# Image Noise Reduction by Means of Bootstrapping-Based Fuzzy Numbers
## 1 Introduction  
An essential task in image processing is the use of image noise reduction filters, which is necessary for any computer vision system that uses images for automated analysis. In fact, noise can cause problems in image processing tasks (such as edge detection, pattern detection, etc.). Therefore, we should try to reduce the noise.  
In recent years, the interest in using color images in various applications such as medical diagnostics, GIS, etc., has increased significantly. Therefore, the use of filtering and image noise reduction has been considered as a new field for applied research. It is widely observed that there is a correlation between image channels. Therefore, color images should be processed taking into account these correlations.  
The vector approach is one of the most famous methods used in this field. Among the methods that are used based on a strong statistical theory, we can mention the vector median filter (VMF) and vector directional filter (VDF), which show good performance in image filtering. Despite the good results in general, these techniques do not perform properly on the edges and some details of the image and cause blurring of the image in these areas. To solve this problem, recently, adaptive vector processing solutions have been used.  
The concept of the fuzzy metric space was first defined by [PERSON] and [PERSON] [1], which gave a generalization of [PERSON]'s probabilistic metric space [2]. This definition was modified by [PERSON] and [PERSON] [3]. This concept has been used in applied sciences such as image noise reduction [4; 5] and the perceptual color difference in [6; 7]. [PERSON] et al. used a fuzzy metric as a distance measure to reduce image noise. This fuzzy metric uses two measures of similarity between color vectors and the spatial proximity of pixels in the image to compare two pixels simultaneously.  
In recent years, much research has been performed in this field. [PERSON] et al. [8] used two different distance measures based on the fuzzy metric definition to filter image noise. Furthermore, they [9] reduced the effect of noise in the images by using fuzzy averaging.  
Using fuzzy thresholding, [PERSON] et al. [10] detected the noisy region of the image. The fuzzy T-metric is another algorithm that was introduced by [PERSON] et al. [11] and showed good results in reducing image noise.  
In this paper, we want to generalize [PERSON]'s method. Our most important tool to achieve this goal is to convert image pixel values to fuzzy numbers. To do this, we used the mean of the neighborhoods per pixel (as will be described in Section 6). We used bootstrap resampling to calculate the mean to reduce the scatter created by the outlier data (noise entered in the image). The bootstrapping resampling method introduced by Efron is used for a variety of estimation problems. Since the strong law of large numbers (SLLN) is particularly important in the bootstrap method, many researchers have studied and researched this field (see [PERSON] [12] and [PERSON] et al. [13]). [PERSON] et al. [14] provided a fuzzy metric space for the fuzzy set, which is a generalization of the fuzzy metric space presented by [PERSON] and [PERSON] [3]. Furthermore, they generalized the SLLN into the fuzzy metric space of the bootstrap mean. We used the SLLN concept in the fuzzy metric space for the bootstrap mean introduced by [PERSON] et al. [14], in image noise reduction, which is a generalization of [PERSON]'s approach [15].  
In the following, some preliminaries are presented. In the next step, we introduce the space of the fuzzy metric. In Section 4, we study the bootstrap mean in the fuzzy metric space. The filtering images by the fuzzy metric is performed in Section 5. In the next step, the proposed filter is introduced and examples are provided to compare the performance accuracy with other methods.