On the efficiency of random walk approach to noise reduction in color images

We present a novel approach to the problem of impulsive noise reduction for colorimages. The new image-filtering 
technique is based on the maximization of the similarities 
between pixels in the filtering window. Themethod is able to 
remove the noise component, while adapting itself to the local 
image structure. In this way, the proposed algorithm eliminates 
impulsive noise while preserving edges and fine image details. 
Since the algorithm can be considered as a modification of the 
vector median filter driven by fuzzy membership functions, it is 
fast, computationally efficient, and easy to implement. 
Experimental results indicate that the new method is superior, in 
terms of performance, to algorithms commonly used for impulsive 
noise reduction.

The orientation difference between two vectors can also be used as their distance measure.This so-called vector angle criterion is used by the vector directional filters (VDF) to remove vectors with atypical directions, [3,14].
The basic vector directional filter (BVDF) is a ranked-order, nonlinear filter which parallelizes the VMF operation.However, a distance criterion, different from the L 1 , L 2 norms used in VMF, is utilized to rank the input vectors.The output of the BVDF is that vector from the input set, which minimizes the sum of the angles with the other vectors.In other words, the BVDF chooses the vector most centrally located without considering the magnitudes of the input vectors.
To improve the efficiency of the directional filters, another method called directional distance filter (DDF) was proposed, [3].This filter retains the structure of the BVDF but utilizes a generalized distance criterion to order the vectors inside the processing window.
Another efficient rank-ordered technique called hybrid directional filter (HDF) was presented in [2].This filter operates in the direction and magnitude of the color vectors independently and then combines them to produce a unique final output.
All standard filters detect and replace well noisy pixels, but their property of preserving pixels which were not corrupted by the noise process is far from ideal.In this paper, we show the construction of a simple, efficient, and fast filter which removes noisy pixels, but has the ability of preserving original image-pixel values.
Our construction starts with the introduction of the similarity function µ : [0;∞) → R. We will need the following assumptions for µ: In the construction of our filter, the central pixel in the window W is replaced by that one, which maximizes the sum of similarities between all its neighbors.Our basic assumption is that a new pixel must be taken from the window W (introducing pixels which do not occur in the image is prohibited like in the VMF and BVDF).
For this purpose, µ must be convex, which can be easily shown.For the gray scale images, we define the following fuzzy measure of similarity between two pixels F k and F l (see [12,13]): (2.1) We now assume that F 0 is the center pixel in the window W and the pixels F 1 ,F 2 ,...,F n−1 B. Smolka et al. 81 Illustration of the construction of the new filtering technique for the 4-neighborhood case.If the center pixel F 0 is replaced by its neighbor F 2 , then the similarity measure M 2 = µ{F 2 ,F 1 } + µ{F 2 ,F 3 } + µ{F 2 ,F 4 } between F 2 (new center pixel) and its neighbors in W is calculated.If the total similarity M 2 is greater than M 0 = µ{F 0 ,F 1 } + µ{F 0 ,F 2 } + µ{F 0 ,F 3 } + µ{F 0 ,F 4 }, then the center pixel is replaced, otherwise it is retained.are surrounding F 0 (Figure 2.1).In the first step of the algorithm, the total sum M 0 of the similarities between the central pixel F 0 , which is suspected to be noisy, and its neighbors F i , i = 1,...,n − 1, is calculated.In the second step, each of the neighbors of the central pixel is moved to the center of the filtering window and the central pixel F 0 is rejected from W. For each pixel F i of the neighborhood, which is being placed in the center of W, the total sum of similarities M i is calculated and then compared with M 0 .It has to be stressed that in the second step the total sum of similarities is calculated without taking into account the original central pixel F 0 , which is rejected from the filter window.
In this way, the central pixel F 0 is replaced by that F i from the neighborhood, for which the total similarity function M i , which is a sum of all values of similarities between the central pixel and its neighbors, reaches its maximum.In other words, if where δ i, j is defined as is larger than then the center pixel is replaced by F i .Generally, the pixel F 0 is given by the value F i * , where i * = arg max i M i , This approach can be easily extended to color images.In this case, we use the similarity function defined by , where • denotes the specific vector norm.Now, in exactly the same way, we can maximize the total similarity function M for the vector case.
In finding the maximum in (2.5), we obtain (n − 1) nonzero components in M 0 .If we replace the central pixel by one of its neighborhoods (by F 2 in Figure 2.1(a)), then we obtain only (n − 2) nonzero components in M, as the pixel which has been put into the center disappears from the filter window (Figure 2.1(b)).In this way the filter tends to replace the original pixel only when it is really noisy and preserves in this way the image structures.
The BASIC code which can be used for the fast computer implementation is presented in Algorithm 2.1.

Results
The performance of the new algorithm was compared with the standard procedures of noise reduction used in color image processing.
The color image Lena has been contaminated by impulsive salt and pepper noise (pixel channel values are randomly replaced by 0 or 255 with equal probability), and the root of the mean square error (RMSE), peak signal-to-noise ratio (PSNR), and normalized mean square error (NMSE) have been used as quantitative measures of quality for evaluation purposes: where N 1 , N 2 are the image dimensions and F(i, j) and F(i, j) denote the original pixel vector and the restored vector, respectively.We investigated the behavior of the proposed filter using various convex functions.The new filter is then compared, in terms of performance, with various filters listed in Table 3.1.The following set of membership functions is considered in this work (Figure 3.1): Experimental analysis revealed that all these similarity functions can be used effectively in the new filtering structure.Table 3.2 summarizes the values of the parameter β i used in the various functions µ i when a test image Lena distorted by impulsive salt and pepper noise up to 10% is considered.Table 3.3 summarizes the results obtained for the test image Lena distorted by 4% impulsive noise.In order to obtain the results reported in Table 3.3, the L 2 norm was used to calculate the differences between the color vectors while the parameter β i values are those reported in Table 3.2.All proposed functions µ give very good results, although the best results are those obtained using the µ 1 , µ 5 , and µ 7 functions.In a second set of experiments, we tested the effect of the distance measure (norm) on the performance of the new filtering algorithm.In these second experiments, only three filters implemented using the "best" similarity functions, namely µ 1 , µ 5 , and µ 7 , are considered.Resulting RMSE values presented in Table 3.4 reveal that the best performance is obtained, as expected, when the Euclidean distance (L 2 norm) is used.The efficiency of the new filtering technique is shown in Figures 3.2, 3.3, and 3.4.Figure 3.2 depicts the result of noise reduction using the new method applied to a gray scale image Lena in comparison with the standard median filter.The test image was contaminated by 4% salt and pepper noise and a 3×3 filtering mask was used.shows the results of image filtering using the new method in comparison with the VMF.
For the comparison color test image Lena was used and the image pixels were distorted by 4% salt and pepper impulsive noise.As can be seen, the new class of filters eliminates efficiently impulsive noise, while preserving important image structures like edges, corners, lines, and fine texture (Figures 3.2 and 3.3).Another interesting property of the presented method of noise attenuation is shown in Figures 3.5 and 3.6.After a relatively small number of iterations, the filter converges to the root signal, meaning that in further iterations no changes are introduced to the image.
According to the results presented in this section, the best performance is achieved when a similarity function is inversely proportional to the distance between the vector signals (µ 7 ).Although someone can argue that the shape and the parameters of the "optimal" similarity function are application-dependent, determined mainly by the nature of the image and the type of noise corruption, we claim that the functions introduced here are easy to build and implement, require minimum user intervention in terms of parameter tuning, provide acceptable results for a wide range of input images, and are robust to suboptimal β i parameters.

Conclusions
In this paper, a new class of filters has been presented.Experimental results indicate that the new method of noise reduction significantly outperforms standard procedures used to restore gray scale and color images contaminated with impulsive noise.The new technique is fast and very easy to implement.The BASIC code is given in Algorithm 2.1 so that the filter can be easily evaluated by the image processing community.

Figure 3 .
4 depicts the efficiency of the new filter in comparison with the VMF, BVDF, and DDF for different percentages of impulsive noise.

Figure 3 . 6
Figure 3.6 Dependence of the noise reduction efficiency of the new filter on the number of iterations for different percentages of impulsive noise (Lena color image, β 1 = 5.04×10 −3 ).

Table 3 .
3Comparison of the new filter with the standard techniques from Table3.1.For the evaluation, the color image Lena is contaminated with 4% salt and pepper noise.

Table 3 .
4Comparison of the new filter results (RMSE) using different vector norms (Lena contaminated with 4% salt and pepper noise). ).