Fast Directional Weighted Median Filter for Removal of Random-Valued Impulse Noise

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1 Fast Directional Weighted Median Filter for Removal of Random-Valued Impulse Noise Waqas Nawaz 1, Arfan Jaffar 2, and Ayyaz Hussain 3 Department of Computer Sciences, FAST-National University of Computer and Emerging Sciences, Islamabad, Pakistan wicky786@gmail.com 1, arfan.jaffar@nu.edu.pk 2, ayyaz.hussain@nu.edu.pk 3 Abstract The restoration process of many known median based algorithms is effective for the images corrupted by high random valued impulse noise, but not efficient especially for real-time applications. We proposed a new modus-operandi; by utilizing the competence of the fast median filter into modified Directional Weighted Median filter (DWM), which can be used in real-time applications to remove random valued impulse noise efficiently and effectively from a highly corrupted image without conciliation on the result s excellence. The simulation fallout depict that the anticipated technique performs better, in terms of time complexity and PSNR, as compared to the existing methods as well as directional weighted median filter. Keywords: fast median filter, histogram, staged search, random valued impulse noise, directional weighted median filter, time complexity, four directional neighbourhood I. INTRODUCTION uring the acquisition or transmission of an image, noise D may appear in it due to some reasons. On the basis of the value, the noise can be categorized into two types; it could be fixed valued, like salt and pepper, or random valued impulse noise [3]. Random valued impulse noise detection and removal is very difficult as compared to the salt and pepper noise. In this letter, we will consider both random and fixed valued impulse noise. Numerous methods have been proposed to remove impulse noise; but a robust technique has to suppress the noise and also preserve the natural information present in the image. A large number of linear and non-linear filters [5] have been introduced to remove the noise present in the image and enhance the quality of the image. Most of the linear filters utilize the neighborhood averaging mechanism to remove impulse noise and tend to destroy all high intensity details like edges, lines and other fine details. This led to the development of the non-linear median based filters such as stack filters [6], multistage median [7], weighted median filter [8, 9], rank conditioned median [10], and relaxed median [11]. However, most of these filters are employed uniformly across the image and thus tend to alter both noisy and noise-free pixels. Consequently, the removal of impulse noise is often accomplished at the cost of blurred and imprecise features, thus removing fine details in the image. Therefore, a noise-detection process, to differentiate between the corrupted and uncorrupted pixels prior to applying nonlinear filtering, is highly desirable. To overcome the deficit of standard median filter, many filters with an impulse detector are proposed, such as multistate median (MSM) filter [12], adaptive center weighted median (ACWM) filter [13],and signal dependent rank order mean (SD-ROM)filter [14], the pixel-wise MAD (PWMAD) filter [15], and iterative median filter [16]. These filters usually perform well, but as the noise level is more than thirty percent, they tend to remove various features from the images, or maintain too much impulse noise. To prevail over this problem, Yiqiu Dong and Shufang Xu in [2] proposed a new directional weighted median filter which first detect the impulse noise then remove high impulse noise and also preserve the details of the image. It outperforms all of the above mentioned methods, but they all utilized the median property [4] for the removal of the impulse noise of the image. The time complexity of the standard median filter is O(n2), when we used the simple sorting algorithm like bubble sort, or O(nlogn), using time efficient sorting algorithm e.g. Quick sort, which is not acceptable for the real-time applications. In this paper, we proposed a strategy to efficiently compute the median by utilizing the competence of the fast median filter algorithms in [1] into directional weighted median filter [2] for random valued impulse noise. We have used the fast median filter, whose time complexity is almost linear O(M) as compared to the standard median filter which uses the efficient sorting algorithm to sort the pixel values, and then pick the middle value as median for that window. The organization of this paper is as follows. In section II, standard median filter is briefly explained. In subsequent section the fast median filter algorithms have been discussed in details. Noise model explained in Section VI. Section V specifies the proposed methodology of the fast directional weighted median filter. Experimental results and conclusion is also given in section VI and VII respectively. II. STANDARD MEDIAN FILTER Median filter has the characteristics to remove the impulse noise and preserve fine detail in an image, that s why it is used in many signal and image processing applications. Even though the process of calculating median is simple and straightforward, but its time complexity is not tolerable for real-time systems/applications. Median can be calculated as follows: Sort all the values (pixels) using any sorting /10/$ IEEE

2 technique Then pick the middle value from it, which will be the median. The key problem with this technique is its time complexity. In first step, we need to sort all the values in ascending or descending order. No matter which algorithm you use, its time complexity will be O(n2) or if you are using faster technique then at maximum, without any other dependency, you can achieve O(nlogn). III. FAST MEDIAN FILTERS To overcome the deficiency of the median algorithm, with respect to time, Quanhua Tang and et al [1] presented a new and efficient technique for calculating median linear time. They have utilized the statistical information of histogram and multilevel staged search to get the median value as early as possible. It can be applicable in diverse applications because it is an independent method. We are focusing on the two algorithms in [1], which are: Median computation based on histogram Median computation based on histogram and staged search Median computation based on histogram utilizes the statistical information of the histogram to calculate the median, on the other hand the second method, which is the extension of the first technique, uses the smaller histogram to efficiently calculate the median as compared to just histogram based computation. Consider an array of values with limited or predetermined range; let V = [v1, v2, v3 vp], be array for calculating the median value, where 0 vi Q and vi is an integer. The algorithms for the above mentioned methods are as follows: In this algorithm, there are mainly two steps, first is to calculate statistic histogram of V, the other is to account the first order summation of HIST. The computing quantity of statistic histogram is P, and the computing quantity of first order summation is Q, so the whole maximum computing quantity is P+Q. When Q/P is smaller, the benefit of this algorithm is evident. B. ALGORITHM FOR MEDIAN COMPUTATION BASED ON HISTOGRAM AND STAGED SEARCH Principle of meter ruler is used to improve the median calculation strategy based on histogram statistics. Measuring scope of 1 meter ruler is from 1 millimeter to 1 meter, the division span is one thousand. Meter ruler s scale, even without numeral, still any value can be found quickly. In this scenario, they have used a strategy: first count decimeter scale, then count the scale of centimeter, and lastly count millimeter scale, by using this strategy we can get any value just by less than twenty-seven operation times. For searching the histogram, we can get clue from above method, i.e. if the summation value, which is found in one big part of the histogram, is smaller than the median then we do not need to consider the inner distribution of that big scope, we can jump to the next big scope. So, we will divide the histogram into N child intervals with some predefined length L. The algorithm in figure 3 is based on this strategy. A. ALGORITHM FOR MEDIAN COMPUTATION BASED ON HISTOGRAM st Qtr 2nd Qtr 3rd Qtr 4th Qtr FIGURE 2: MEDIAN BASED ON HISTOGRAM [1] FIGURE 3: MEDIAN BASED ON HISTOGRAM AND STAGED SEARCH [1]

3 IV. IMPULSE NOISE MODEL Consider an image Img and an observation image Y of same size differences for each pixel with the centered pixel in a particular direction, and the value of the weights depends on the closeness of the pixel Pixik from the center pixel Pix0k. If the spatial distance for two pixels is small then their gray level values should be close to each other. Thus, we have (1) Where 1 i limit1, 1 j limit2and 0<prob<1. Imgi,j and NImgij indicate the pixel values at index (i,j) of the original image and the noisy image, respectively and NImgija noise value independent from Iij. For gray level images, where each pixel value of the image can only be stored in 8 bits, when the images are polluted by fixed value impulse noise, NImgi,j the corrupted pixel is equal to 0 or 255 each with equivalent probability (prob/2). V. FAST DIRECTED WEIGHTED MEDIAN FILTER In directional weighted median filter, the author proposed a new impulse detector [2] which is used to identify the noisy pixel then using all four directional information of the selected pixel to calculate the median so that to preserve the natural details present in the image as removing noise. In this approach, there are two major steps: Detect noisy pixel using new impulse detector Utilize weighted directional information to calculate the median for removing impulse noise and preserve details Now consider 5x5 window centered at (i, j), we calculate the sum of all the absolute weighted differences of gray level values in a specific direction, Diff(k) is used to define the differences where k specify the direction. Noisy image's pixel values for calculating median value, e.g. 3x3 window impulse detection using weighted difference along four directons in its neighborhood and threshold value determine direction on the basis of standard deviation and add pixel value to the window apply the fast median filter on the updated window and return the median value replace the central value of the considered window with median value FIGURE 4: STEPS FOR FAST DIRECTIONAL WEIGHTED MEDIAN FILTER The weights are multiplied at the time of calculating the Where (2) We used Diffk as a direction index and each direction index is receptive to the edge aligned with a given direction. To identify the impulse noise, we use minimum value from all four direction indexes, if the selected value is greater than predefined threshold T, as its value is 510 in [2], then pixel is noisy otherwise noise-free, as shown in Equation 3. and (3) Where T is threshold, Min is the operator to identify the minimum value from all four Diffk values. Now we can determine the noise by employing a threshold T, no matter we are dealing with an edge, flat region or thin line. After detecting the impulse noise, many researchers apply the standard median filter for the reduction of the noise. The details preservation is not possible with standard median filter, so to overcome this problem, in [2], they suggested a new directional weighted median filter in which information of the four directions in incorporated to effectively preserve the detail and remove the impulse noise. We all know that the standard deviation is used to determine how tightly all values are clustered to a specific value. The steps of modified directional weighted median filter are as follows: The standard deviation is calculated for each direction and we choose the direction where the standard deviation is minimum, Pick pixel values from this direction and, add them to the existing window [2], add them twice to the existing window to increase the possibility of the nearest to the exact median value. This has an evident effect on the quality, PSNR, of the restored image. After that we apply the standard median filter to the new updated window. In this paper we have proposed a new efficient idea to calculate the directional weighted median. In real time applications, where we need to remove the high impulse noise and also try to preserve the details, we need to have an

4 efficient and effective technique. In such scenarios fast directional weighted median filter is sufficient to fulfill the requirements. In our proposed technique, we have utilized the competence of the fast median filters into the existing directional weighted median filter for time efficiency. In figure 5 we have listed the major steps of our proposed idea. In figure 5, the first step is simply the input pixel values of the noisy image; secondly we used the new impulse detection algorithm, in [2], to detect the impulse noise, then after this we determine the standard deviation in all the four directions of the current window and add the pixel values from the direction where the standard deviation is minimum. Now after the addition of these pixel values we calculate the median of this window using fast median algorithms using histogram and stage search in [1] instead using the standard median filter. VI. SIMULATION The experiment has performed using Matlab version R2009a, on Intel dual Core 1.60 GHz, with 2GB Ram. During simulation, we have used standardd testing pictures like Peppers, Elaine, Airplane, Goldhill, and TestPat. The results in the Table (1, 2) depict that modification of the existing directional weighted median filter produce good quality image with higher PSNR value, as compared to the existing methods for uniform and salt & pepper noise including directional weighted median filter approach. TABLE 1 COMPARISON WITH EXISTING METHOD ON THE BASISS OF PSNR VALUE AFTER APPLYING FILTER ON 512X512 RESOLUTION STANDARD IMAGES CONTAMINATED WITH 40% SALT & PEPPER NOISE Filters Salt & Pepper Noise 40% Peppers Elaine Airplane Goldhill Med3x DUMMY TSM ACWMF SDROM PSM CSAM PWMAD Trilateral FIDRM FRINR DWM UINFGP Proposed During experimentation we have considered 3x3 window for our proposed technique. As we are dealing median pixel value of the considered window so if we increase the size of the window then resultant image will have blur effect. From results mentioned in Table 1 and 2, we can conclude that whenever we are dealing with already discussed images, like peppers, Elaine, airplane, and goldhill, our proposed technique is better to restore the image with salt & pepper or uniform impulse noise. TABLE 2 COMPARISON WITH EXISTING METHOD ON THE BASIS OF PSNR VALUE AFTER APPLYING FILTER ON 512X512 RESOLUTION STANDARD IMAGES CONTAMINATED WITH 40% UNIFORM IMPULSE NOISE Filters Uniform Impulse Noise 40% Peppers Elaine Airplane Goldhill Med3x DUMMY TSM ACWMF SDROM PSM CSAM PWMAD Trilateral FIDRM FRINR DWM UINFGP Proposed The major contribution of our proposed strategy is to reduce the time complexity of the restoration process without compromising on the quality of the resultant image. The time complexity of proposed technique depends on the noise ratio contained in the image. It means that when the noise ratio is smaller, e.g. >15% %, then it perform better otherwise its takes almost equal time as compared to the other methods. All of the above experiments have done considering only 3x3 size window. To avoid the biasness we have simulated our technique with varying windoww size, i.e. 3x3, 5x5, 7x7; and results are shown in figure(5). (A) (B)

5 (C) FIGURE 5: TIME COMPLEXITY OF OUR PROPOSED METHOD (IN SECONDS) W.R.T WIDE RANGE OF IMPULSE NOISE CONTAMINATION (5% 70%) TEST IMAGE TEXTPAT, RESOLUTION 128X128, SALT & PEPPER NOISE; (A) 3X3 WINDOW (B) 5X5 WINDOW (C) 7X7 WINDOW VII. CONCLUSION In this paper, we proposed a strategy to efficiently compute the median by utilizing the competence of the fast median filter algorithms [1], median filter based on histogram (MBH) and median based on histogram and staged search (MBHSS), into directional weighted median filter [2] to improve its time complexity without compromising on the quality of the resultant image. The experimental results show that the proposed idea works and time complexity of the directional weighted median algorithms has been improved. The modified DWM filter also performed well, in terms of PSNR, as compared to the existing DWM filter. Now in real-time application this technique can be used to efficiently reduce the impulse noise from an image. [9] R. Yang, L. Yin, M. Gabbouj, J. Astola, and Y. Neuvo, Optimal weighted median filters under structural constraints, IEEE Trans.Signal Processing, vol. 43, pp , Mar [10] R. C. Hardie and K. E. Barner, Rank conditioned rank selection filters for signal restoration, IEEE Trans.Image Processing, vol. 3, pp , Mar [11] A. Ben Hamza, P. Luque, J. Martinez, and R. Roman, Removing noise and preserving details with relaxed median filters, J. Math. Imag. Vision, vol. 11, no. 2, pp , Oct [12] T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Transactions on Circuits and Systems II, vol. 48, pp , [13] T. Chen and H. R. Wu, Adaptive impulse detection using center- Processing Letters, vol. 8, pp. weighted median filters, IEEE Signall 1 3, [14] E. Abreu, M. Lightstone, S. K. Mitra and K. Arakawa, A new efficient approach for the removal of impulse noise from highly corrupted images, IEEE Transactionss on Image Processing, vol. 5, pp , [15] V. Crnojevi c, V. ˇSenk and ˇ Z. Trpovski, Advanced impulse detection based on pixel-wise MAD, IEEE Signal Processing Letters, vol. 11, pp , [16] W. Luo, A new efficient impulse detection algorithm for the removal of impulse noise, IEICE Transactions on Fundamentals, vol. E88-A, no. 10, pp , October REFERENCES [1] Tang. Quanhua, Zhou. Yan, Lei. Jine, Fast median filters based on histogram and multilevel staged search, IEEE Fourth International Conference on Image and Graphics, [2] Dong. Yiqiu, Xu. Shufang, A New Directional Weighted Median Filter for Removal of Random Valued Impulse Noise IEEE Signal Processing Letters. VOL. 14, No. 3, March [3] T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Transactions on Circuits and Systems II, vol. 48, pp , [4] W. K. Pratt, Median filtering, Image Proc. Institute, University of Southern California, Los Angeles, Tech. Rep., September [5] R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd Edition, Prentice Hall, [6] P. D. Wendt, E. J. Coyle, and N. C. Gallagher, Stack filters, IEEE Trans. Acoust Speech Signal Processing, vol. ASSP-34, pp ,Aug [7] G. R. Arce and R. E. Foster, Detail preserving ranked-order based filters for image processing, IEEE Trans. Acoust Speech Signal Processing, ol. 37, pp , Jan [8] S. J. Ko and Y. H. Lee, Center weighted median filters and their applications to image enhancement IEEE Trans. Circuits Syst., vol. 38, no.9, pp , Sep