Fast Directional Weighted Median Filter for Removal of Random-Valued Impulse Noise Waqas Nawaz1, Arfan Jaffar2, and Ayyaz Hussain3 Department of Computer Sciences, FAST-National University of Computer and Emerging Sciences, Islamabad, Pakistan
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[email protected] 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
D
uring the acquisition or transmission of an image, noise 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
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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
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). •
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
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:
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 100
50
0 1s t Qtr
2nd Qtr
3rd Qtr
4th Qtr
FIGURE 2: MEDIAN BASED ON HISTOGRAM [1]
FIGURE 3: MEDIAN BASED ON HISTOGRAM AND STAGED SEARCH [1]
IV.
IMPULSE NOISE MOD DEL
Consider an image Img and an observation iimage Y of same size
differences for each pixel with the t centered pixel in a particular direction, and the value of o the weights depends on the closeness of the pixel Pixik from m the center pixel Pix0k. If the spatial distance for two pixells is small then their gray level values should be close to each h other. Thus, we have (1)
Where 1 ≤ i ≤ limit1, 1 ≤ j≤ limit2and 00
