Similafuty Based Impulsive Noise Removal in Color Images

In this paper a novel approach to the problem of impulsive noise reduction in color images based on the nonparametric density estimation is presented. The basic idea behind the new image filtering technique is the maximization of the similarities between pixels in apredefined filtering window. The new method is faster than the standard vector median filter (VMF) and preserves better edges and fine image details. Simulation results show that the proposed method outperforms standard algorithms of the reduction of impulsive noise in color images. 1. NOISE REMOVAL IN COLOR lMAGES A number of nonlinear, multichannel filters, which utilize correlation among multivariate vectors using various distance measures, have been proposed [ 1-71. The most popular nonlinear, multichannel filters are based on the ordering of vectors in a predefined moving window. The output of these filters is defined as the lowest ranked vector according Let F(z) represents a multichannel image and let W be a window of finite size 11 + 1, (filter length). The noisy image vectors inside the filtering window W are denoted as Fj , ,f = 0,1, ..., n . If the distance between two vectors F i , Fj is denoted as p ( F i , Fj) then the scalar quantity Ri = E,”=, p ( F i , Fj) is the distance associated with the noisy vector F i . The ordering of the Ri ’s: R p ) 5 R ( , ) 5 ... 5 implies thc same ordering to the corresponding vectors Fi : F(0) 5 F(,) 5 ... 5 F(n). Nonlinear ranked type multichannel estimators define the vector F(”) as the filter output. However, the concept of input ordering, initially applicd to scalar quantities is not easily extended lo multichannel data, since there is no universal way to define ordering in vector spaces. To overcome this problem, distance functions are often utilized to order vectors, [1,2]. The majority of standard filters detect and replace well noisy pixels, hut their propeny of preserving pixels which were not corrupted hy the noise process is far from the ideal. In this paper we show the construction of a simple, efficient and fast filter which removes noisy pixels, but ha? the ahility of preserving original image pixel values. vector ordering technique. 2. PROPOSED ALGORITHM 2.1. Gray-scale Images Let us assume a filtering window W containing nf 1 image pixels, {Fo, F l , . . . , F,,}, where n is the number of neighbors of the central pixel Fo, (see Fig. 2a) and let us define the similarity function p : 10; cu) R which is nonascending in [O; 03) , convex in 10; 00) and satisfies p(0 ) = 1, p ( m ) = 0 . The similarity between two pixels of the same intensity should be 1, and the similarity between pixels with far distant gray scale values should he very close to 0. The function p(F;, F j ) defined as p(F?, F j ) = ~ ~ ( l f i ; Fj l ) satisfies the three above conditions. Let us additionally define the cumulated sum M of similarities between the pixel Fk and all its neighbors. For the central pixel and its neighhors we define 11.1, and hfk as ” fifo=Clr(~OII.;), A J ~ = C ~ . ( F ~ , P ~ ) , ( i j which mcans that for Fk which are neighbors of FO we do not take into account the similarity between Fk and Fo, which is the main idea hehind the new algorithm. The omission of the similarity p(Fk , Fo) privileges the central pixel, as in the calculation of MO we have TI similarities p(F0, Fk) , k = 1,2, . . . ~ 71, and for A4,, I; > 0 we have only n 1 similarity values, as the central pixel FO is excluded from Af,. In the construction of the new filter the reference pixel Fo in the window H’ is replaced by one of its neighbors if hfO < A{,, k = 1 , . . . , n. If this is the case, then FO is replaced hy that Fk for which I; = arg tnax Mi, i = 1 , . . . ,T I . . In other words FO is detected as being cormpted if A40 < A t k , I; = 1, . . . , ? I and is replaced by its neighhors Fi which maximizes the sum of similarities A t between all its neighbors excluding the central pixel. This is illustrated in Figs. 2 and 5. The basic assumption is that a new pixel must he taken from the window W , (introducing pixels which do not occur in the image is prohibited like in VMF). For this purpose p must be convex, which means that in order to find a maximum of the sum of similarity functions A4 it is sufficient to calculate the values of A l only in points PO:. . ~ F,?, [7]. j=1 j = 1 . j #b 2.2. Color Images 'The presented approach can be applied in a straightforward way to color images. We usc the similarity function defincd by b{Fi, Fj} = / I ( llFi -F,j)ll where 1 ) . 1 1 denotes the specific vector norm. Now, in exactly the same way we maximize the total similarity function A 4 for the vector case. We have checked several convex functions in order to compare our approach with the standard filters uscd in color image processing presented in Tab. 1 and we have obtained very good results (Tab. 21, when applying the following similarity functions, which can he treated as kernels ofnonparametric density estimation, 17-91, (.see Fig. 4). Applying the linear similarity fuuction / L , we obtain 1 -p(Fi lFk) /h for p (Fi ,Fk) < / I , 0 otherwise. /L(FI, Fk) = Then we have from (2), A40 = n p(F0, F 3 ) , and ? = I If this condition is satisfied, then the central pixel is considered a,, not disturhed by the noise process, otherwise the pixel Fi for which the cumulative similarity valuc achieves maximum, replaces the central noisy pixel. In this way the filter replaces the central pixcl only when it is really noisy and preserves the original undistorted image structures. The parameter / I can he set experimentally or can be determined adaptively using the technique described in [71. a') b) Fig. 2. Illustration of the construction of the new filtering technique. First the cumulativesimilarityvalue MO between the central pixel F0 and its neighbors is calculated (.a), then the pixel Fo is rejected from the filter window and the cumulative similarity values M I : k = 1: . . . , n of the pixels Fig. 1. Illustration of the efficiency ofthc new algorithm of impulsive noise reduction in color images: a ) tcst iinase, h) image compted by 4% impulsive salt & pepper noise, e) new filter output, d ) effect of median filtering (3 x 3 mask). FI, . . . , F, are determined, (b). It is interesting to note, that the best results were achieved for the simplest similarity function p?(:c), which allows to construct a Sast noise reduction algorithm. In the multichannel case, we have j = I j=l.j#" Table 1. I'ilters taken Soor comparisons, [l-31 wherep{F;.Fe} = llFk Ft)11 and 1 1 . 1 1 is the LZ n o m .