I. INTRODUCTION:

In the process of imaging and transmission¹, it’s hard to avoid the interference of different kinds of noise. So, in the presence of noise, pre-processing steps such as image enhancement are widely used. The objectives of image enhancement are to remove impulsive noise, to smooth non impulsive noise, and to enhance the edges or other salient structures in the input image. In the techniques of image enhancement, image smoothing and image sharpening are two important methods. Images can be contaminated² with different types of noise, for different reasons. For example, noise can occur because of the circumstances of recording, transmission, or storage, copying, scanning etc. Impulse noise and additive noise are most commonly found. It is a great challenge to develop algorithms that can remove noise from the image without disturbing its content. The neighborhood averaging and smoothing by image averaging are the classical image processing techniques for noise removal. Because fuzzy set theory³ has the potential capability to efficiently represent input/output relationships of dynamic systems, this theory has gained popularity, especially in pattern recognition and computer vision applications. In the well-known rule-based approach, for image processing one, may use human knowledge expressed heuristically in linguistic terms.

Yang and Tou applied heuristic fuzzy rules to improve the performance of the traditional multilevel median filter. Noise filtering can be viewed as replacing the gray-level value of every pixel in the image with a new value depending on the local context. Ideally, the filtering algorithm should vary from pixel to pixel based on the local context. We use a fuzzy technique to achieve this goal. Fuzzy techniques have already been applied in several domains of image processing (e.g., filtering, interpolation, and morphology), and have numerous practical applications (e.g., in industrial and medical image processing). Already several fuzzy filters for noise reduction have been developed, e.g., the well-known FIREfilter from^5,6, the weighted fuzzy mean filter from^7,8, and the iterative fuzzy control based filter from⁹. Most fuzzy techniques in image noise reduction mainly deal with fat-tailed noise like impulse noise. These fuzzy filters are able to outperform rank-order filter schemes (such as the median filter). The fuzzy weighted mean filter¹ is an extension of the adaptive weighted mean filter. The idea behind the FWM filter is that the weights should take values in [0,1], instead of only the crisp values 0 , 1, and that the weights should not depend on a threshold value, but should be determined by means of fuzzy rules. The idea behind this filter is good, but it does not use enough parameters. In our approach we extend WFM filter by considering other important parameters to the fuzzy rule based system.

II FUZZY PROCESSING:

Fuzzy image processing has three main stages: image fuzzification, modification of membership values, and, if necessary, image defuzzification.

Figure 1: General Structure of Fuzzy processing

The fuzzification and defuzzification steps are due to the fact that we do not possess fuzzy hardware. Therefore, the coding of image data (fuzzification) and decoding of the results (defuzzification) are steps that make possible to process images with fuzzy techniques. The main power of fuzzy image processing is in the middle step. After the image data are transformed from gray-level plane to the membership plane (fuzzification), appropriate fuzzy techniques modify the membership values. This can be a fuzzy clustering, a fuzzy rule-based approach, a fuzzy integration approach and so on. In the following sections we describe the process in more detail for a specific application of image restoration.

III RELATED WORK:

Most fuzzy techniques in image noise reduction mainly deal with fat-tailed noise like impulse noise. These fuzzy filters are able to outperform rank-order filter schemes (such as the median filter). Nevertheless, most fuzzy techniques are not specifically designed for Gaussian noise or do not produce convincing results when applied to handle this type of noise. In this section we focus on a new technique for filtering narrow-tailed and medium narrow-tailed noise by a fuzzy filter⁴. Two important features are presented: first, the filter estimates a “fuzzy derivative” in order to be less sensitive to local variations due to image structures such as edges; second, the membership functions are adapted accordingly to the noise level to perform “fuzzy smoothing.” The general idea behind the smooth filter is to average a pixel using other pixel values from its neighborhood, but simultaneously to take care of important image structures such as edges. In order to accomplish this, for each pixel we derive a value that expresses the degree in which the derivative in a certain direction is small. Such a value is derived for each direction corresponding to the neighboring pixels of the processed pixel by a fuzzy rule⁸. The further construction of the filter is then based on the observation that a small fuzzy derivative most likely is caused by noise, while a large fuzzy derivative most likely is caused by an edge in the image. Consequently, for each direction we will apply two fuzzy rules that take this observation into account (and thus distinguish between local variations due to noise and due to image structure), and that determine the contribution of the neighboring pixel values. The result of these rules (16 in total) is defuzzified and a “correction term” is obtained for the processed pixel value⁷.

IV PROPOSED SYSTEM:

The adaptive weighted mean filter replaces the gray value of a pixel (i,j) by a weighted average of the gray values in a neighborhood of that pixel. The choice of the weights is based on the gray value differences |f(x,y) – f(x-k,y-l)|: if this difference exceeds a certain threshold, one defines wij(k,l)=0; in the other case wij(k,l)=1. In our approach the weights take values in the range [0,1], and they do not depend on a threshold value, but they are determined by means of fuzzy rules¹¹. The proposed approach has two stages; each of them uses a fuzzy rule based system. In the first stage, we try to determine whether a pixel is a noisy pixel or not. For this purpose, we use a fuzzy rule based system to determine a degree for each pixel of the image. The degree is a real number in the range [0,1]. If the degree of a pixel is equals to 1, we’ll assume that the pixel is not corrupted, and if it is less than 1, we’ll assume the pixel is noisy¹². The nearer the degree of a pixel to zero, the more it is considered as a noisy pixel. After finishing this stage, we’ll have a degree matrix as the same size as the corrupted image. We use this matrix in the next stage which performs fuzzy smoothing by weighting the contributions of neighborhood pixel values. In the rest of this section, we describe these two stages with more detail⁹.

Fuzzy Noise Estimation:

In this part we want to determine whether a pixel is corrupted or not. For this, the following criteria are considered:

1. If a pixel is severely noisy, there aren’t any similar gray level value in its neighborhood pixels, so the minimum gray value difference of that pixel and its 8-neighborhood pixels is large. Reversely, if minimum gray level difference of a pixel and its neighborhood pixels is small, one assumes that the pixel is not categorized as a noisy pixel. Hence we use minimum gray level differences as the first parameter of our fuzzy rule based system:

dif = = min |f (x, y) − f (x′, y′)| ,where (x′, y′) is an 8-neighborhood pixel of (x,y).

2. If a pixel has many similar pixels in its neighborhood, one assumes that it is uncorrupted, so we can use number of similar pixels to an assumed pixel in its 8-neighborhood as an important parameter to realize whether the pixel is corrupted or not[10]. For this, we determine the number of pixels in the 8- neighborhood of a given pixel that their gray level differences with central pixel is less than a predefined threshold. We exploit this number as the second parameter of our fuzzy rule based system:

Number Of Similar = {Number of (x′, y′) | ( x’,y’ ) ∈ N₈ (x, y) & |f (x, y) − f (x′, y′) |< Threshold }

In this paper we set threshold statically equal to 5, but it may determined dynamically for each image to gain better results. The output of the fuzzy system is a degree associated to each pixel that is a real number between 0 and 1. It denotes the degree which a pixel is considered as an uncorrupted pixel. Fuzzy membership functions are illustrated in Fig 1. The rules of the fuzzy system are as follows:

1. If (dif is low) and (num is none) then (deg is moderate)

2. If (dif is low) and (num is few) then (deg is big)

3. If (dif is low) and (num is many) then (deg is very big)

4. If (dif is med) and (num is none) then (deg is small)

5. If (dif is med) and (num is few) then (deg is moderate)

6. If (dif is med) and (num is many) then (deg is big)

7. If (dif is high) and (num is none) then (deg is small)

8. If (dif is high) and (num is few) then (deg is moderate)

9. If (dif is high) and (num is many) then (deg is moderate)

We use Mamdani inference engine, max fuzzifier, and centroid defuzzifier¹³.

B. Fuzzy Smoothing:

To compute the correction term Δ for the processed pixel value, we use a pair of fuzzy rules for each direction. The idea behind the rules is the following: if no edge is assumed to be present in a certain direction, the (crisp) derivative value in that direction can and will be used to compute the correction term. The first part (edge assumption) can be realized by using the fuzzy derivative value, for the second part (filtering) we will have to distinguish between positive and negative values. Therefore we define two fuzzy membership functions 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 and 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 with linear membership functions similar to Figure 2

Figure2: Membership functions of positive and negative

V. RESULT:

Figure 3 Input Image

Figure 4 Image After Restoration

VI. CONCLUSION:

This paper proposed a new fuzzy filter for additive impulse noise reduction. The main feature of the proposed filter is that it tries to determine corrupted pixels to reduce their contribution in smoothing process. Hence it performs fuzzy smoothing by the previous knowledge of the pixels. We performed some experiments in order to demonstrate the effectiveness of the proposed filtering approach. The fuzzy filter is able to compete with state-of-the-art filter techniques for noise reduction. A numerical measure, such as MSE, and visual observation show convincing results. In fuzzy noise estimation stage, we used a fixed threshold for number of similar pixels in a neighborhood. One can investigate on determining the threshold value dynamically to gain better results. The histogram of the image or part of the image can be used in this process.

VII. REFERENCES:

1. K. Arakawa, Fuzzy rule-based image processing with optimization, Fuzzy techniques in image processing, Springer-Verlag, 2000.

2. D. Van De Ville, M. Nachtegael, D. Van der Weken, E. Kerre, W. Philips, and I. Lemahieu, Noise Reduction by Fuzzy Image Filtering, IEEE transactions on fuzzy systems, vol. 11, No. 4, August 2003.

3. F.Farbiz, M. Menhaj, S. Motamedi, Fixed Point Filter Design for Image Enhancement using Fuzzy Logic, In Proc. IEEE, 1998.

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Received on 12.09.2011 Accepted on 10.10.2011

Int. J. Tech. 1(2): July-Dec. 2011; Page 87-89