhancement method and the automated image enhancement method of [12] and the proposed method.

The original image is a bright image, its pixels has occupied a narrow range of gray levels in the histogram. The proposed algorithm has distributed the histogram of the image over the total range of gray levels, whereas the histograms of the images enhanced by other methods are not distributed completely, hence, the results are not enhanced as significant as when the proposed method is applied.

Image\Method -enhancement Method of [12] The proposed methodCameraman

Lena

Peppers

Table 1. The optimal parameters obtained in the experiments.

Fig. 8. Histograms of (a) the initial low contrasted image of Peppers , the image of Lena processed by (b) the -enhancement method, (c) the automated image enhancement method of [12] and (d) the proposed algorithm .

The parameters obtained in enhancement of images of Cameraman, Lena and Peppers by the -enhancement method, the automated image enhancement method of [12] and the proposed method are shown in Table 1.

It should be mentioned that due to the positioning the corresponding to the location of the first order fuzzy

moment, where in the histogram the pixels are concentrated, thus, it may not lead to over-exposure (under-exposure).

IX. DISCUSSIONS AND CONCLUSION

In this paper, an automated fuzzy enhancement method based on the first-order fuzzy moment, the S-function as the transformation function and a novel PIF was presented. In order to tune the PIF we utilized the first order fuzzy moment. Parameter set of the S-function was found by genetic algorithm as an optimization search tool aiming to maximize the PIF. Comprehensive experiments were made to demonstrate the efficiency of the proposed algorithm. It has been seen that the proposed method enhanced both the dark and bright images comprehensively. Moreover, the proposed method delivers contrast-enhanced images well adapted to the human brightness perception. Finally, we demonstrated that the concepts of principle indices of fuzziness and parameter-dependent complements

successfully improve the performance of the existing contrast-enhancement algorithms and provide a more flexible framework for dealing with image processing problems.

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