# Impulsive Noise Removal with an Adaptive Weighted Arithmetic Mean Operator for Any Noise Density  
[PERSON]  
1 Soft Computing, Image Processing and Aggregation Research Group (SCOPIA), Department of Mathematics and Computer Science, University of the Balearic Islands, E07122 Palma, Spain; [EMAIL_ADDRESS] (S.M. ); armau.mir@uibbes. (A.M.); [EMAIL_ADDRESS] (D.R.-A. )12 Health Research Institute of the Balearic Islands (dslBa), E07010 Palma, Spain2  
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In addition to the well-known standard median filter, many different impulsive noise filters can be found in the literature. In [1] a method based on the detection of impulsive noise as a first stage is proposed. After deciding if one pixel is noisy or not, it is replaced by the median of the neighboring values of a \(3\times 3\) 2-D window in case the pixel is corrupted and it is left unchanged otherwise. Another approach can be found in [2] where the stage of detecting corrupted pixels is made using mathematical morphology, more specifically by comparing the result of the opening and closing filters using a flat structuring element. Another method that first identifies corrupted pixels is developed in [3]. This identification is based on the idea that corrupted pixels can be considered to be local extremes of the intensity function with respect to their neighbors. The next step is to filter corrupted pixels as the median of the values of the non-noisy pixels in a window centred at that pixel. A similar technique is performed in [4] where the median or the mean are used to aggregate the neighbor's values of the noisy pixels to restore the image in windows of different sizes. In [5], a more complex approach is proposed. The maximum absolute luminance difference is considered to classify pixels as non-noisy pixels, pixels lightly affected by noise and heavily noisy pixels. In the filtering phase, the corrupted pixels are reconstructed using a weighted mean value of (i) the weighted mean of the uncorrupted pixels of the neighborhood and (ii) the original pixel value, with weights which depend on the corruption level of the pixel and the distances to neighboring uncorrupted pixels. The theory of fuzzy logic and fuzzy sets has also been considered to propose a set of novel methods to filter corrupted images with salt-and-pepper noise. For example, in [6] a fuzzy two-step color filter is proposed to reduce impulsive noise in color images. This filter is based first on the computation of fuzzy gradient values and then fuzzy reasoning is applied to determine three different membership functions that will be used in the filtering stage. Another two-step scheme relying on an adaptive neural-fuzzy inference system can be found in [7], where a local fuzzy membership function is used to improve the rate of noise detection. Also, a two-stage method is developed in [8], where the histogram of the image is considered in the detection stage to identify pixels affected by noise. Then fuzzy reasoning is applied to adequately manage the unavoidable uncertainty, caused by noise, when one deals with local information. Besides these methods, algorithms based on fuzzy mathematical morphology are becoming increasingly popular. These methods generalise the binary morphology (see [9]) using fuzzy sets theory (we refer the reader to [10,11]). For example, in [12], an improvement of the method introduced in [2] is presented using a fuzzy mathematical morphology approach. An even better improvement of the previous filter is proposed in [13] by using another function to identify noisy pixels and a window with an adaptive size. These methods have a drawback: images corrupted by a high-density noise (\(\geq\)90%) are not restored successfully.