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Fuzzy Based Directional Weighted Median Filter for Impulse Noise Detection and Reduction

Fuzzy Based Directional Weighted Median Filter for Impulse Noise Detection and Reduction Fuzzy Inf. Eng. (2012) 4: 351-369 DOI 10.1007/s12543-012-0120-2 ORIGINAL ARTICLE Fuzzy Based Directional Weighted Median Filter for Impulse Noise Detection and Reduction Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy Received: 6 February 2012/ Revised: 27 September 2012/ Accepted: 12 November 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The known median-based denoising methods tends to work well for restor- ing the images corrupted by random-valued impulse noise with low noise level, but it fails in denoising highly corrupted images. In this paper, a new noise reduction method based on directional weighted median based fuzzy impulse noise detection and reduction method (DWMFIDRM) has been proposed, which has been specially developed for denoising all categories of impulse noise. The contribution of this paper is threefold. The main contribution of the novel impulse noise reduction tech- nique lies in the unification of three different methods; the impulse noise detection phase utilizing the concept of fuzzy gradient values, edge-preserving noise reduc- tion phase based on the directional weighted median of the neighboring pixels and a final filtering step in order to deal with noisy pixels of non-zero degree. Such a unique combination has improved the efficiency of this method for high density noise removal. The experimental results of our proposed method have a significant im- provement when compared to other existing filters for high density noise removal. This paper utilizes the concept of fuzzy gradient values. The noise reduction phase that preserves edge sharpness is based on the directional weighted median of neigh- boring pixels. Final filtering phase is performed only when there is non-zero degree of noise pixels. This phase makes our method more efficient in high noise density. Experimental results show that DWMFIDRM provides a significant improvement on other existing filters. Keywords Fuzzy filter · Directional weighted median · Impulse noise · Noise re- duction 1. Introduction Riji R () · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy Department of Computer Science, University of Kerala, Kariavattom, Thiruvananthapuram, 695581, India email: rijir kmr@yahoo.com 352 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) Digital images play a vital role in our daily lives as well as in an inevitable explo- ration of scientific research areas like satellite television, geographical information systems, astronomy, and in medical imaging technologies like magnetic resonance imaging (MRI), and computed tomography (CT). But more often challenging situa- tions are faced in all image processing domains where the image sensors suffer from noise contamination. Besides this, the issues caused during the data acquisition pro- cess, wrong machine readings, interference causedby natural phenomena, bit errors occurred during transmission and the transmission errors caused in the compressed image degrades the data of interest. This makes it difficult for analyzing and process- ing the images in a better way. Image denoising is an ultimate solution to deal with such problems and hence is taken as the first and necessary step before image anal- ysis. But all methodologies for image denoising must deal with efficient recovery of the corrupted image without causing any loss of information. Our method is robust against high-density corruptions caused in images either through natural phenomena or through the interferences caused during machine acquisition. The need for image denoising is encountered in many medical applications like MRI, ultrasound and CT scan images. Among these modalities, more emphasis is given to MRI image denoising in the medical field, as they are affected by random noise generated during the image acquisition process. Hence, denoising has to be done more effectively, since the presence of noise may cause damage to its visual quality and its true contrast as well by making it difficult to analyze the low contrast objects in the image. Hence, a novel image denoising scheme has to recover even the fine image details hidden within inside the objects in the image. The main property of a good image denoising model is its edge preservation af- ter the noise removal process. Traditionally, linear models have been used for noise removal. But they lack its ability for the better preservation of edges in a good and ef- fective manner. Instead the discontinuities in the image regarded as edges are smeared out by damaging its visual quality. Fuzzy filters are introduced in our approach to deal with the issues incurred in the earlier approaches. Fuzzy filters have succeeded in bringing out an edge-preserving impulse noise removal scheme which preserves even the fine image details. The main property of this filter is its ability to adapt to the local characteristics of a given neighborhood or window by making it well suited for removing the impulse noise while preserving the fine image details. Digital images play an important role both in our daily life lives applications such as satellite television, magnetic resonance imaging, computer tomography as well as in areas of research and technology such as geographical information systems and as- tronomy. Data sets collected by image sensors are generally contaminated by noise. Imperfect instruments, problems with the data acquisition process, and interfering natural phenomena can all degrade the data of interest. Furthermore, noise can be introduced by transmission errors and compression. Thus, denoising is often a neces- sary and the first step to be taken before the images data is analyzed. It is necessary to apply an efficient denoising technique to compensate for such data corruption. Image denoising has become an essential exercise in medical imaging especially the magnetic resonance imaging. Medical images obtained from MRI are the most common tool for diagnosis in medical field. These images are often affected by ran- Fuzzy Inf. Eng. (2012) 4: 351-369 353 dom noise arising in the image acquisition process. The presence of noise not only produces undesirable visual quality but also lowers the visibility of low contrast ob- jects. Noise removal is essential in medical imaging applications in order to enhance and recover fine details that may be hidden in the data. The main property of a good image denoising model is that it will remove noise while preserving edges. Traditionally, linear models have been used. But a back draw of the linear models is that they are not able to preserve edges in a good manner: edges, which are recognized as discontinuities in the image, are smeared out. Fuzzy filters have been suggested as a means of solving the above problem. It succeeded in removing impulse noise while preserving image detail such as edge. The property of fuzzy filters to adapt to the local characteristics in a given window makes it well suited for removing impulse noise while preserving image details. A number of approaches have been proposed for the impulse noise removal. Tukey [1], Astola et al [2] and Pitas et al [3] have utilized median filtering to remove im- pulse noise from the corrupted images. Other filters for removal of impulse noise include histogram based fuzzy filter (HFF) [4], novel fuzzy filter (NFF) [5] by Lee et al, genetic based fuzzy image filter (GFIF) [6], fuzzy impulse noise detection and reduction method (FIDRM) [7], fuzzy random impulse noise reduction method (FRINR) [8], universal impulse noise filter based on genetic programming [9] by Ne- manja et al, a new directional weighted median (DWM) filter [10] for removal of random valued impulse noise and detail preserving fuzzy filter (DPFF) [11] are the examples of the most recent filters. Other approaches based on fuzzy-based impulse noise reduction are fuzzy impulse noise detection and reduction method (FIDRM) [13] and directional weighted median based fuzzy filter for random-valued impulse noise removal [14]; nevertheless, all of these methods face the same major drawback, such that higher the noise level, lower is its noise reduction capability. To overcome this drawback, we propose a more efficient noise reduction method for images having high noise density. The proposed method consists of two phases: noise detection and reduction step. The detection method is based on fuzzy gradient values which determine whether a pixel is noisy or not. The reduction phase uses the concept of directional weighted median of the neighboring pixels which has been used to calculate the membership degree of each noisy pixel. After that a fuzzy averaging method which makes use of the membership degree is used for the noise reduction step. The paper is organized as follows. For clarity, we have first given a short descrip- tion of different noise models in Section 2. The details of the detection phase are given in Section 3. In Section 4, the proposed filtering method is discussed. Performance measures, experimental results and conclusions are finally presented in Sections 5, 6 and 7. 2. Noise Models To understand the noise detection and reduction method more clearly, we first give a brief description of different noise models. Four impulse noise models are imple- mented for extensively examining the performance of our proposed filter with con- sideration of practical situations. Each model is described in detail as follows. 354 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) 1) Noise Model 1: Noise is modeled as salt-and-pepper impulse noise as prac- ticed. Pixels are randomly corrupted by two fixed extreme values, 0 and 255 (for 8-bit monochrome image), generated with the same probability. That is, for each im- age pixel at location (i, j) with the intensity value s , the corresponding pixel of the i, j noisy image will be x in which the probability density function of x is i, j i, j ⎪ , for x = 0, ⎪ 2 f (x) = (1) 1− p, for x = s , ⎪ i, j , for x = 255. 2) Noise Model 2: For Model 2, it is similar to Model 1, except that each pixel might be corrupted by either “pepper” noise (i.e., 0) or “salt” noise with unequal probabilities. That is p1 , for x = 0, ⎪ 2 f (x) = 1− p, for x = s , (2) ⎪ i, j p2 , for x = 255, where p = p1+ p2 is the noise density and p1 not equal to p2. 3) Noise Model 3: Instead of two fixed values, impulse noise could be more realisti- cally modeled by two fixed ranges that appear at both ends with a length of m each, respectively. For example, if m is 10, noise will equal likely be any values in the range of either [0, 10] or [245, 255]. That is ⎪ , for 0 ≤ x < m, ⎪ 2m f (x) = (3) 1− p, for x = s , ⎪ i, j , for (255− m) < x ≤ 255, 2m where p is the noise density. 4) Noise Model 4: Model 4 is similar to Model 3, except that the densities of low- intensity impulse noise and high-intensity impulse noise are unequal. That is ⎪ p1 , for 0 ≤ x < m, ⎪ 2m f (x) = 1− p, for x = s , (4) ⎪ i, j p2 , for (255− m) < x ≤ 255, 2m where p = p1+ p2 is the noise density and p1 not equal to p2. 3. Fuzzy Impulse Noise Detection In the noise detection phase, a fuzzy impulse noise detection method has been com- bined with FIDRM [13]. This detection method uses fuzzy gradient values, intro- duced with the GOA filter [15, 16], to determine whether a certain pixel is corrupted with impulse noise or not. Fuzzy Inf. Eng. (2012) 4: 351-369 355 The detection method first calculates the basic gradient values and relates gradient values for each pixel to determine if a central pixel is corrupted with impulse noise or not. Finally, the method defines eight fuzzy gradient values for each of the eight directions. These values indicate in which degree the central pixel can be seen as an impulse noise pixel. The fuzzy gradient value∇ A(i, j) for direction R (R ∈{NW, N, NE, E, SE, S, SW, W}) is calculated by the following fuzzy rule. IF|∇ A(i, j) | is large AND|∇ A(i, j) | is small OR IF|∇ A(i, j) | is large AND|∇ A(i, j) | is small OR IF∇ A(i, j) is big positive AND ∇ A(i, j) AND∇ A(i, j) are big negative R R OR IF∇ A(i, j) is big negative AND ∇ A(i, j) AND∇ A(i, j) are big positive R R THEN∇ A(i, j) is large, where ∇ A(i, j) is the basic gradient value and ∇ A(i, j) and ∇ A(i, j) are the two R R related gradient values for a direction R. To decide if a central pixel (a no border pixel of course) is an impulse noise pixel, the following (fuzzy) rule has been used: IF most of the eight ∇ A(i, j) are large THEN the central pixel A(i, j) is an Impulse noise pixel. After detecting a pixel as noisy based on the above fuzzy rule, a histogram is con- structed to find the presence of impulse noise density. Thus, we conclude that the presence of peak values in a histogram indicates the presence of impulse noise in an image or noise-corrupted; otherwise the image is noise-free. The entire procedure of the detection phase is shown below in Fig.1. If a pixel is detected as noisy by using the above rule, then use the histogram to find the presence of impulse noise. Construct histogram using the pixels which is detected as noisy by the fuzzy rule. If the histogram contains peaks, then conclude that the image is noisy otherwise the image is free from impulse noise. Finally, this detection phase ends with the procedure pictured in Fig.1. If the detection phase has detected impulse noise, then we perform the proposed filtering method explained in the next section. Otherwise, we leave the image un- changed. Our proposed filtering method based on “fuzzy based directional weighted me- dian filter for impulse noise detection and reduction” is used after the impulse noise detection phase for effective denoising by improving the visual quality of the noisy image with better fine detail and edge preservation. However, if the detection phase has estimated the presence of a noise-free image, then the image is left unchanged. The proposed filtering method is explained in the next section. 4. Proposed Filtering Method We introduce a new filtering method which uses the concepts of directional weighted median introduced in [17]. This section is structured as follows. The parameter cal- culation (a, b, c, d), which have been used to construct the fuzzy set as more or less 356 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) Input: HIST : the noise histogram matrix HIST(i) : indicates the amount of selected noise pixels with gray scale value i. If (i < 0) or (i < 255) HIST(i)=0 TOT : the total number of detected noise pixels HIST(m) : the maximum histogram value assumed to be located at gray scale value m = [0, 255] 1) IF (HIST(m)/TOT)×100 < THR1) 2) NO IMPULSE NOISE DETECTED 3) ELSE 4) IF (HIST(i) / HIST(i− 1) + HIST(i) + HIST(i+ 1) ≥ 0.95 5) Perform Filtering For Noise Reduction 6) END IF 7) END IF Fig. 1 Pseudo-code for decision procedure impulse noise. After this, a first filtering operation is iterated based on the member- ship function that represents the fuzzy set w(). Finally, this section is terminated with the explanation of other iterations and some stopping criteria. First, we explain the calculation of the parameters (a, b, c, d), which are used to construct the fuzzy set more or less impulse noise. Afterwards, we construct our first filtering iteration based on the membership function that represents this fuzzy set w(). The explanation of the other iterations and some stopping criteria finally terminate this section. A. Configuration of the Parameters Before introducing a new fuzzy filtering method, a basic assumption has been re- viewed, which is a noise-free image consisting of locally smoothly varying areas sep- arated by edges. More focus is given to the edges aligned with four main directions as shown in Fig.2. Before introducing the new fuzzy filtering method, we review a basic assumption, that is, a noise-free image consists of locally smoothly varying areas separated by edges. Here, we only focus on the edges aligned with four main directions shown in Fig.2. Let S (k = 1 to 4) denotes a set of coordinates aligned with the kth direction cen- tered at (0, 0), i.e., S ={(−2,−2), (−1,−1), (0, 0), (1, 1), (2, 2)}, S ={(0,−2), (0,−1), (0, 0), (0, 1), (0, 2)}, (5) S ={(2,−2), (1,−1), (0, 0), (−1, 1), (−2, 2)}, S ={(−2, 0), (−1, 0), (0, 0), (1, 0), (2, 0)}. 4 Fuzzy Inf. Eng. (2012) 4: 351-369 357 In order to perform the filtering, we consider a window of size 5× 5 and construct trapezoidal shaped fuzzy membership function for the fuzzy set more or less impulse noise as shown in Fig.3. The parameter [a, b, c, d] of the membership function de- pends on the value of directional weighted median as shown in Equation (6). Fig. 2 Four directions used for impulse detection maxX(i+ x, j+ y)− minX(i+ x, j+ y) dev(i, j)= x, y∈{−2,··· ,+2}, a = med − 1.1dev(i, j), dir (6) b = med − dev(i, j), dir c = med + dev(i, j), dir d = med + 1.1dev(i, j). dir To determine directional weighted median, first of all we calculate the standard deviation σ of the pixel intensities aligned with each of the four directions D (k k k =1, ··· , 4) [17]. Since the standard deviation describes how tightly all the values are clustered around the mean in the set of pixels shows that the four pixels aligned with this direction are closest to each other. Therefore, the center value should also be close to them in order to keep the edges intact. Thus, pixels aligned with the direction having minimum standard deviation are repeated in the considered window and then the median is calculated. This median is called the med , which is used in dir Equation (6) to compute the parameters of the fuzzy membership function more or less impulse noise. The use of med in fuzzy set construction makes the fuzzy set dir more or less impulse noise robust against noisy pixels and only the pixel values which are tightly close together under the considered window will take part in estimating the 358 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) restored value of the noisy pixel. The use of fuzzy membership more or less impulse noise gives appropriate importance to the pixels in estimating the non-noisy value of the considered pixel based on their homogeneity with the neighbors. Therefore, the restored value will be more accurate in better preservation of fine details including edge and texture information. Therefore, the restored value will be more accurate and helps in preserving the small image details such as edge and texture information. Fig. 3 Membership function “more or less impulse noise” After constructing the fuzzy set more or less impulse noise, the weights (degree of membership in fuzzy set more or less impulse noise) for the pixel values under the window of size 5 × 5 are determined. Finally, these weights are used for filtering which is explained in the first iteration. B. First Iteration The filtering step of the first iteration is given in the pseudo code presented in Fig.4. This method is based on the membership function “more or less impulse noise” (w). The corresponding membership degree of a certain intensity value X(i, j) is denoted as w(i, j). w(i, j) is computed based on the fuzzy membership function shown in Fig.3. A degree one (zero) indicates that the intensity value is noisy for sure (not noisy for sure). When the degree is between one and zero, then there is some kind of uncertainty. We only filter pixels which are part of the support of the fuzzy set more or less impulse noise (pixels which have a nonzero membership degree in this fuzzy set). Otherwise, we leave the pixel unchanged. As shown in line three of Fig.4, we use, a 3 × 3 window around the filtered pixel. In addition, we use the standard negation to express the membership degrees in the fuzzy set noise free [5, 6].The Fuzzy Inf. Eng. (2012) 4: 351-369 359 weighted average AVG is then computed as follows: 2 2 X(i+ k, j+ l)w(i+ k, j+ l) k=−2 l=−2 AVG = + w(i, j)X(i, j). (7) 2 2 w(i+ k, j+ l) k=−2 l=−2 C. Next Iterations For special cases where an image is contaminated by high density impulse noise, there is a possibility for a side effect after the first iteration where by the impulse noise remains clustered around one or more pixels. In order to deal with this problem, few more iterations are performed similar to the first iteration for noise reduction. During each iteration, the modified image in the previous iteration has been used. After the first iteration, it is possible as a side effect (especially with high initial impulse noise) that the impulse noise is clustered around one or more pixels. To reduce these noisy pixels, we will provide some more iteration that are similar to the first one. In each iteration, we use the modified image of the previous performed iteration. Input : A: The noisy image with impulse noise w(A(i, j)) : the membership degree of the fuzzy set “more or less impulse noise” F: The output image 1) FOR each non-border pixel (i, j)in A 2) IF A(i, j) in supp (“more or less impulse noise”) 3) F(i, j)=(1− w(i, j)) * AVG 4) ELSE 5) F(i, j) = A(i, j) 6) END IF 7) END FOR Fig. 4 Pseudo code for filtering D. Stopping Criteria There are several techniques to improve the efficiency of the filtering phase. We will focus on the stopping criteria and the amount of pixels that must be scanned during an iteration. During the first iteration, we check every pixel. If the pixel value does not belong to the support of the fuzzy set more or less impulse noise, then we do not change this pixel value, not only in this iteration neither in the other ones. By remembering only the positions of pixels, whose pixel value is an element of the support of the fuzzy set more or less impulse noise, we can drastically reduce the scanning amount in the next iterations. A stopping criterion is essential at this instance. For this, # is defined which indicates the amount of pixel values belonging to the support of the fuzzy set more 360 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) or less impulse noise in the eth iteration, then we can apply the following stopping criteria. 1) There are no pixel values in the support of the fuzzy set (# = 0) in any iteration (e ≥ 2). 2) # is equal to # . This indicates that the resulting pixels (pixels which still e e−1 are element of the support of the fuzzy set more or less impulse noise) are not noisy even with the nonzero membership degrees. E. Final Step of Filtering After completing the filtering process based on directional weighted median, the final filtering step is performed. This filtering phase is based on fuzzy averaging [13]. If the amount of pixel values which belong to the support of the fuzzy set more or less impulse noise in the previous iteration is not equal to zero, then only we go for the final step of filtering. This phase also uses the parameters (a , b , c , d ) for constructing the fuzzy set k k k k more or less impulse noise. But the calculation of the parameters is slightly differ- ent from that of the previous filtering phase. The configuration of the parameter is described in the procedure pictured in Fig.5. Input : HIST : The noise histogram matrix TOT : the total number of detected noise pixels 1) cum = 0 2) FOR k =1to5 3) kmax = select the k-th greatest y value of the noise histogram 4) kpos = the corresponding gray scale value where the maximum is reached 5) IF (kmax/TOT) + cum ≥ THR2 6) p =kpos 7) Set the four parameters (a , b , c , d )to(p , p , p , p ) k k k k k k k k 8) cum = cum + (kmax/TOT) 9) ELSE 10) Break 11) END IF 12) END FOR Fig. 5 Pseudo code for configuring parameters in the final filtering phase The filtering step of the final iteration is given in the pseudo code presented in Fig.6. This method is based on the membership function “more or less impulse noise” (μ). The corresponding membership degree of a certain intensity value A(i, j) is de- noted as μ(A(i, j)). A degree one (zero) indicates that the intensity value is noisy for sure (not noisy for sure). When the degree is between one and zero, then there is some Fuzzy Inf. Eng. (2012) 4: 351-369 361 kind of uncertainty. We only filter pixels which are part of the support of the fuzzy set more or less impulse noise (pixels which have a nonzero membership degree in this fuzzy set). Otherwise, the pixel values are left unchanged. Otherwise, we leave the pixel unchanged. 5. Performance Measures The following performance measures are used for the evaluation of our method. 1) PSNR (peak signal to noise ratio), 2) SSIM (structural similarity index measure), 3) IQI (image quality index). As a measure of objective dissimilarity between a filtered image and the original one, we use the mean square error (MSE) and the peak signal to noise ratio (PSNR) in decibels N M [Org(i, j)− Img(i, j)] i=1 j=1 MS E(Img, Org) = , (8) MN PS NR(MS E(Img, Org)) = 10 log , (9) MS E(Img, Org) where Org is the original image, Img is the filtered image of size M× N, and S is the maximum possible pixel value (with 8-bit integer values the maximum will be 255). Input : A: the noisy image with impulse noise μ(A(i, j)) : the membership degree for the fuzzy set “more or less impulse noise” F : the output image 1) FOR each non-border pixel (i, j)in A 2) IF A(i, j) in supp (“more or less impulse noise”) 1 1 1−μ(A(i+h, j+l))A(i+h, j+l) h=−1 l=−1 3) F(i, j) = 1 1 1−μ(A(i+h, j+l)) h=−1 l=−1 4) ELSE 5) F(i, j) = A(i, j) 6) END IF 7) END FOR Fig. 6 Pseudo code for final filtering The SSIM is a method for measuring the similarity between two images. The SSIM index is a full reference metric, in other words, the measuring of image quality based on an initial uncompressed or distortion-free image as reference. The SSIM metric is calculated on various windows of an image. The measure between two 362 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) windows x and y of common size N × N is (2μ μ + c )(2σ + c ) x y 1 xy 2 SS IM(x, y) = . (10) 2 2 2 2 (μ +μ + c )(σ +σ + c ) 1 2 x y x y 6. Experimental Results In this section, we compare the efficiency and performance of DWMFIDRM for grayscale images with other well-known filters for impulse noise reduction. The first group of filters consists of the FIRE filters (fuzzy inference rule by else-action filters). Table 1 : PSNR results for the (256× 256) Lena image for Type 4 impulse noise with noise densities 20%, 30% and 50%. PSNR 20% 30% 50% Original 13.68 11.78 8.75 CWM(3X3) 27.07 23.48 15.18 CWM(7X7) 26.38 26.01 24.32 TSM(3X3) 27.08 23.74 24.75 TSM(7X7) 26.53 23.83 15.47 LUM 26.74 23.74 18.43 MED(3X3) 26.74 23.74 15.05 MED(5X5) 26.35 26.23 22.43 MED(7X7) 25.03 25.34 23.89 ATMAV 21.37 21.02 20.17 FSD 26.52 23.25 14.78 HAF 24.35 22.01 25.56 AWFM 27.83 26.43 26.87 SFCF 25.32 21.12 14.69 EIFCF 26.74 24.53 17.43 MIFCF 26.03 23.05 17.32 IFCF 26.72 24.32 17.32 FIRE 24.01 20.53 13.48 FMF 27.93 23.54 15.67 DSFIRE 22.56 21.48 17.85 PWLFIRE 24.39 19.49 12.73 FIDRM 32.64 28.67 22.08 DWFIDRM 36.90 34.90 31.01 The idea behind these filters is that they try to calculate positive and negative correc- tion terms in order to express the degree of noise for a certain pixel. We distinguish Fuzzy Inf. Eng. (2012) 4: 351-369 363 three FIRE filters: the normal FIRE [18] (fuzzy inference rule by else-action filters), the DS-FIRE [19] (dual step FIRE), and the PWLFIRE [20] (piecewise linear FIRE). A second group of filters contains filters which are extensions of classical median (de- noted by MED) and weighted filters. We use the FMF [21, 22] (fuzzy median filter), AWFM [23, 24] (adaptive weighted fuzzy mean) and the ATMAV [25] (asymmetri- cal triangular fuzzy filter with moving average center). The FCF (fuzzy control-based filters) constitute a third group of filters. These filters correct a certain central pixel value according to some features of some luminance (pixel values) differences be- tween the central pixel value and some neighbor pixel values. In the literature, we know that the IFCF [26] (iterative fuzzy control based filter), the MIFCF [26] (mod- ified IFCF), the EIFCF [26] (extended IFCF), and the SFCF [27] (smoothing fuzzy control based filter). Furthermore, there exist many other types of filters, such as, e.g., the histogram adaptive fuzzy filter which of course uses the histogram of an im- age [28] (HAF) or the fuzzy similarity filter [29] (FSB), where the local similarity between some patterns is used. Besides all these fuzzy filters, some other popular filters exist. We use the CWM [30] (center weighted median), the TSM [31] (tri-state median filter), and the LUM [32] (lower-upper-middle filter) for comparison. We also compare our method with FIDRM. We used the well-known Lena, Baboon and circuit images of size, 256× 256, 204 × 204 and 280 × 272 respectively, to compare DWMFIDRM with the other filters. From Tables 1 and 2, we can see that DWMFIDRM outperforms all the conventional as well as advanced filters, especially at high noise levels. In all the experiments, we have used the noise Model 4, since the other noise Models (1, 2 and 3) are special cases of noise Model 4. Table 2: PSNR results for the (256× 256) Lena image for Type 4 impulse noise with different corruption rates (10% - 70%). Noisy FIDRM DWFIDRM PSNR SSIM IQI PSNR SSIM IQI PSNR SSIM IQI 10% 20.41 0.9509 0.3495 40.85 0.9993 0.9605 42.62 0.9994 0.9790 20% 17.57 0.9059 0.2263 37.86 0.9986 0.8989 39.29 0.9988 0.9607 30% 15.69 0.8620 0.1684 34.05 0.9971 0.8033 37.45 0.9983 0.9405 40% 14.56 0.8239 0.1334 31.69 0.9952 0.7166 35.41 0.9976 0.9227 50% 13.44 0.7851 0.1072 29.43 0.9920 0.6307 33.58 0.9963 0.8915 60% 12.73 0.7514 0.0844 26.96 0.9860 0.5441 32.36 0.9952 0.8599 70% 12.06 0.7182 0.0696 25.30 0.9801 0.4760 30.53 0.9929 0.8238 As one can see in Fig.7, DWMFIDRM gives better results for higher impulse noise densities based on the three performance measures, PSNR, SSIM and IQI. Figs.11, 12 and 13 show the one dimensional signal representing a specific row of the corrupted (10%, 30% & 60%) and restored Lena, Baboon and circuit images. 364 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) (a) (b) (c) Fig. 7 Comparison of DWMFIDRM with FIDRM based on (a) PSNR (b) IQI (c) SSIM Fuzzy Inf. Eng. (2012) 4: 351-369 365 Fig. 8 (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWM- FIDRM (g) Edgemap of original (h) Edgemap of noisy image (i) Edgemap of mean (j) Edgemap of median (k) Edgemap of FIDRM (l) Edgemap of DWFIDRM (m) Signal of original (n) IQI of noise (o) IQI of mean (p) Signal of median (q) IQI of FIDRM (r) IQI of DWFIDRM(noise density 10%) Fig. 9 (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWM- FIDRM (g) Edgemap of original (h) Edgemap of noisy image (i) Edgemap of mean (j) Edgemap of median (k) Edgemap of FIDRM (l) Edgemap of DWFIDRM (m) IQI of original (n) IQI of noise (o) IQI of mean (p) IQI of median (q) IQI of FIDRM (r) IQI of DWFIDRM(noise density 30%) 366 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) Fig. 10 (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWM- FIDRM (g) Edgemap of original (h) Edgemap of noisy image (i) Edgemap of mean (j) Edgemap of median (k) Edgemap of FIDRM (l) Edgemap of DWFIDRM (m) IQI of original (n) IQI of noise (o) IQI of mean (p) IQI of median (q) IQI of FIDRM (r) IQI DWFIDRM(noise density 60%) Fig. 11 Signal of (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWMFIDRM with noise density 10%(Lena image) Fuzzy Inf. Eng. (2012) 4: 351-369 367 Fig. 12 Signal of (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWMFIDRM with noise density 30%(Baboon image) (a) (b) (c) (d) (f) (e) Fig. 13 Signal of (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWMFIDRM with noise density 60%(circuit image) 368 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) 7. Conclusion A new two-step iterative filtering algorithm, based on fuzzy based directional weighted median filter for impulse noise detection and reduction, has been presented. This fil- ter combination is found effective in reducing four types of impulse noise models. The attractive feature of this work is that all the noise-free pixels remain unchanged. Thus the overall visual quality and fine details of the noisy image are preserved af- ter the proposed denoising operation. This filter is especially developed for reducing four types of impulse noise models. Its main feature is that it leaves the pixels which are noise-free unchanged. Experimental results show the feasibility of the new filter. A numerical measure, such as the PSNR, SSIM, IQI and visual observations (Figs.8, 9,10,11,12 and 13) show convincing results for grayscale images. Acknowledgements We express our deep sense of gratitude and thanks to all the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. References 1. Tukey J W (1971) Exploratory data analysis. Addison-Wesley. Reading 2. Astola J, Kuosmanen P (1997) Fundamentals of nonlinear digital filtering. Boca Raton. CRC Press 3. Pitas I, Venetsanopoulos A (1990) Nonlinear digital filters: principles and application. Norwell. Kluwer Academic Publishers 4. Wang J H, Liu W J, Lin L D (2002) Histogram-based fuzzy filter for image restoration. IEEE Trans- actions on Systems, Man, and Cybernetics 32(2): 230-238 5. Lee C S, Guo S M, Hsu C Y (2004) A novel fuzzy filter for impulse noise removal. Lecture Notes in Computer Science: 375-380 6. Lee C S, Guo S M, Hsu C Y (2005) Genetic-based fuzzy image filter and its applications to image processing. IEEE Transactions on Systems, Man, and Cybernetics 35 (4): 694-711 7. Schulte S, de Witte V, Nachtegael M, Van der Weken D, Kerre E E (2007). Fuzzy random impulse noise reduction method. Fuzzy Sets and Systems 158(3): 270-283 8. Petrovic N I, Crnojevic V (2008) Universal impulse noise filter based on genetic programming. IEEE Transactions on Image Processing 17(7): 1109-1120 9. Dong Y, Xu S (2007) A new directional weighted median filter for removal of random-valued impulse noise. IEEE Signal Processing Letters 14(3): 193-196 10. Hussain A, Jaffar A, Mirza A M (2009) Detail preserving fuzzy filter for impulse noise removal. International Journal of Innovative Computing, Information and Control 5(10(B)): 3583-3591 11. Farbiz F, Menhaj M B (2000) A fuzzy logic control based approach for image filtering. In: E. E. Kerre, M. Nachtegael (Eds.), Fuzzy Techniques in Image Processing, Physica-Verlag, Heidelberg, Germany 52: 194-221 12. Farbiz F, Menhaj M B, Motamedi S A (1998) Edge preserving image filtering based on fuzzy logic. Proceedings of the 6th European Congress on Intelligent Techniques and Soft Computing (EUFIT) Conference: 1417-1421 13. Schulte S, Nachtegael M, de Witte V, Van der Weken D, Kerre E E (2006) A fuzzy impulse noise detection and reduction method. IEEE Trans. on Image Processing 15(5): 1153-1162 14. AyyazHussain, ArfanJaffar M, Zia Ul-Qayyum, Anwar M Mirza (2010) Directional weighted median based fuzzy filter for random-valued impulse noise removal. ICIC Express Letters, Part B: Applica- tions 1(1): 9-14 15. Van De Ville D, Nachtegael M, Van derWeken D, Kerre E E, Philips W (2001) Noise reduction by fuzzy image filtering. IEEE Trans. Fuzzy Systems 11(4): 429-436 16. Nachtegael M, Van der Weken D, Kerre E E (2002) Fuzzy techniques in image processing: three case studies. Int. J. Comput. Anticipatory Systems 12: 89-104 Fuzzy Inf. Eng. (2012) 4: 351-369 369 17. Dong Y, Xu S (2007) A new directional weighted median filter for removal of random-valued impulse noise. IEEE Signal Processing Letters 14(3): 193-196 18. Russo F, Ramponi G (1996) A fuzzy filter for images corrupted by impulse noise. IEEE Signal Processing Letters 3(6): 168-170 19. Russo F, Ramponi G (1996) Removal of impulse noise using a FIRE filter. Proc. 3rd IEEE Int. Conf. Image Processing: 975-978 20. Russo F (1999) Fire operators for image processing. Fuzzy Sets Syst. 103: 265-275 21. Arakawa K (1996) Median filter based on fuzzy rules and its application to image restoration. Fuzzy Sets Syst. 77: 3-13 22. Kerre E E, Nachtegael M (2000) Fuzzy rule-based image processing with optimization. Fuzzy Tech- niques in Image Processing 52: 222-247 23. Lee C S, Kuo Y H, Yu P T (1997) Weighted fuzzy mean filters for image processing. Fuzzy Sets Syst.89: 157-180 24. Lee C S, Kuo Y H (2000) Adaptive fuzzy filter and its application to image enhancement. In: Kerre E E, Nachtegael M (Eds.), Fuzzy Techniques in Image Processing, Physica-Verlag, Heidelberg, Ger- many 52: 172-193 25. Kwan H K (2003) Fuzzy filters for noise reduction in images. in: Nachtegael M, Van der Weken D, Van De Ville D, Kerre E E (Eds.), Fuzzy Filters for Image Processing, Physica-Verlag, Heidelberg, Germany 122: 25-53 26. Farbiz F, Menhaj (2000) M B. A fuzzy logic control based approach for image filtering. In: Kerre E E, Nachtegael M (Eds.), Fuzzy Techniques in Image Processing, Physica-Verlag, Heidelberg, Germany 52: 194-221 27. Farbiz F, Menhaj M B, Motamedi S A (1998) Edge preserving image filtering based on fuzzy logic. Proc. 6th EUFIT Conf.: 1417-1421 28. Wang J H, Chiu H C (1999) An adaptive fuzzy filter for restoring highly corrupted images by his- togram estimation. Proc. Natl. Sci. Council, ROC(A) 23(5): 630-643 29. Kalaykov I, Tolt G (2003) Real-time image noise cancellation based on fuzzy similarity. In: Nachte- gael M, Van der Weken D, Van De Ville D, Kerre E E (Eds.). Fuzzy Filters for Image Processing, Physica-Verlag, Heidelberg, Germany 122: 54-71 30. Ko S J, Lee Y H (1991) Center weighted median filters and their applications to image enhancement. IEEE Trans. Circuits Syst. 38(9): 984-993 31. Chen T, Ma K K, Chen L H (1999) Tri-state median filter for image denoising. IEEE Trans. Image Processing 8(12): 1834-1838 32. Hardie R C, Boncelet C G(1993) LUM filters: A class of rank-order-based filters for smoothing and sharpening. IEEE Trans. Signal Processing 41(3): 1834-1838 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Fuzzy Based Directional Weighted Median Filter for Impulse Noise Detection and Reduction

Fuzzy Based Directional Weighted Median Filter for Impulse Noise Detection and Reduction

Abstract

AbstractThe known median-based denoising methods tends to work well for restoring the images corrupted by random-valued impulse noise with low noise level, but it fails in denoising highly corrupted images. In this paper, a new noise reduction method based on directional weighted median based fuzzy impulse noise detection and reduction method (DWMFIDRM) has been proposed, which has been specially developed for denoising all categories of impulse noise. The contribution of this paper is...
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Fuzzy Inf. Eng. (2012) 4: 351-369 DOI 10.1007/s12543-012-0120-2 ORIGINAL ARTICLE Fuzzy Based Directional Weighted Median Filter for Impulse Noise Detection and Reduction Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy Received: 6 February 2012/ Revised: 27 September 2012/ Accepted: 12 November 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract The known median-based denoising methods tends to work well for restor- ing the images corrupted by random-valued impulse noise with low noise level, but it fails in denoising highly corrupted images. In this paper, a new noise reduction method based on directional weighted median based fuzzy impulse noise detection and reduction method (DWMFIDRM) has been proposed, which has been specially developed for denoising all categories of impulse noise. The contribution of this paper is threefold. The main contribution of the novel impulse noise reduction tech- nique lies in the unification of three different methods; the impulse noise detection phase utilizing the concept of fuzzy gradient values, edge-preserving noise reduc- tion phase based on the directional weighted median of the neighboring pixels and a final filtering step in order to deal with noisy pixels of non-zero degree. Such a unique combination has improved the efficiency of this method for high density noise removal. The experimental results of our proposed method have a significant im- provement when compared to other existing filters for high density noise removal. This paper utilizes the concept of fuzzy gradient values. The noise reduction phase that preserves edge sharpness is based on the directional weighted median of neigh- boring pixels. Final filtering phase is performed only when there is non-zero degree of noise pixels. This phase makes our method more efficient in high noise density. Experimental results show that DWMFIDRM provides a significant improvement on other existing filters. Keywords Fuzzy filter · Directional weighted median · Impulse noise · Noise re- duction 1. Introduction Riji R () · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy Department of Computer Science, University of Kerala, Kariavattom, Thiruvananthapuram, 695581, India email: rijir kmr@yahoo.com 352 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) Digital images play a vital role in our daily lives as well as in an inevitable explo- ration of scientific research areas like satellite television, geographical information systems, astronomy, and in medical imaging technologies like magnetic resonance imaging (MRI), and computed tomography (CT). But more often challenging situa- tions are faced in all image processing domains where the image sensors suffer from noise contamination. Besides this, the issues caused during the data acquisition pro- cess, wrong machine readings, interference causedby natural phenomena, bit errors occurred during transmission and the transmission errors caused in the compressed image degrades the data of interest. This makes it difficult for analyzing and process- ing the images in a better way. Image denoising is an ultimate solution to deal with such problems and hence is taken as the first and necessary step before image anal- ysis. But all methodologies for image denoising must deal with efficient recovery of the corrupted image without causing any loss of information. Our method is robust against high-density corruptions caused in images either through natural phenomena or through the interferences caused during machine acquisition. The need for image denoising is encountered in many medical applications like MRI, ultrasound and CT scan images. Among these modalities, more emphasis is given to MRI image denoising in the medical field, as they are affected by random noise generated during the image acquisition process. Hence, denoising has to be done more effectively, since the presence of noise may cause damage to its visual quality and its true contrast as well by making it difficult to analyze the low contrast objects in the image. Hence, a novel image denoising scheme has to recover even the fine image details hidden within inside the objects in the image. The main property of a good image denoising model is its edge preservation af- ter the noise removal process. Traditionally, linear models have been used for noise removal. But they lack its ability for the better preservation of edges in a good and ef- fective manner. Instead the discontinuities in the image regarded as edges are smeared out by damaging its visual quality. Fuzzy filters are introduced in our approach to deal with the issues incurred in the earlier approaches. Fuzzy filters have succeeded in bringing out an edge-preserving impulse noise removal scheme which preserves even the fine image details. The main property of this filter is its ability to adapt to the local characteristics of a given neighborhood or window by making it well suited for removing the impulse noise while preserving the fine image details. Digital images play an important role both in our daily life lives applications such as satellite television, magnetic resonance imaging, computer tomography as well as in areas of research and technology such as geographical information systems and as- tronomy. Data sets collected by image sensors are generally contaminated by noise. Imperfect instruments, problems with the data acquisition process, and interfering natural phenomena can all degrade the data of interest. Furthermore, noise can be introduced by transmission errors and compression. Thus, denoising is often a neces- sary and the first step to be taken before the images data is analyzed. It is necessary to apply an efficient denoising technique to compensate for such data corruption. Image denoising has become an essential exercise in medical imaging especially the magnetic resonance imaging. Medical images obtained from MRI are the most common tool for diagnosis in medical field. These images are often affected by ran- Fuzzy Inf. Eng. (2012) 4: 351-369 353 dom noise arising in the image acquisition process. The presence of noise not only produces undesirable visual quality but also lowers the visibility of low contrast ob- jects. Noise removal is essential in medical imaging applications in order to enhance and recover fine details that may be hidden in the data. The main property of a good image denoising model is that it will remove noise while preserving edges. Traditionally, linear models have been used. But a back draw of the linear models is that they are not able to preserve edges in a good manner: edges, which are recognized as discontinuities in the image, are smeared out. Fuzzy filters have been suggested as a means of solving the above problem. It succeeded in removing impulse noise while preserving image detail such as edge. The property of fuzzy filters to adapt to the local characteristics in a given window makes it well suited for removing impulse noise while preserving image details. A number of approaches have been proposed for the impulse noise removal. Tukey [1], Astola et al [2] and Pitas et al [3] have utilized median filtering to remove im- pulse noise from the corrupted images. Other filters for removal of impulse noise include histogram based fuzzy filter (HFF) [4], novel fuzzy filter (NFF) [5] by Lee et al, genetic based fuzzy image filter (GFIF) [6], fuzzy impulse noise detection and reduction method (FIDRM) [7], fuzzy random impulse noise reduction method (FRINR) [8], universal impulse noise filter based on genetic programming [9] by Ne- manja et al, a new directional weighted median (DWM) filter [10] for removal of random valued impulse noise and detail preserving fuzzy filter (DPFF) [11] are the examples of the most recent filters. Other approaches based on fuzzy-based impulse noise reduction are fuzzy impulse noise detection and reduction method (FIDRM) [13] and directional weighted median based fuzzy filter for random-valued impulse noise removal [14]; nevertheless, all of these methods face the same major drawback, such that higher the noise level, lower is its noise reduction capability. To overcome this drawback, we propose a more efficient noise reduction method for images having high noise density. The proposed method consists of two phases: noise detection and reduction step. The detection method is based on fuzzy gradient values which determine whether a pixel is noisy or not. The reduction phase uses the concept of directional weighted median of the neighboring pixels which has been used to calculate the membership degree of each noisy pixel. After that a fuzzy averaging method which makes use of the membership degree is used for the noise reduction step. The paper is organized as follows. For clarity, we have first given a short descrip- tion of different noise models in Section 2. The details of the detection phase are given in Section 3. In Section 4, the proposed filtering method is discussed. Performance measures, experimental results and conclusions are finally presented in Sections 5, 6 and 7. 2. Noise Models To understand the noise detection and reduction method more clearly, we first give a brief description of different noise models. Four impulse noise models are imple- mented for extensively examining the performance of our proposed filter with con- sideration of practical situations. Each model is described in detail as follows. 354 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) 1) Noise Model 1: Noise is modeled as salt-and-pepper impulse noise as prac- ticed. Pixels are randomly corrupted by two fixed extreme values, 0 and 255 (for 8-bit monochrome image), generated with the same probability. That is, for each im- age pixel at location (i, j) with the intensity value s , the corresponding pixel of the i, j noisy image will be x in which the probability density function of x is i, j i, j ⎪ , for x = 0, ⎪ 2 f (x) = (1) 1− p, for x = s , ⎪ i, j , for x = 255. 2) Noise Model 2: For Model 2, it is similar to Model 1, except that each pixel might be corrupted by either “pepper” noise (i.e., 0) or “salt” noise with unequal probabilities. That is p1 , for x = 0, ⎪ 2 f (x) = 1− p, for x = s , (2) ⎪ i, j p2 , for x = 255, where p = p1+ p2 is the noise density and p1 not equal to p2. 3) Noise Model 3: Instead of two fixed values, impulse noise could be more realisti- cally modeled by two fixed ranges that appear at both ends with a length of m each, respectively. For example, if m is 10, noise will equal likely be any values in the range of either [0, 10] or [245, 255]. That is ⎪ , for 0 ≤ x < m, ⎪ 2m f (x) = (3) 1− p, for x = s , ⎪ i, j , for (255− m) < x ≤ 255, 2m where p is the noise density. 4) Noise Model 4: Model 4 is similar to Model 3, except that the densities of low- intensity impulse noise and high-intensity impulse noise are unequal. That is ⎪ p1 , for 0 ≤ x < m, ⎪ 2m f (x) = 1− p, for x = s , (4) ⎪ i, j p2 , for (255− m) < x ≤ 255, 2m where p = p1+ p2 is the noise density and p1 not equal to p2. 3. Fuzzy Impulse Noise Detection In the noise detection phase, a fuzzy impulse noise detection method has been com- bined with FIDRM [13]. This detection method uses fuzzy gradient values, intro- duced with the GOA filter [15, 16], to determine whether a certain pixel is corrupted with impulse noise or not. Fuzzy Inf. Eng. (2012) 4: 351-369 355 The detection method first calculates the basic gradient values and relates gradient values for each pixel to determine if a central pixel is corrupted with impulse noise or not. Finally, the method defines eight fuzzy gradient values for each of the eight directions. These values indicate in which degree the central pixel can be seen as an impulse noise pixel. The fuzzy gradient value∇ A(i, j) for direction R (R ∈{NW, N, NE, E, SE, S, SW, W}) is calculated by the following fuzzy rule. IF|∇ A(i, j) | is large AND|∇ A(i, j) | is small OR IF|∇ A(i, j) | is large AND|∇ A(i, j) | is small OR IF∇ A(i, j) is big positive AND ∇ A(i, j) AND∇ A(i, j) are big negative R R OR IF∇ A(i, j) is big negative AND ∇ A(i, j) AND∇ A(i, j) are big positive R R THEN∇ A(i, j) is large, where ∇ A(i, j) is the basic gradient value and ∇ A(i, j) and ∇ A(i, j) are the two R R related gradient values for a direction R. To decide if a central pixel (a no border pixel of course) is an impulse noise pixel, the following (fuzzy) rule has been used: IF most of the eight ∇ A(i, j) are large THEN the central pixel A(i, j) is an Impulse noise pixel. After detecting a pixel as noisy based on the above fuzzy rule, a histogram is con- structed to find the presence of impulse noise density. Thus, we conclude that the presence of peak values in a histogram indicates the presence of impulse noise in an image or noise-corrupted; otherwise the image is noise-free. The entire procedure of the detection phase is shown below in Fig.1. If a pixel is detected as noisy by using the above rule, then use the histogram to find the presence of impulse noise. Construct histogram using the pixels which is detected as noisy by the fuzzy rule. If the histogram contains peaks, then conclude that the image is noisy otherwise the image is free from impulse noise. Finally, this detection phase ends with the procedure pictured in Fig.1. If the detection phase has detected impulse noise, then we perform the proposed filtering method explained in the next section. Otherwise, we leave the image un- changed. Our proposed filtering method based on “fuzzy based directional weighted me- dian filter for impulse noise detection and reduction” is used after the impulse noise detection phase for effective denoising by improving the visual quality of the noisy image with better fine detail and edge preservation. However, if the detection phase has estimated the presence of a noise-free image, then the image is left unchanged. The proposed filtering method is explained in the next section. 4. Proposed Filtering Method We introduce a new filtering method which uses the concepts of directional weighted median introduced in [17]. This section is structured as follows. The parameter cal- culation (a, b, c, d), which have been used to construct the fuzzy set as more or less 356 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) Input: HIST : the noise histogram matrix HIST(i) : indicates the amount of selected noise pixels with gray scale value i. If (i < 0) or (i < 255) HIST(i)=0 TOT : the total number of detected noise pixels HIST(m) : the maximum histogram value assumed to be located at gray scale value m = [0, 255] 1) IF (HIST(m)/TOT)×100 < THR1) 2) NO IMPULSE NOISE DETECTED 3) ELSE 4) IF (HIST(i) / HIST(i− 1) + HIST(i) + HIST(i+ 1) ≥ 0.95 5) Perform Filtering For Noise Reduction 6) END IF 7) END IF Fig. 1 Pseudo-code for decision procedure impulse noise. After this, a first filtering operation is iterated based on the member- ship function that represents the fuzzy set w(). Finally, this section is terminated with the explanation of other iterations and some stopping criteria. First, we explain the calculation of the parameters (a, b, c, d), which are used to construct the fuzzy set more or less impulse noise. Afterwards, we construct our first filtering iteration based on the membership function that represents this fuzzy set w(). The explanation of the other iterations and some stopping criteria finally terminate this section. A. Configuration of the Parameters Before introducing a new fuzzy filtering method, a basic assumption has been re- viewed, which is a noise-free image consisting of locally smoothly varying areas sep- arated by edges. More focus is given to the edges aligned with four main directions as shown in Fig.2. Before introducing the new fuzzy filtering method, we review a basic assumption, that is, a noise-free image consists of locally smoothly varying areas separated by edges. Here, we only focus on the edges aligned with four main directions shown in Fig.2. Let S (k = 1 to 4) denotes a set of coordinates aligned with the kth direction cen- tered at (0, 0), i.e., S ={(−2,−2), (−1,−1), (0, 0), (1, 1), (2, 2)}, S ={(0,−2), (0,−1), (0, 0), (0, 1), (0, 2)}, (5) S ={(2,−2), (1,−1), (0, 0), (−1, 1), (−2, 2)}, S ={(−2, 0), (−1, 0), (0, 0), (1, 0), (2, 0)}. 4 Fuzzy Inf. Eng. (2012) 4: 351-369 357 In order to perform the filtering, we consider a window of size 5× 5 and construct trapezoidal shaped fuzzy membership function for the fuzzy set more or less impulse noise as shown in Fig.3. The parameter [a, b, c, d] of the membership function de- pends on the value of directional weighted median as shown in Equation (6). Fig. 2 Four directions used for impulse detection maxX(i+ x, j+ y)− minX(i+ x, j+ y) dev(i, j)= x, y∈{−2,··· ,+2}, a = med − 1.1dev(i, j), dir (6) b = med − dev(i, j), dir c = med + dev(i, j), dir d = med + 1.1dev(i, j). dir To determine directional weighted median, first of all we calculate the standard deviation σ of the pixel intensities aligned with each of the four directions D (k k k =1, ··· , 4) [17]. Since the standard deviation describes how tightly all the values are clustered around the mean in the set of pixels shows that the four pixels aligned with this direction are closest to each other. Therefore, the center value should also be close to them in order to keep the edges intact. Thus, pixels aligned with the direction having minimum standard deviation are repeated in the considered window and then the median is calculated. This median is called the med , which is used in dir Equation (6) to compute the parameters of the fuzzy membership function more or less impulse noise. The use of med in fuzzy set construction makes the fuzzy set dir more or less impulse noise robust against noisy pixels and only the pixel values which are tightly close together under the considered window will take part in estimating the 358 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) restored value of the noisy pixel. The use of fuzzy membership more or less impulse noise gives appropriate importance to the pixels in estimating the non-noisy value of the considered pixel based on their homogeneity with the neighbors. Therefore, the restored value will be more accurate in better preservation of fine details including edge and texture information. Therefore, the restored value will be more accurate and helps in preserving the small image details such as edge and texture information. Fig. 3 Membership function “more or less impulse noise” After constructing the fuzzy set more or less impulse noise, the weights (degree of membership in fuzzy set more or less impulse noise) for the pixel values under the window of size 5 × 5 are determined. Finally, these weights are used for filtering which is explained in the first iteration. B. First Iteration The filtering step of the first iteration is given in the pseudo code presented in Fig.4. This method is based on the membership function “more or less impulse noise” (w). The corresponding membership degree of a certain intensity value X(i, j) is denoted as w(i, j). w(i, j) is computed based on the fuzzy membership function shown in Fig.3. A degree one (zero) indicates that the intensity value is noisy for sure (not noisy for sure). When the degree is between one and zero, then there is some kind of uncertainty. We only filter pixels which are part of the support of the fuzzy set more or less impulse noise (pixels which have a nonzero membership degree in this fuzzy set). Otherwise, we leave the pixel unchanged. As shown in line three of Fig.4, we use, a 3 × 3 window around the filtered pixel. In addition, we use the standard negation to express the membership degrees in the fuzzy set noise free [5, 6].The Fuzzy Inf. Eng. (2012) 4: 351-369 359 weighted average AVG is then computed as follows: 2 2 X(i+ k, j+ l)w(i+ k, j+ l) k=−2 l=−2 AVG = + w(i, j)X(i, j). (7) 2 2 w(i+ k, j+ l) k=−2 l=−2 C. Next Iterations For special cases where an image is contaminated by high density impulse noise, there is a possibility for a side effect after the first iteration where by the impulse noise remains clustered around one or more pixels. In order to deal with this problem, few more iterations are performed similar to the first iteration for noise reduction. During each iteration, the modified image in the previous iteration has been used. After the first iteration, it is possible as a side effect (especially with high initial impulse noise) that the impulse noise is clustered around one or more pixels. To reduce these noisy pixels, we will provide some more iteration that are similar to the first one. In each iteration, we use the modified image of the previous performed iteration. Input : A: The noisy image with impulse noise w(A(i, j)) : the membership degree of the fuzzy set “more or less impulse noise” F: The output image 1) FOR each non-border pixel (i, j)in A 2) IF A(i, j) in supp (“more or less impulse noise”) 3) F(i, j)=(1− w(i, j)) * AVG 4) ELSE 5) F(i, j) = A(i, j) 6) END IF 7) END FOR Fig. 4 Pseudo code for filtering D. Stopping Criteria There are several techniques to improve the efficiency of the filtering phase. We will focus on the stopping criteria and the amount of pixels that must be scanned during an iteration. During the first iteration, we check every pixel. If the pixel value does not belong to the support of the fuzzy set more or less impulse noise, then we do not change this pixel value, not only in this iteration neither in the other ones. By remembering only the positions of pixels, whose pixel value is an element of the support of the fuzzy set more or less impulse noise, we can drastically reduce the scanning amount in the next iterations. A stopping criterion is essential at this instance. For this, # is defined which indicates the amount of pixel values belonging to the support of the fuzzy set more 360 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) or less impulse noise in the eth iteration, then we can apply the following stopping criteria. 1) There are no pixel values in the support of the fuzzy set (# = 0) in any iteration (e ≥ 2). 2) # is equal to # . This indicates that the resulting pixels (pixels which still e e−1 are element of the support of the fuzzy set more or less impulse noise) are not noisy even with the nonzero membership degrees. E. Final Step of Filtering After completing the filtering process based on directional weighted median, the final filtering step is performed. This filtering phase is based on fuzzy averaging [13]. If the amount of pixel values which belong to the support of the fuzzy set more or less impulse noise in the previous iteration is not equal to zero, then only we go for the final step of filtering. This phase also uses the parameters (a , b , c , d ) for constructing the fuzzy set k k k k more or less impulse noise. But the calculation of the parameters is slightly differ- ent from that of the previous filtering phase. The configuration of the parameter is described in the procedure pictured in Fig.5. Input : HIST : The noise histogram matrix TOT : the total number of detected noise pixels 1) cum = 0 2) FOR k =1to5 3) kmax = select the k-th greatest y value of the noise histogram 4) kpos = the corresponding gray scale value where the maximum is reached 5) IF (kmax/TOT) + cum ≥ THR2 6) p =kpos 7) Set the four parameters (a , b , c , d )to(p , p , p , p ) k k k k k k k k 8) cum = cum + (kmax/TOT) 9) ELSE 10) Break 11) END IF 12) END FOR Fig. 5 Pseudo code for configuring parameters in the final filtering phase The filtering step of the final iteration is given in the pseudo code presented in Fig.6. This method is based on the membership function “more or less impulse noise” (μ). The corresponding membership degree of a certain intensity value A(i, j) is de- noted as μ(A(i, j)). A degree one (zero) indicates that the intensity value is noisy for sure (not noisy for sure). When the degree is between one and zero, then there is some Fuzzy Inf. Eng. (2012) 4: 351-369 361 kind of uncertainty. We only filter pixels which are part of the support of the fuzzy set more or less impulse noise (pixels which have a nonzero membership degree in this fuzzy set). Otherwise, the pixel values are left unchanged. Otherwise, we leave the pixel unchanged. 5. Performance Measures The following performance measures are used for the evaluation of our method. 1) PSNR (peak signal to noise ratio), 2) SSIM (structural similarity index measure), 3) IQI (image quality index). As a measure of objective dissimilarity between a filtered image and the original one, we use the mean square error (MSE) and the peak signal to noise ratio (PSNR) in decibels N M [Org(i, j)− Img(i, j)] i=1 j=1 MS E(Img, Org) = , (8) MN PS NR(MS E(Img, Org)) = 10 log , (9) MS E(Img, Org) where Org is the original image, Img is the filtered image of size M× N, and S is the maximum possible pixel value (with 8-bit integer values the maximum will be 255). Input : A: the noisy image with impulse noise μ(A(i, j)) : the membership degree for the fuzzy set “more or less impulse noise” F : the output image 1) FOR each non-border pixel (i, j)in A 2) IF A(i, j) in supp (“more or less impulse noise”) 1 1 1−μ(A(i+h, j+l))A(i+h, j+l) h=−1 l=−1 3) F(i, j) = 1 1 1−μ(A(i+h, j+l)) h=−1 l=−1 4) ELSE 5) F(i, j) = A(i, j) 6) END IF 7) END FOR Fig. 6 Pseudo code for final filtering The SSIM is a method for measuring the similarity between two images. The SSIM index is a full reference metric, in other words, the measuring of image quality based on an initial uncompressed or distortion-free image as reference. The SSIM metric is calculated on various windows of an image. The measure between two 362 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) windows x and y of common size N × N is (2μ μ + c )(2σ + c ) x y 1 xy 2 SS IM(x, y) = . (10) 2 2 2 2 (μ +μ + c )(σ +σ + c ) 1 2 x y x y 6. Experimental Results In this section, we compare the efficiency and performance of DWMFIDRM for grayscale images with other well-known filters for impulse noise reduction. The first group of filters consists of the FIRE filters (fuzzy inference rule by else-action filters). Table 1 : PSNR results for the (256× 256) Lena image for Type 4 impulse noise with noise densities 20%, 30% and 50%. PSNR 20% 30% 50% Original 13.68 11.78 8.75 CWM(3X3) 27.07 23.48 15.18 CWM(7X7) 26.38 26.01 24.32 TSM(3X3) 27.08 23.74 24.75 TSM(7X7) 26.53 23.83 15.47 LUM 26.74 23.74 18.43 MED(3X3) 26.74 23.74 15.05 MED(5X5) 26.35 26.23 22.43 MED(7X7) 25.03 25.34 23.89 ATMAV 21.37 21.02 20.17 FSD 26.52 23.25 14.78 HAF 24.35 22.01 25.56 AWFM 27.83 26.43 26.87 SFCF 25.32 21.12 14.69 EIFCF 26.74 24.53 17.43 MIFCF 26.03 23.05 17.32 IFCF 26.72 24.32 17.32 FIRE 24.01 20.53 13.48 FMF 27.93 23.54 15.67 DSFIRE 22.56 21.48 17.85 PWLFIRE 24.39 19.49 12.73 FIDRM 32.64 28.67 22.08 DWFIDRM 36.90 34.90 31.01 The idea behind these filters is that they try to calculate positive and negative correc- tion terms in order to express the degree of noise for a certain pixel. We distinguish Fuzzy Inf. Eng. (2012) 4: 351-369 363 three FIRE filters: the normal FIRE [18] (fuzzy inference rule by else-action filters), the DS-FIRE [19] (dual step FIRE), and the PWLFIRE [20] (piecewise linear FIRE). A second group of filters contains filters which are extensions of classical median (de- noted by MED) and weighted filters. We use the FMF [21, 22] (fuzzy median filter), AWFM [23, 24] (adaptive weighted fuzzy mean) and the ATMAV [25] (asymmetri- cal triangular fuzzy filter with moving average center). The FCF (fuzzy control-based filters) constitute a third group of filters. These filters correct a certain central pixel value according to some features of some luminance (pixel values) differences be- tween the central pixel value and some neighbor pixel values. In the literature, we know that the IFCF [26] (iterative fuzzy control based filter), the MIFCF [26] (mod- ified IFCF), the EIFCF [26] (extended IFCF), and the SFCF [27] (smoothing fuzzy control based filter). Furthermore, there exist many other types of filters, such as, e.g., the histogram adaptive fuzzy filter which of course uses the histogram of an im- age [28] (HAF) or the fuzzy similarity filter [29] (FSB), where the local similarity between some patterns is used. Besides all these fuzzy filters, some other popular filters exist. We use the CWM [30] (center weighted median), the TSM [31] (tri-state median filter), and the LUM [32] (lower-upper-middle filter) for comparison. We also compare our method with FIDRM. We used the well-known Lena, Baboon and circuit images of size, 256× 256, 204 × 204 and 280 × 272 respectively, to compare DWMFIDRM with the other filters. From Tables 1 and 2, we can see that DWMFIDRM outperforms all the conventional as well as advanced filters, especially at high noise levels. In all the experiments, we have used the noise Model 4, since the other noise Models (1, 2 and 3) are special cases of noise Model 4. Table 2: PSNR results for the (256× 256) Lena image for Type 4 impulse noise with different corruption rates (10% - 70%). Noisy FIDRM DWFIDRM PSNR SSIM IQI PSNR SSIM IQI PSNR SSIM IQI 10% 20.41 0.9509 0.3495 40.85 0.9993 0.9605 42.62 0.9994 0.9790 20% 17.57 0.9059 0.2263 37.86 0.9986 0.8989 39.29 0.9988 0.9607 30% 15.69 0.8620 0.1684 34.05 0.9971 0.8033 37.45 0.9983 0.9405 40% 14.56 0.8239 0.1334 31.69 0.9952 0.7166 35.41 0.9976 0.9227 50% 13.44 0.7851 0.1072 29.43 0.9920 0.6307 33.58 0.9963 0.8915 60% 12.73 0.7514 0.0844 26.96 0.9860 0.5441 32.36 0.9952 0.8599 70% 12.06 0.7182 0.0696 25.30 0.9801 0.4760 30.53 0.9929 0.8238 As one can see in Fig.7, DWMFIDRM gives better results for higher impulse noise densities based on the three performance measures, PSNR, SSIM and IQI. Figs.11, 12 and 13 show the one dimensional signal representing a specific row of the corrupted (10%, 30% & 60%) and restored Lena, Baboon and circuit images. 364 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) (a) (b) (c) Fig. 7 Comparison of DWMFIDRM with FIDRM based on (a) PSNR (b) IQI (c) SSIM Fuzzy Inf. Eng. (2012) 4: 351-369 365 Fig. 8 (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWM- FIDRM (g) Edgemap of original (h) Edgemap of noisy image (i) Edgemap of mean (j) Edgemap of median (k) Edgemap of FIDRM (l) Edgemap of DWFIDRM (m) Signal of original (n) IQI of noise (o) IQI of mean (p) Signal of median (q) IQI of FIDRM (r) IQI of DWFIDRM(noise density 10%) Fig. 9 (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWM- FIDRM (g) Edgemap of original (h) Edgemap of noisy image (i) Edgemap of mean (j) Edgemap of median (k) Edgemap of FIDRM (l) Edgemap of DWFIDRM (m) IQI of original (n) IQI of noise (o) IQI of mean (p) IQI of median (q) IQI of FIDRM (r) IQI of DWFIDRM(noise density 30%) 366 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) Fig. 10 (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWM- FIDRM (g) Edgemap of original (h) Edgemap of noisy image (i) Edgemap of mean (j) Edgemap of median (k) Edgemap of FIDRM (l) Edgemap of DWFIDRM (m) IQI of original (n) IQI of noise (o) IQI of mean (p) IQI of median (q) IQI of FIDRM (r) IQI DWFIDRM(noise density 60%) Fig. 11 Signal of (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWMFIDRM with noise density 10%(Lena image) Fuzzy Inf. Eng. (2012) 4: 351-369 367 Fig. 12 Signal of (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWMFIDRM with noise density 30%(Baboon image) (a) (b) (c) (d) (f) (e) Fig. 13 Signal of (a) Original (b) Noisy image (c) Mean (d) Median (e) FIDRM (f) DWMFIDRM with noise density 60%(circuit image) 368 Riji R · Keerthi A S Pillai · Madhu S. Nair · M. Wilscy (2012) 7. Conclusion A new two-step iterative filtering algorithm, based on fuzzy based directional weighted median filter for impulse noise detection and reduction, has been presented. This fil- ter combination is found effective in reducing four types of impulse noise models. The attractive feature of this work is that all the noise-free pixels remain unchanged. Thus the overall visual quality and fine details of the noisy image are preserved af- ter the proposed denoising operation. This filter is especially developed for reducing four types of impulse noise models. Its main feature is that it leaves the pixels which are noise-free unchanged. Experimental results show the feasibility of the new filter. A numerical measure, such as the PSNR, SSIM, IQI and visual observations (Figs.8, 9,10,11,12 and 13) show convincing results for grayscale images. Acknowledgements We express our deep sense of gratitude and thanks to all the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. References 1. Tukey J W (1971) Exploratory data analysis. Addison-Wesley. Reading 2. Astola J, Kuosmanen P (1997) Fundamentals of nonlinear digital filtering. Boca Raton. CRC Press 3. Pitas I, Venetsanopoulos A (1990) Nonlinear digital filters: principles and application. Norwell. Kluwer Academic Publishers 4. Wang J H, Liu W J, Lin L D (2002) Histogram-based fuzzy filter for image restoration. IEEE Trans- actions on Systems, Man, and Cybernetics 32(2): 230-238 5. Lee C S, Guo S M, Hsu C Y (2004) A novel fuzzy filter for impulse noise removal. Lecture Notes in Computer Science: 375-380 6. Lee C S, Guo S M, Hsu C Y (2005) Genetic-based fuzzy image filter and its applications to image processing. IEEE Transactions on Systems, Man, and Cybernetics 35 (4): 694-711 7. Schulte S, de Witte V, Nachtegael M, Van der Weken D, Kerre E E (2007). Fuzzy random impulse noise reduction method. Fuzzy Sets and Systems 158(3): 270-283 8. Petrovic N I, Crnojevic V (2008) Universal impulse noise filter based on genetic programming. IEEE Transactions on Image Processing 17(7): 1109-1120 9. Dong Y, Xu S (2007) A new directional weighted median filter for removal of random-valued impulse noise. IEEE Signal Processing Letters 14(3): 193-196 10. Hussain A, Jaffar A, Mirza A M (2009) Detail preserving fuzzy filter for impulse noise removal. International Journal of Innovative Computing, Information and Control 5(10(B)): 3583-3591 11. Farbiz F, Menhaj M B (2000) A fuzzy logic control based approach for image filtering. In: E. E. Kerre, M. Nachtegael (Eds.), Fuzzy Techniques in Image Processing, Physica-Verlag, Heidelberg, Germany 52: 194-221 12. Farbiz F, Menhaj M B, Motamedi S A (1998) Edge preserving image filtering based on fuzzy logic. Proceedings of the 6th European Congress on Intelligent Techniques and Soft Computing (EUFIT) Conference: 1417-1421 13. Schulte S, Nachtegael M, de Witte V, Van der Weken D, Kerre E E (2006) A fuzzy impulse noise detection and reduction method. IEEE Trans. on Image Processing 15(5): 1153-1162 14. AyyazHussain, ArfanJaffar M, Zia Ul-Qayyum, Anwar M Mirza (2010) Directional weighted median based fuzzy filter for random-valued impulse noise removal. ICIC Express Letters, Part B: Applica- tions 1(1): 9-14 15. Van De Ville D, Nachtegael M, Van derWeken D, Kerre E E, Philips W (2001) Noise reduction by fuzzy image filtering. IEEE Trans. Fuzzy Systems 11(4): 429-436 16. Nachtegael M, Van der Weken D, Kerre E E (2002) Fuzzy techniques in image processing: three case studies. Int. J. Comput. Anticipatory Systems 12: 89-104 Fuzzy Inf. Eng. (2012) 4: 351-369 369 17. Dong Y, Xu S (2007) A new directional weighted median filter for removal of random-valued impulse noise. IEEE Signal Processing Letters 14(3): 193-196 18. Russo F, Ramponi G (1996) A fuzzy filter for images corrupted by impulse noise. IEEE Signal Processing Letters 3(6): 168-170 19. Russo F, Ramponi G (1996) Removal of impulse noise using a FIRE filter. Proc. 3rd IEEE Int. Conf. Image Processing: 975-978 20. Russo F (1999) Fire operators for image processing. Fuzzy Sets Syst. 103: 265-275 21. Arakawa K (1996) Median filter based on fuzzy rules and its application to image restoration. Fuzzy Sets Syst. 77: 3-13 22. Kerre E E, Nachtegael M (2000) Fuzzy rule-based image processing with optimization. Fuzzy Tech- niques in Image Processing 52: 222-247 23. Lee C S, Kuo Y H, Yu P T (1997) Weighted fuzzy mean filters for image processing. Fuzzy Sets Syst.89: 157-180 24. Lee C S, Kuo Y H (2000) Adaptive fuzzy filter and its application to image enhancement. In: Kerre E E, Nachtegael M (Eds.), Fuzzy Techniques in Image Processing, Physica-Verlag, Heidelberg, Ger- many 52: 172-193 25. Kwan H K (2003) Fuzzy filters for noise reduction in images. in: Nachtegael M, Van der Weken D, Van De Ville D, Kerre E E (Eds.), Fuzzy Filters for Image Processing, Physica-Verlag, Heidelberg, Germany 122: 25-53 26. Farbiz F, Menhaj (2000) M B. A fuzzy logic control based approach for image filtering. In: Kerre E E, Nachtegael M (Eds.), Fuzzy Techniques in Image Processing, Physica-Verlag, Heidelberg, Germany 52: 194-221 27. Farbiz F, Menhaj M B, Motamedi S A (1998) Edge preserving image filtering based on fuzzy logic. Proc. 6th EUFIT Conf.: 1417-1421 28. Wang J H, Chiu H C (1999) An adaptive fuzzy filter for restoring highly corrupted images by his- togram estimation. Proc. Natl. Sci. Council, ROC(A) 23(5): 630-643 29. Kalaykov I, Tolt G (2003) Real-time image noise cancellation based on fuzzy similarity. In: Nachte- gael M, Van der Weken D, Van De Ville D, Kerre E E (Eds.). Fuzzy Filters for Image Processing, Physica-Verlag, Heidelberg, Germany 122: 54-71 30. Ko S J, Lee Y H (1991) Center weighted median filters and their applications to image enhancement. IEEE Trans. Circuits Syst. 38(9): 984-993 31. Chen T, Ma K K, Chen L H (1999) Tri-state median filter for image denoising. IEEE Trans. Image Processing 8(12): 1834-1838 32. Hardie R C, Boncelet C G(1993) LUM filters: A class of rank-order-based filters for smoothing and sharpening. IEEE Trans. Signal Processing 41(3): 1834-1838

Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Dec 1, 2012

Keywords: Fuzzy filter; Directional weighted median; Impulse noise; Noise reduction

References