# Image Noise Reduction by Means of Bootstrapping-Based Fuzzy Numbers
## 3 Introduction to Fuzzy Metric Space  
As we know, one of the important concepts in data mining, image processing, and multivariate data analysis is the distance measurement criteria. Some distance measures of the exact number are well established in the literature. Distances are calculated in an imprecise framework, which creates logical problems due to ambiguity. In such cases, crisp numbers are converted to fuzzy numbers. Various definitions of the fuzzy metric space have been proposed by various authors [19; 20; 21]. [PERSON] and [PERSON] in [3] modified the concept of the fuzzy metric space introduced by [PERSON] and [PERSON] [1] and defined a Hausdorff topology on this fuzzy metric space. In this definition, which we refer to in the following, \(M(x,y,t)\) can be thought of as the degree of nearness between \(x\) and \(y\) with respect to \(t\). Assuming that \(\mathcal{X}\) is an arbitrary non-empty set, \(M\) is a fuzzy metric (fuzzy set) on \(\mathcal{X}\times\mathcal{X}\times(0,\infty)\) and \(\ast\) is a continuous t-norm. In this case, the three-tuple \((\mathcal{X},M,\ast)\) is said to be a fuzzy metric space [3]. Here, if \(M(x,y,t)\) is symmetric with respect to \(x\) and \(y\), continuous on \(t\), and satisfies the following conditions for all:  
* \(M(x,y,t)=1\) if and only if \(x=y\);
* \(M(x,z,t+s)\geq M(x,y,t).M(y,z,s)\). \(M(x,y,t)\) is called a fuzzy metric and indicates the degree of closeness or similarity of \(x\) and \(y\) according to \(t\). **Example 1** ([3]). : _Suppose that \((\mathcal{X},d)\) is a metric space, and we define \(a\ast b=\min\{a,b\}\) or \(a\ast b=ab\) and \(\forall x,y\in\mathcal{X}\),_  
\[M(x,y,t)=\frac{kt^{n}}{kt^{n}+md(x,y)},\;\;k,m\in\mathbb{R}^{+},n\geq 1.\]  
_Then, \((\mathcal{X},M,\ast)\) is a fuzzy metric space. Let us say now that \(M(x,y,t)\) is the standard fuzzy metric induced by the \(d\) metric, whenever \(k=n=m=1\):_  
\[M(x,y,t)=\frac{t}{t+d(x,y)}, \tag{1}\]  
For the first time, the fuzzy normed space was defined by [PERSON] and [PERSON] [22]. Assuming that \(\mathcal{X}\) is a vector space, \(N\) is a fuzzy set (fuzzy norm) on \(\mathcal{X}\times(0,\infty)\), and \(\ast\) is a continuous t-norm, then the three-tuple \((X,N,\ast)\) is called a fuzzy normed space. Detecting and reducing the effect of noise in an image can be considered one of the important applications of the fuzzy metric. [PERSON] et al. [15] used this metric to detect and reduce noise in images and showed that this method has a better effect than other methods. Actually, [PERSON] used the standard fuzzy metric as follows:  
\[R(F_{i},F_{j},t)=\frac{t}{t+\|F_{i}-F_{j}\|},\]  
where \(F_{i}=(F_{i}^{1},F_{i}^{2},F_{i}^{3})\) are red (R), green (G), and blue (B) for each pixel of the image. Now, if we have a fuzzy view of the colors and consider the value of each pixel as a fuzzy number, the above method cannot be effective. In fact, the method presented by [PERSON] et al. [15] is suitable for when the value of each pixel is a crisp number, and if the value of each pixel can be defined as a fuzzy number, this method cannot be used anymore. In this case, the fuzzy metric for fuzzy sets should be used. [PERSON] et al. in [14] presented a definition of the fuzzy metric space for fuzzy sets, which are a generalization of the fuzzy metric space [3]. The definition is as follows:  
**Definition 2** ([14]). : _Let \(\mathcal{X}\) be an arbitrary non-empty set, \(\mathcal{F}(\mathcal{X})\) be the collection of those fuzzy sets on \(\mathcal{X}\), and \(\ast\) be a continuous t-norm.