Fuzzy metric and its applications in removing the image noise

Abstract

This paper introduces the notions of the fuzzy T-metric and fuzzy S-metric and considers their duality and their links to standard metrics. Numerous examples of fuzzy T-metrics are given in the paper, and some of their properties are proved. A procedure for constructing new fuzzy metrics as a composition of the given fuzzy metrics and norms is provided. An application of the fuzzy T-metrics in image filtering is given. The proposed filtering algorithm is a modification of the algorithm presented by Valentin, Morillas and Sapena in Valentin G et al. (Fuzzy Sets Syst 170:95–111, 2011). Instead of the fuzzy metric that is used in the paper of Valentin G et al. 2011, a new fuzzy T -metric is applied in the algorithm. The aim was to improve the sharpness and the quality of the image, measured by the image quality index UIQI. It is shown that the image filtered by this modified algorithm has better quality and greater sharpness than the image filtered by the median filter. Fuzzy T-metric parameters that give the best image quality and sharpness are determined experimentally.

1 Introduction

The notion of a distance over a non-empty set X comprises a wide mapping class d from $X^2$ to $[0,+\,\infty )$. Depending on the set X,  the notions of d are introduced as required to satisfy some important characteristics. The notion of a distance can be generalized so that the distance between two elements of the set does not have to be a nonnegative number; for example, it can be the fuzzy set. Thus, in this paper, the notion of the fuzzy metric is introduced. The aim of the paper is to modify the algorithm for image filtering, presented in Valentin et al. (2011), by using the defined fuzzy T-metric, with some predefined characteristics. The properties of the defined fuzzy T-metrics, which are used in removing the image noise, are presented.

The paper is organized as follows. The second section gives the basic notions of distance, t-norm and t-conorms and their properties which will be needed when proving the properties of T-metrics and S-metrics. In this section, the theorems and definitions are from Klement et al. (2000) and Klir and Yuan (1995). The third section defines the fuzzy S-metric and the fuzzy T-metric; the notion of the dual fuzzy metric space is given with respect to the fuzzy complement and the relationship between them. Some examples of the fuzzy T-metric and the fuzzy S-metric are given. The way in which a standard metric can be generated from the fuzzy T-metric and the fuzzy S-metric is presented. The procedure of defining a new fuzzy T-metric from several fuzzy T-metrics with respect to the same norm is provided. An analogous assertion for the fuzzy S-metric is proved. This section is mostly referenced from Bloch (1999), Deza and Deza (2009), Gregori and Romaguera (2000) and Morillas et al. (2007).

In the fourth section, some explanations on the manner of forming the fuzzy T-metric, that will be used in filtering the image, are given. Annotations for the fuzzy T-metrics, that will be used for filtering, are presented. Some general notions about filtering the images are provided. The fifth section gives specific fuzzy T-metric used for filtering. The algorithm for filtering the images is a modification of the algorithm in Valentin et al. (2011) obtained by using a fuzzy T-metric instead of the multiplication. The values of parameters, which are the part of fuzzy T-metrics and which give the best performance in filtering images, are experimentally determined. The quality of images filtered by the modified algorithm and images filtered using median filter is measured by using the image quality metric defined in Wang and Bovik (2002). In this section were used the notions of fuzzy filtering that are defined in Astola et al. (1990), Morillas et al. (2005a, b) and Smolka et al. (2001). Specific image quality indexes of the image filtered by the modified algorithm, and by median filter, are given. Comparing these quality indexes, the conclusion about performance of these algorithms is made.

2 Preliminaries

In the classical sense, the distance is most often defined by means of mappings that are metrics, pseudo-metrics, semi-metrics, and similarities, and are defined in the following way.

Definition 1

If $X\ne \emptyset ,$ a function $d{:}\,X^2 \rightarrow \mathbb {R}^+_0$ which satisfies the following properties:

• 1. $\forall x \in X,\; d (x,x)=0,$

• 2. $\forall x,y \in X, \; d(x,y)=d(y,x),$

is called a distance, and the ordered pair (X, d) is a space with distance. If only the first property holds, it is called a quasi-distance. If

• 3. $\forall x,y \in X,\; d(x,y)=0\Rightarrow x=y,$

• 4. $\forall x,y,z\in X, \; d(x,z)\le d(x,y)+d(y,z),$

then d is a metric, and the ordered par (X, d) is a metric space. If only the properties 1., 2., and 4. are valid for d, d is called a pseudo-metric. If only 1., 3., and 4. are valid for d, it is referred to as a quasi-metric.

If 1., 2., and 3. are valid, d is called a semi-metric. For a semi-metric in which instead of the inequality of the triangle (the property 4.) one of the following inequalities is valid:

• 4.’ $\forall x,y,z\in X,\; d(x,z)\ge T(d(x,y),d(y,z)),$

• 4.’’ $\forall x,y,z\in X, \;d(x,z)\le S(d(x,y),d(y,z)),$

(T is a t-norm, and S is a t-conorm), d is called a similarity.

The next section will be limited to distances which are fuzzy T-metric and fuzzy S-metric, and the distance is determined between sets which are “crisp” sets. Definitions of triangular norms and conorms and some of their properties are given (see Klement et al. 2000; Klir and Yuan 1995).

Definition 2

The triangular norm (shorter t-norm) is a binary operation $T{:}\,[0,1]^2\rightarrow [0,1]$ which satisfies the following axioms for all $a, b, c, b_1 \in [0,1]$:

• 1. $T(a,1)=a$ (boundary condition);

• 2. $b\le b_1 \Rightarrow T(a,b)\le T(a,b_1)$ (monotonicity);

• 3. $T(a,b)=T(b,a)$ (commutativity);

• 4. $T(a,T(b,c))=T(T(a,b),c)$ (associativity).

The t-norm is said to be an Archimedeant-norm, if in addition to the previous axioms two more axioms are satisfied:

• 5. T is a continuous function;

• 6. $\forall a\in (0,1),\; T(a,a)<a.$

Remark 1

From the conditions given in the definition of the t-norm follows the monotonicity by coordinates, i.e., for all $a_1, a_2, b_1, b_2 \in [0,1]$

$$\begin{aligned} a_1\le a_2 \wedge b_1\le b_2\Rightarrow T(a_1,b_1)\le T(a_2,b_2). \end{aligned}$$

(*)

Replacing the monotonic condition in Definition 2 by condition (*), an equivalent definition of the t-norm is obtained.

If, in definition of the t-norm, instead of the axiom of monotonicity, a strict monotonicity is valid, i.e.,

$$\begin{aligned} a_1< a_2 \wedge b_1<b_2\Rightarrow T(a_1,b_1)< T(a_2,b_2), \end{aligned}$$

for all $a_1,a_2,b_1,b_2 \in [0,1],$ then the t-norm is strict.

It can be shown that 0 is the annihilator for t-norm, i.e., for all $a \in [0,1]$ holds

$$\begin{aligned} T(a,0)=T(0,a)=0. \end{aligned}$$

Definition 3

Decreasing generatorg is a continuous and strictly increasing function from [0, 1] to ${\mathbb {R}}$, such that $g(1)=0.$

Pseudo-inverse function for the decreasing generator g, denoted by $g^{(-1)}$, is a function from ${\mathbb {R}}$ to [0, 1] defined with

$$\begin{aligned} g^{(-1)}(a)= \left\{ \begin{array}{ll} 1, &{}\quad a\in (-\,\infty ,0)\\ g^{-1}(a),&{}\quad a\in [0,g(0)]\\ 0, &{}\quad a\in (g(0),+\,\infty ), \end{array}\right. \end{aligned}$$

where $g^{-1}$ is the usual inverse function for g.

For a decreasing generator g and its pseudo-inverse function $g^{(- 1)}$, the equality $g^{(-1)}(g (a)) = a,$ is satisfied for all $a\in [0,1]$ and it holds

$$\begin{aligned} g(g^{(-1)}(a))= \left\{ \begin{array}{ll} 0, &{}\quad a\in (-\,\infty ,0)\\ a,&{}\quad a\in [0,g(0)]\\ g(0), &{} \quad a\in (g(0),+\,\infty ). \end{array}\right. \end{aligned}$$

Definition 4

The power of thet-norm is given by formulas:

$$\begin{aligned}&T^1(a_1,a_2)=T(a_1,a_2),\\&T^n(a_1,\ldots ,a_n,a_{n+1})=T(T^{n-1}(a_1,\ldots ,a_n),a_{n+1}). \end{aligned}$$

Remark 2

Because of the associativity, it holds

$$\begin{aligned} T^n(a_1,\ldots ,a_n,a_{n+1})=T\left( a_1,T^{n-1}(a_2,\ldots ,a_{n},a_{n-1})\right) . \end{aligned}$$

For example:

$$\begin{aligned}&T^2(a_1,a_2,a_3)=T(T(a_1,a_2),a_3)=T(a_1,T(a_2,a_3)),\\&T^3(a_1,a_2,a_3,a_4)= T(T^2(a_1,a_2,a_3),a_4)\\&\quad = T(T(T(a_1,a_2),a_3),a_4) =T(T(a_1,a_2),T(a_3,a_4))\\&\quad =T(a_1,T(a_2,T(a_3,a_4))) =T(T(a_1,T(a_2,a_3)),a_4)\\&\quad =T(a_1,T(T(a_2,a_3),a_4)), \mathrm{etc}. \end{aligned}$$

Lemma 1

For the power of t-norm, holds:

$$\begin{aligned} T^{n-1}(a_1,\ldots ,a_n)=T^{n-1}(a_{i_1},\ldots ,a_{i_n}), \end{aligned}$$

where $a_ {i_1}, \ldots , a_ {i_n}$ is an arbitrary permutation of elements $a_1, \ldots , a_n.$

Remark 3

If T is a t-norm, then:

$$\begin{aligned}&T(a_1,a_2)=1\Leftrightarrow \;a_1=a_2=1,\\&T^n(a_1,a_2,\ldots ,a_{n+1})=1\Leftrightarrow \;a_1=\cdots =a_{n+1}=1. \end{aligned}$$

Indeed, $a_1\le 1 \Rightarrow 1=T(a_1,a_2)\le T(1,a_2)=a_2$ and $a_2\le 1 \Rightarrow 1=T(a_1,a_2)\le T(a_1,1)=a_1$, i.e., $a_1=a_2=1.$

The opposite direction follows directly from the axiom of the boundary condition, i.e., $T(a_1,a_2)=T(1,1)=1.$

The proof for $T^n$ follows by induction.

Definition 5

The function $f(a_1,\ldots ,a_n)$ is strict if:

$$\begin{aligned} a_1<b_1\wedge \cdots \wedge a_n<b_n\Rightarrow \;f(a_1,\ldots ,a_n)<f(b_1,\ldots ,b_n). \end{aligned}$$

Lemma 2

If T is a strict triangular norm, then $T^n$ is a strictly increasing function.

Remark 4

If T is a strict t-norm, then:

$$\begin{aligned}&T(a_1,a_2)=0\Leftrightarrow \;a_1=0\vee a_2=0,\\&T^n(a_1,a_2,\ldots ,a_{n+1})=0\Leftrightarrow \;a_1=0\vee \cdots \vee a_{n+1}=0. \end{aligned}$$

Suppose the opposite that $T(a_1,a_2)=0$ implies $\lnot (a_1=0\vee a_2=0) \Leftrightarrow \;a_1>0\wedge a_2>0.$ Then from strict monotonicity $0<a_1\wedge 0<a_2,$ it follows that $T(0,0)<T(a_1,a_2)=0,$ which is a contradiction.

The opposite direction follows from the fact that 0 is the annihilator for the t-norm.

The proof for $T^n$ follows by induction.

Theorem 1

The mapping $T{:}\,[0,1]^ 2 \rightarrow [0,1]$ is an Archimedean t-norm if and only if there exists a decreasing generator g so that $T(a, b)=g^{(- 1)}(g (a) + g (b)),\,a,b\in [0,1]$.

The most commonly used triangular norms are:

• 1. $T(a,b)=\min (a,b)$ (standard intersection);

• 2. $T(a,b)=ab$ (algebraic product);

• 3. $T(a,b)=\max (a+b-1,0)$ (bounded difference);

• 4. $T(a,b)=\left\{ \begin{array}{ll} a, &{}\quad b=1\\ b, &{}\quad a=1\\ 0, &{}\quad \text{ otherwise } \end{array}\right.$ (drastic intersection).

Definition 6

The triangular conorm (shortly t-conorm) is a binary operation $S{:}\,[0,1]^2\rightarrow [0,1]$ satisfying the following axioms for all $a, b, c, b_1 \in [0,1]$:

• 1. $S(a,0)=a$ (boundary condition);

• 2. $b\le b_1 \Rightarrow S(a,b)\le S(a,b_1)$ (monotonicity);

• 3. $S(a,b)=S(b,a)$ (commutativity);

• 4. $S(a,S(b,c))=S(S(a,b),c)$ (associativity).

The t-conorm is said to be an Archimedeant-conorm, if in addition to the previous axioms two more axioms are satisfied:

• 5. S is a continuous function;

• 6. $\forall a\in (0,1),\; S(a,a)>a.$

Remark 5

From the conditions given in the definition of the t-conorm follows the monotonicity by coordinates, i.e., for all $a_1,a_2,b_1,b_2\in [0,1]$

$$\begin{aligned} a_1\le a_2 \wedge b_1\le b_2\Rightarrow S(a_1,b_1)\le S(a_2,b_2). \end{aligned}$$

Replacing the given condition with the condition of monotonicity in the definition of t-conorms, an equivalent definition of t-conorm is obtained.

If in the definition of t-conorm, instead of the axiom of monotonicity, the strict monotonicity holds, i.e.,

$$\begin{aligned} a_1< a_2 \wedge b_1<b_2\Rightarrow S(a_1,b_1)< S(a_2,b_2), \end{aligned}$$

for all $a_1,a_2,b_1,b_2\in [0,1],$ then the t-conorm is strict.

It can be shown that 1 is an annihilator for the t-conorm, i.e., for all $a \in [0,1]$

$$\begin{aligned} S(a,1)=S(1,a)=1. \end{aligned}$$

Definition 7

Increasing generatorg is a continuous and strictly increasing function from [0, 1] to ${\mathbb {R}}$, such that $g (0)=0.$Pseudo-inverse function for an increasing generator g, denoted by $g^{(-1)}$, is the function from ${\mathbb {R}}$ to [0, 1], defined by

$$\begin{aligned} g^{(-1)}(a)= \left\{ \begin{array}{ll} 0, &{}\quad a\in (-\,\infty ,0)\\ g^{-1}(a),&{}\quad a\in [0,g(1)]\\ 1, &{}\quad a\in (g(1),+\,\infty ), \end{array}\right. \end{aligned}$$

where $g^{-1}$ is the usual inverse function for g.

For an increasing generator g and its pseudo-inverse function $g^{(- 1)}$, the equality $g^{(-1)}(g(a))=a$ is satisfied for all $a\in [0,1]$ and

$$\begin{aligned} g(g^{(-1)}(a))= \left\{ \begin{array}{ll} 0, &{}\quad a\in (-\,\infty ,0)\\ a,&{} \quad a\in [0,g(1)]\\ g(1), &{} \quad a\in (g(1),+\,\infty ). \end{array}\right. \end{aligned}$$

Definition 8

The power of thet-conorm is given by formulas:

$$\begin{aligned}&S^1(a_1,a_2)=S(a_1,a_2),\\&S^{n}(a_1,\ldots ,a_n,a_{n+1})=S(S^{n-1}(a_1,\ldots ,a_{n}),a_{n+1}). \end{aligned}$$

Remark 6

Because of the associativity of S, 

$$\begin{aligned} S^n(a_1,\ldots ,a_n,a_{n+1})=S(a_1,S^{n-1}(a_2,\ldots ,a_{n},a_{n-1})). \end{aligned}$$

For example

$$\begin{aligned}&S^2(a_1,a_2,a_3)=S(S(a_1,a_2),a_3)=S(a_1,S(a_2,a_3)),\\&S^3(a_1,a_2,a_3,a_4)=S(S^2(a_1,a_2,a_3),a_4)\\&\quad =S(S(S(a_1,a_2),a_3),a_4) =S(S(a_1,a_2),S(a_3,a_4))\\&\quad =S(a_1,S(a_2,S(a_3,a_4))) =S(S(a_1,S(a_2,a_3)),a_4)\\&\quad =S(a_1,S(S(a_2,a_3),a_4)), \mathrm{etc.} \end{aligned}$$

Lemma 3

For a t-conorm S, holds:

$$\begin{aligned} S^{n-1}(a_1,\ldots ,a_n)=S^{n-1}(a_{i_1},\ldots ,a_{i_n}), \end{aligned}$$

where $a_{i_1},\ldots ,a_{i_n}$ is an arbitrary permutation of elements $a_1,$$\ldots ,$$a_n.$

Lemma 4

If S is a strict triangular conorm, then $S^n$ is a strictly increasing function.

Remark 7

If S is a t-conorm, then:

$$\begin{aligned}&S(a_1,a_2)=0\Leftrightarrow \;a_1=a_2=0.\\&S^n(a_1,a_2,\ldots ,a_{n+1})=0\Leftrightarrow \;a_1=\cdots =a_{n+1}=0. \end{aligned}$$

Indeed, $0\le a_1 \Rightarrow a_2=S(0,a_2)\le S(a_1,a_2)=0$ and $0\le a_2 \Rightarrow a_1=S(a_1,0)\le S(a_1,a_2)=0$, i.e., $a_1=a_2=0.$

The opposite direction follows directly from the axiom of the boundary condition, i.e., $S(a_1,a_2)=S(0,0)=0.$ The proof for $S^n$ follows by induction.

Remark 8

If S is a strict t-conorm, then:

$$\begin{aligned}&S(a_1,a_2)=1\Leftrightarrow \;a_1=1\vee a_2=1.\\&S^n(a_1,a_2,\ldots ,a_{n+1})=1\Leftrightarrow \;a_1=1\vee \cdots \vee a_{n+1}=1. \end{aligned}$$

Suppose the opposite that from $S(a_1, a_2) = 1$ follows $\lnot (a_1 = 1 \vee a_2 = 1) \Leftrightarrow \; a_1<1 \wedge a_2 <1.$ But then $a_1<1 \wedge a_2<1 \Rightarrow S(a_1, a_2) < S (1,1) = 1,$ which is a contradiction.

The opposite direction follows from the fact that 1 is the annihilator for the t-conorm.

The proof for $S^n$ follows by induction.

Theorem 2

The mapping $S{:}\,[0,1]^2\rightarrow [0,1]$ is an Archimedean t-conorm if and only if there exists an increasing generator g so that it holds

$$\begin{aligned} S(a,b)=g^{(-1)}(g(a)+g(b)),\quad a,b\in [0,1]. \end{aligned}$$

The most common triangular conorms are:

• 1. $S(a,b)=\max (a,b)$ (standard union);

• 2. $S(a,b)=a+b-ab$ (algebraic sum);

• 3. $S(a,b)=\min (1,a+b)$ (bounded sum);

• 4. $S(a,b)=\left\{ \begin{array}{ll} a, &{}\quad b=0\\ b, &{}\quad a=0\\ 1, &{}\quad \text{ otherwise } \end{array}\right.$ (drastic union).

Definition 9

The function $c{:}\,[0,1] \rightarrow [0,1]$ is a fuzzy complement, if the following conditions are satisfied:

($c_1$):

$c(0)=1$ i $c(1)=0,$ (boundary conditions)

($c_2$):

$(\forall a,b\in [0,1])\;a\le b\Rightarrow c(a)\ge c(b)$ (monotonicity).

If $c(c(a))=a$ holds for all $a\in [0,1]$, then a function c is involutive.

If c is a continuous function, then we say that cis a continuous fuzzy complement.

If $c{:}\,[0,1] \rightarrow [0,1]$ is an involutive monotonic non-increasing function, it follows that c is a continuous bijective function for which boundary conditions are valid (see Klir and Yuan 1995).

The triangular norm T and the triangular conorm S are dual with respect to the fuzzy complement c iff

$$\begin{aligned} c(T(a,b))\,{=}\,S(c(a),c(b))\; \text{ and } \;c(S(a,b))\,{=}\,T(c(a),c(b)). \end{aligned}$$

(T, S, c) is called a dual triple.

For the triangular norm T and the involutive fuzzy complement c, the binary operation S on [0, 1] defined with

$$\begin{aligned} S(a,b))=c(T(c(a),c(b))) \end{aligned}$$

for all $a,b\in [0,1]$ is a triangular conorm S such that (T, S, c) is a dual triple.

3 Fuzzy metrics

In the fuzzy framework, many different definitions of a distance were proposed, satisfying various properties, depending on applications (see Deza and Deza 2009). On a set of fuzzy sets, defined over a set, distance can be considered according to certain properties and applications (see, e.g., Bloch 1999). In this section, the fuzzy S-metric and the fuzzy T-metric are considered. Some of the new characteristics of the T-fuzzy metrics are presented in this section. The well-known notions and characteristics are given and proven in the papers: Bloch (1999), Deza and Deza (2009), Gregori and Romaguera (2000) and Morillas et al. (2007).

Definition 10

FuzzyS-metric space is a triple $(X,{\varvec{s}},S)$ such that X is a non-empty set, S is a continuous t-conorm, and ${\varvec{s}}$ is a fuzzy set defined on $X\times X\times (0,+\,\infty )$ that satisfies the following conditions for all $x,y,z \in X,\alpha , \beta > 0$:

• 1. ${\varvec{s}}(x,y,\alpha )\in [0,1);$

• 2. ${\varvec{s}}(x,y,\alpha )=0 \Leftrightarrow x=y;$

• 3. ${\varvec{s}}(x,y,\alpha )={\varvec{s}}(y,x,\alpha );$

• 4. $S({\varvec{s}}(x,y,\alpha ),{\varvec{s}}(y,z,\beta ))\ge {\varvec{s}}(x,z,\alpha +\beta );$

• 5. ${\varvec{s}}(x,y,{}_{-}){:}\,(0,+\,\infty )\rightarrow [0,1] \text{ is } \text{ a } \text{ continuous } \text{ function. }$

The fuzzy set ${\varvec{s}}$ is called a fuzzyS-metric. If instead of 1., ${\varvec{s}}(x,y,\alpha )\in [0,1],$ the fuzzy set ${\varvec{s}}$ is a fuzzyS-metric in the broader sense, and $(X,{\varvec{t}},S)$ is a fuzzyS-metric space in the broader sense.

Definition 11

FuzzyT-metric space is an ordered triple $(X,{\varvec{t}},T)$ such that X is a non-empty set, T is a continuous t-norm and ${\varvec{t}}$ is a fuzzy set defined on $X\times X\times (0,+\,\infty )$ that satisfies the following conditions for all $x,y,z \in X,\alpha , \beta > 0$:

• 1. ${\varvec{t}}(x,y,\alpha )\in (0,1];$

• 2. ${\varvec{t}}(x,y,\alpha )=1 \Leftrightarrow x=y;$

• 3. ${\varvec{t}}(x,y,\alpha )={\varvec{t}}(y,x,\alpha );$

• 4. $T({\varvec{t}}(x,y,\alpha ),{\varvec{t}}(y,z,\beta ))\le {\varvec{t}}(x,z,\alpha +\beta );$

• 5. ${\varvec{t}}(x,y,{}_{-}):(0,+\,\infty )\rightarrow [0,1] \text{ is } \text{ a } \text{ continuous } \text{ function. }$

The fuzzy set ${\varvec{t}}$ is called fuzzyT-metric. If instead of 1., ${\varvec{t}}(x,y,\alpha )\in [0,1],$ the fuzzy set ${\varvec{t}}$ is a fuzzyT-metric in the broader sense, and $(X,{\varvec{t}},T)$ is a fuzzyT-metric space in the broader sense.

Definition 12

Fuzzy S-metric ${\varvec{s}}$ (T-metric ${\varvec{t}}$) is stationary on X if ${\varvec{s}}$ (${\varvec{t}}$) does not depend of $\alpha$, i.e., if for all fixed $x,y \in X,$ the function ${\varvec{s}}_{x,y}(\alpha )={\varvec{s}}(x,y,\alpha )$ (${\varvec{t}}_{x,y}(\alpha )={\varvec{t}}(x,y,\alpha )$) is a constant.

Theorem 3

If $(X,{\varvec{s}},S)$ is a fuzzy S-metric space and the T is a t-norm dual to the t-conorm S with respect to the continuous involutive fuzzy complement c,  then $(X,c\circ {\varvec{s}},T)$ is a fuzzy T-metric space.

If $(X,{\varvec{t}},T)$ is a fuzzy T-metric space and S is a t-conorm dual to the norm T with respect to a continuous involutive fuzzy complement c,  then $(X,c\circ {\varvec{t}},S)$ is a fuzzy S-metric space.

Proof

First of all, let us note that c is injective (bijective) and therefore strictly monotone.

• 1. $0\le {\varvec{s}}(x,y,\alpha )<1$

  $\Rightarrow 1=c(0)\ge c({\varvec{s}}(x,y,\alpha ))>c(1)=0.$

• 2. $x=y \Leftrightarrow {\varvec{s}}(x,y,\alpha )=0$

  $\Leftrightarrow \; c({\varvec{s}}(x,y,\alpha ))=c(0)=1.$

• 3. ${\varvec{s}}(x,y,\alpha )={\varvec{s}}(y,x,\alpha )$

  $\Rightarrow \;c({\varvec{s}}(x,y,\alpha ))=c({\varvec{s}}(y,x,\alpha )).$

• 4. $S({\varvec{s}}(x,y,\alpha ),{\varvec{s}}(y,z,\beta ))\ge {\varvec{s}}(x,z,\alpha +\beta )$

  $\Rightarrow \;c(S({\varvec{s}}(x,y,\alpha ),{\varvec{s}}(y,z,\beta )))\le c({\varvec{s}}(x,z,\alpha +\beta ))$

  $\Leftrightarrow \;T(c({\varvec{s}}(x,y,\alpha )),c({\varvec{s}}(y,z,\beta )))\le c({\varvec{s}}(x,z,\alpha +\beta )).$

• 5. $c{:}\,[0,1]\rightarrow [0,1]$ is continuous and ${\varvec{s}}(x,y,{}_{-}){:}\,(0,+\,\infty )\rightarrow [0,1]$ is continuous, therefore a composed function $c\circ {\varvec{s}}(x,y,{}_{-}){:}\,(0,+\,\infty )\rightarrow [0,1]$ is also continuous.

For the second part of the theorem, the proof is analogous.

$\square$

The theorem is also valid for fuzzy metric spaces in the broader sense.

Example 1

The mapping ${} \mathbf{t}_K{:}\,{\mathbb {R}}^+\times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}$ defined by $\mathbf{t}_K(x,y)=\frac{\min \{x,y\}+K}{\max \{x,y\}+K},$ where $K>0$, is a fuzzy T-metric with respect to multiplication, and ${} \mathbf{s}_K(x,y)=\frac{|x-y|}{\max (x,y)+K}$ is a fuzzy S-metric with respect to the algebraic sum, $S (x, y) = 1- (1-x) (1-y) = x + y-xy,$ dual to T with respect to the standard fuzzy complement.

• 1. $x,y\in {\mathbb {R}}^+, K> 0\Rightarrow 0<\min \{x,y\}+K\le \max \{x,y\} \Rightarrow 1\ge \mathbf{t}_K(x,y)=\frac{\min \{x,y\}+K}{\max \{x,y\}+K}>0.$

• 2. ${} \mathbf{t}_K(x,y)=\frac{\min \{x,y\}+K}{\max \{x,y\}+K}=1\Leftrightarrow \min \{x,y\}+K=\max \{x,y\}+K\Leftrightarrow \min \{x,y\}=\max \{x,y\}\Leftrightarrow x=y$

• 3. $\mathbf{t}_K(x,y)=\frac{\min \{x,y\}+K}{\max \{x,y\}+K}=\frac{\min \{y,x\}+K}{\max \{y,x\}+K}=\mathbf{t}_K(y,x)$

• 4. Let’s prove inequality

  $$\begin{aligned} \mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)\le \mathbf{t}_K(x,z). \end{aligned}$$

  There are six cases: $(1)\;x\le y\le z,$$(2)\;x\le z\le y,$$(3)\;y\le x\le z,$$(4)\;z\le y\le x,$$(5)\;z\le x\le y,$$(6)\;y\le z\le x$; it is enough to examine the first three because changing the place of x and z: $\mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)\le \mathbf{t}_K(x,z)\;\Leftrightarrow \mathbf{t}_K(z,y) \cdot \mathbf{t}_K(y,x)\le \mathbf{t}_K(z,x)$ the remaining three cases follow.

  • (1) $\mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)=\frac{\min \{x,y\}+K}{\max \{x,y\}+K}\cdot \frac{\min \{y,z\}+K}{\max \{y,z\}+K} =\frac{x+K}{y+K}\cdot \frac{y+K}{z+K}=\frac{x+K}{z+K}=\frac{\min \{x,z\}+K}{\max \{x,z\}+K}=\mathbf{t}_K(x,z),$

  • (2) $\mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)= \frac{\min \{x,y\}+K}{\max \{x,y\}+K}\cdot \frac{\min \{y,z\}+K}{\max \{y,z\}+K} =\frac{x+K}{y+K}\cdot \frac{z+K}{y+K}\le \frac{x+K}{z+K}\cdot \frac{z+K}{z+K} =\frac{\min \{x,z\}+K}{\max \{x,z\}+K}=\mathbf{t}_K(x,z),$

  • (3) $\mathbf{t}_K(x,y) \cdot {} \mathbf{t}_K(y,z)=\frac{\min \{x,y\}+K}{\max \{x,y\}+K}\cdot \frac{\min \{y,z\}+K}{\max \{y,z\}+K}=\frac{y+K}{x+K}\cdot \frac{y+K}{z+K}\le \frac{x+K}{x+K}\cdot \frac{x+K}{z+K}=\frac{\min \{x,z\}+K}{\max \{x,z\}+K}=\mathbf{t}_K(x,z).$

  The function $f(K)=\frac{a+K}{b+K},$ where $a,b,K>0$, is monotonously increasing, so it is

  $$\begin{aligned} \mathbf{t}_{K_1}(x,y) \le \mathbf{t}_{K_1+K_2}(x,y),\;\;\;\mathbf{t}_{K_2}(y,z)\le \mathbf{t}_{K_1+K_2}(y,z), \end{aligned}$$

  i.e.,

  $$\begin{aligned} \mathbf{t}_{K_1}(x,y) \cdot \mathbf{t}_{K_2}(y,z)\le & {} \mathbf{t}_{K_1+K_2}(x,y) \cdot \mathbf{t}_{K_1+K_2}(y,z)\\\le & {} \mathbf{t}_{K_1+K_2}(x,z). \end{aligned}$$

• 5. The mapping $\mathbf{t}_K$ is obviously a continuous function on the parameter K.

Example 2

The mapping $\mathbf{t}_K{:}\,{\mathbb {R}}^+\times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}$ defined by $\mathbf{t}_K(x,y)=\frac{\frac{x+y}{2}+K}{\max \{x,y\}+K},$ where $K>0,$ is a fuzzy T-metric with respect to multiplication, and $\mathbf{s}_K(x,y)=\frac{|x-y|}{2(\max (x,y)+K)}$ is the fuzzy S-metric with respect to the algebraic sum, dual to T with respect to standard fuzzy complement.

• 1. $x,y\in {\mathbb {R}}^+, K> 0.$ Without loss of generality, let $x\le y.$ Then

  $x+y\le y+y=2y=2\max \{x,y\}$

  $\Rightarrow \;\frac{x+y}{2}\le \max \{x,y\}$

  $\Rightarrow \;0<\frac{x+y}{2}+K\le \max \{x,y\}+K$

  $\Rightarrow \;1\ge \mathbf{t}_K(x,y)=\frac{\frac{x+y}{2}+K}{\max \{x,y\}+K}>0.$

• 2. $(\Leftarrow )\;$ $x=y\Rightarrow \mathbf{t}_K(x,y)=\frac{\frac{x+x}{2}+K}{\max \{x,x\}+K}=\frac{x+K}{x+K}=1$

  $(\Rightarrow )\;\mathbf{t}_K(x,y)=\frac{\frac{x+y}{2}+K}{\max \{x,y\}+K}=1\Leftrightarrow \frac{x+y}{2}+K=\max \{x,y\}+K\Leftrightarrow x+y=2\max \{x,y\}:$

  $x\ge y\Rightarrow x+y=2x\Rightarrow y=x,$

  $x\le y\Rightarrow \; x+y=2y\Rightarrow x=y.$

• 3. $\mathbf{t}_K(x,y)=\frac{\frac{x+y}{2}+K}{\max \{x,y\}+K}=\frac{\frac{y+x}{2}+K}{\max \{y,x\}+K}=\mathbf{t}_K(y,x).$

• 4. Let’s prove inequality

  $$\begin{aligned} \mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)\le \mathbf{t}_K(x,z). \end{aligned}$$

  There are three cases: $(1)\;x\le y\le z,$$(2)\;x\le z\le y,$ and $(3)\;y\le x\le z.$ For simplicity, few replacements are used: $X=x+K, Y=y+K, Z=z+K.$

  (1) $\mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)= \frac{1}{2}\frac{X+Y}{Y}\cdot \frac{1}{2}\frac{Y+Z}{Z}\le \frac{1}{2}\frac{X+Z}{Z}=\mathbf{t}_K(x,z)$

  $\Leftrightarrow \;(X+Y)(Z+Y)\le 2(X+Z)Y$

  $\Leftrightarrow \;Y^2+XY+ZY+XZ\le 2XY+2ZY$

  $\Leftrightarrow \;Y^2-(X+Z)Y+XZ\le 0$

  $\Leftrightarrow \;(Y-X)(Y-Z)\le 0$

  $\Leftrightarrow \top .$ The inequality is correct because two following inequalities hold: $X\le Y$ and $Y\le Z.$

  (2) $\mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)=\frac{1}{2}\frac{x+K+y+K}{y+K}\cdot \frac{1}{2}\frac{y+K+z+K}{y+K}\le \frac{1}{2}\frac{x+K+z+K}{z+K}=\mathbf{t}_K(x,z)$

  $\Leftrightarrow \frac{X+Y}{Y}\cdot \frac{Y+Z}{Y}\le 2\frac{X+Z}{Z}$

  $\Leftrightarrow \;\big (Y^2+(X+Z)Y+XZ\big )Z\le 2(X+Z)Y^2$

  $\Leftrightarrow (X+Z)ZY+XZ^2\le 2XY^2+ZY^2$

  $\Leftrightarrow \;XZY+Z^2Y+XZ^2\le XY^2+XY^2+ZY^2.$ This inequality is valid because:

  $Z\le Y\Rightarrow XZY\le XY^2,$

  $Z\le Y\Rightarrow XZ^2\le XY^2,$

  $Z\le Y\Rightarrow Z^2Y\le ZY^2.$

  (3) $\mathbf{t}_K(x,y) \cdot \mathbf{t}_K(y,z)=\frac{1}{2}\frac{X+Y}{X}\cdot \frac{1}{2}\frac{Y+Z}{Z}\le \frac{1}{2}\frac{X+Z}{Z}=\mathbf{t}_K(x,z)$

  $\Leftrightarrow (X+Y)(Y+Z)\le 2X\cdot (X+Z).$ This inequality is true because the following inequalities hold:

  $Y\le X\Rightarrow X+Y\le 2X,\;\;Y\le X\Rightarrow Y+Z\le X+Z.$ The function $f(K)=\frac{a+K}{b+K},$ where $a,b,K>0$, is monotonously increasing, so

  $\mathbf{t}_{K_1}(x,y) \le \mathbf{t}_{K_1+K_2}(x,y),\;\;\;\mathbf{t}_{K_2}(y,z)\le \mathbf{t}_{K_1+K_2}(y,z),$

  i.e.,

  $$\begin{aligned} \mathbf{t}_{K_1}(x,y) \cdot \mathbf{t}_{K_2}(y,z)\le & {} \mathbf{t}_{K_1+K_2}(x,y) \cdot \mathbf{t}_{K_1+K_2}(y,z)\\\le & {} \mathbf{t}_{K_1+K_2}(x,z). \end{aligned}$$

• 5. The mapping $\mathbf{t}_K$ is obviously a continuous function on the parameter K.

Example 3

The mapping $\mathbf{t}_p{:}\,{\mathbb {R}}^+\times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}, p>0$ defined by $\mathbf{t}_p(x,y)=\frac{\root p \of {\frac{x^p+y^p}{2}}}{\max \{x,y\}},$ is a fuzzy T-metric with respect to multiplication.

• 1. $\;x,y\in {\mathbb {R}}^+.$ Without loss of generality, let $x\le y.$ Then $x^p\le y^p \Rightarrow x^p+y^p\le 2y^p \Rightarrow \frac{x^p+y^p}{2}\le y^p\Rightarrow \root p \of { \frac{x^p+y^p}{2}}\le \root p \of {y^p}=y=\max \{x,y\}$, i.e., $1\ge \mathbf{t}_p(x,y)=\frac{\root p \of {\frac{x^p+y^p}{2}}}{\max \{x,y\}}>0.$

• 2. $\;(\Leftarrow )\; x=y$ $\Rightarrow \;\mathbf{t}_p(x,y)=\frac{\root p \of {\frac{x^p+x^p}{2}}}{\max \{x,x\}}=\frac{\root p \of {\frac{2x^p}{2}}}{x}=1.$

  $\;(\Rightarrow )\;\mathbf{t}_p(x,y)=1 \Rightarrow \root p \of {\frac{x^p+y^p}{2}}=\max \{x,y\}$

  $x\le y\Rightarrow \root p \of {\frac{x^p+y^p}{2}}=y\Rightarrow \frac{x^p+y^p}{2}=y^p\Rightarrow x^p=y^p \;(x,y > 0) \Rightarrow x=y,$

  $y\le x\Rightarrow \root p \of {\frac{x^p+y^p}{2}}=x\Rightarrow \frac{x^p+y^p}{2}=x^p\Rightarrow y^p=x^p \;(x,y > 0) \Rightarrow y=x.$

• 3. $\;\mathbf{t}_p(x,y)=\frac{\frac{\root p \of {x^p+y^p}}{2}}{\max \{x,y\}}=\frac{\frac{\root p \of {y^p+x^p}}{2}}{\max \{y,x\}}=\mathbf{t}_p(y,x).$

• 4. To prove the inequality

  $$\begin{aligned} \mathbf{t}_p(x,y) \cdot \mathbf{t}_p(y,z)\le \mathbf{t}_p(x,z). \end{aligned}$$

  The following cases are examined: $(1)\;x\le y\le z,$$(2)\;x\le z\le y,$$\;3)\;y\le x\le z.$

  (1) $\mathbf{t}_p(x,y) \cdot \mathbf{t}_p(y,z)=\frac{\root p \of {\frac{x^p+y^p}{2}}}{y}\cdot \frac{\root p \of {\frac{y^p+z^p}{2}}}{z}$

  $\le \frac{\root p \of {\frac{x^p+z^p}{2}}}{z}=\mathbf{t}_p(x,z)$

  $\Leftrightarrow \; (x^p+y^p)(y^p+z^p)\le 2y^p(x^p+z^p)$

  $\Leftrightarrow \;(y^p)^2+x^pz^p\le y^px^p+y^pz^p$

  $\Leftrightarrow \;0\le y^p(z^p-y^p)-x^p(z^p-y^p)$

  $\Leftrightarrow \;0\le (y^p-x^p)(z^p-y^p)$

  $\Leftrightarrow \;\top .$

  (2) $\mathbf{t}_p(x,y)\cdot \mathbf{t}_p(y,z)=\frac{\root p \of {\frac{x^p+y^p}{2}}}{y}\cdot \frac{\root p \of {\frac{y^p+z^p}{2}}}{y} \le \frac{\root p \of {\frac{x^p+z^p}{2}}}{\max \{x,z\}}=\mathbf{t}_p(x,z)$

  $\Leftrightarrow \;z^p(x^p+y^p)(y^p+z^p)\le 2(y^p)^2(x^p+z^p)$

  $\Leftrightarrow \;z^px^py^p+x^p(z^p)^2+(z^p)^2y^p\le 2(y^p)^2x^p+(y^p)^2z^p$

  $\Leftrightarrow \;0\le (y^p)^2x^p-x^pz^py^p+(y^p)^2x^p-x^p(z^p)^2+(y^p)^2z^p-y^p(z^p)^2$

  $\Leftrightarrow \;0\le y^px^p(y^p-z^p)+x^p(y^p+z^p)(y^p-z^p)+y^pz^p(y^p-z^p)$

  $\Leftrightarrow \;0\le (y^p-z^p)(2y^px^p+x^pz^p+y^pz^p)$

  $\Leftrightarrow \top .$ (3) $\mathbf{t}_p(x,y) \cdot \mathbf{t}_p(y,z)=\frac{\root p \of {\frac{x^p+y^p}{2}}}{x}\cdot \frac{\root p \of {\frac{y^p+z^p}{2}}}{z}$

  $\le \frac{\root p \of {\frac{x^p+z^p}{2}}}{z}=\mathbf{t}_p(x,z)$

  $\Leftrightarrow \;(x^p+y^p)(y^p+z^p)\le 2x^p(x^p+z^p)$

  $\Leftrightarrow \;x^py^p+(y^p)^2+y^pz^p\le 2(x^p)^2+x^pz^p$

  $\Leftrightarrow \;0\le (x^p)^2-(y^p)^2+(x^p)^2-x^py^p+x^pz^p-y^pz^p$

  $\Leftrightarrow \;0\le (x^p-y^p)(2x^p+y^p+z^p)$

  $\Leftrightarrow \top .$ The function $f(p)=\frac{1}{c}\root p \of {\frac{a^p+b^p}{2}},$ where $a,b,c,p>0$, is monotonously increasing, so it is

  $\mathbf{t}_{p_1}(x,y) \le \mathbf{t}_{p_1+p_2}(x,y),\;\;\;\mathbf{t}_{p_2}(y,z)\le \mathbf{t}_{p_1+p_2}(y,z),$

  i.e.,

  $$\begin{aligned} \mathbf{t}_{p_1}(x,y) \cdot \mathbf{t}_{p_2}(y,z)\le & {} \mathbf{t}_{p_1+p_2}(x,y) \cdot \mathbf{t}_{p_1+p_2}(y,z)\\\le & {} \mathbf{t}_{p_1+p_2}(x,z). \end{aligned}$$

• 5. The mapping $\mathbf{t}_p$ is obviously a continuous function on the parameter p.

Example 4

If (X, d) is a metric space, then the mapping $\mathbf{t}{:}\,X\times X \times {\mathbb {R}}^+\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} \mathbf{t}(x,y,t)=\frac{t}{t+d(x,y)} \end{aligned}$$

is a fuzzy T-metric with respect to the multiplication and its dual (with respect to the standard fuzzy complement) $\mathbf{s}(x,y,t)=1-\mathbf{t}(x,y,t)=\frac{d(x,y)}{t+d(x,y)}$ is a fuzzy S-metric with respect to the algebraic sum.

Only property 4 is proven, because the remaining three follow directly by means of the properties of the standard metric.

$$\begin{aligned}&{} \mathbf{t}(x,y,t_1)\cdot \mathbf{t}(y,z,t_2)=\frac{t_1}{t_1+d(x,y)}\cdot \frac{t_2}{t_2+d(y,z)}\\&\quad \le \frac{t_1+t_2}{t_1+t_2+ d(x,z)} =\mathbf{t}(x,z,t_1+t_2)\\&\quad \Leftrightarrow t^2_1t_2+t_1t^2_2+t_1t_2d(x,z)\le t^2_1t_2+t_1t^2_2\\&\qquad +\, t_2(t_1+t_2)d(x,y)+t_1(t_1+t_2)d(y,z)\\&\qquad +\,(t_1+t_2)d(x,y)d(y,z)\\&\quad \Leftrightarrow t_1t_2d(x,z)\le t_1t_2(d(x,y)+d(y,z))\\&\qquad +\, t_2^2 d(x,y)+ t_1^2 d(y,z)+ (t_1+t_2)d(x,y)d(y,z)\\&\quad \Leftrightarrow \top . \end{aligned}$$

Theorem 4

Let ${\varvec{s}}$ be a stationary fuzzy S-metric (in the broadest sense) with respect to the t-conorm S. If S is an Archimedean t-conorm and g its corresponding increasing generator, then $d=g\circ {\varvec{s}}$ is a standard metric.

Proof

• 1. As g is strictly increasing, from $1>{\varvec{s}}(x,y)\ge 0$ follows

  $g(1)>d(x,y)=g({\varvec{s}}(x,y))\ge g(0)=0.$ For the fuzzy S-metric in the broader sense $g(1)\ge d(x,y)\ge 0.$

• 2. From the strict monotonicity of the function g and its injectivity, it follows that $g(a)=0\Leftrightarrow a=0$, and due to the second condition of fuzzy metric follows:

  $$\begin{aligned} d(x,y)=g({\varvec{s}}(x,y))=0\Leftrightarrow {\varvec{s}}(x,y)=0 \Leftrightarrow x=y. \end{aligned}$$

• 3. ${\varvec{s}}(x,y)={\varvec{s}}(y,x)\Rightarrow$

  $d(x,y)=g({\varvec{s}}(x,y))=g({\varvec{s}}(y,x))=d(y,x).$

• 4. From condition 4 of the definition of the fuzzy S-metric ${\varvec{s}}$ and the theorems of the representation of the t-conorm S, it holds that

  $$\begin{aligned} S({\varvec{s}}(x,y),{\varvec{s}}(y,z))= & {} g^{(-1)}(g({\varvec{s}}(x,y))+g({\varvec{s}}(y,z))\\\ge & {} {\varvec{s}}(x,z), \end{aligned}$$

  and as g is the increasing generator, it follows that:

  $$\begin{aligned} g(g^{(-1)}(g({\varvec{s}}(x,y))+g({\varvec{s}}(y,z)))\ge g({\varvec{s}}(x,z)). \end{aligned}$$

  For $g({\varvec{s}}(x,y))+g({\varvec{s}}(y,z))\in [0,g(1)]$, it holds that $g({\varvec{s}}(x,y))$$+$$g({\varvec{s}}(y,z))$$\ge g({\varvec{s}}(x,z)),$ and for d the inequality of the triangle is valid. In case that $g({\varvec{s}}(x,y)) +g({\varvec{s}}(y,z))$$\ge g(1)$, and from $1> {\varvec{s}}(x,z)$ it follows that $g(1)> g({\varvec{s}}(x,z)),$ i.e., the inequality of the triangle is valid. $\square$

Theorem 5

Let ${\varvec{t}}$ be a stationary fuzzy T-metric (in the broader sense) with respect to the t-norm T. If T is an Archimedean t-norm and g its corresponding decreasing generator, then $d=g\circ {\varvec{t}}$ is a standard metric.

Proof

• 1. Since g is a strictly decreasing function, from $1\ge {\varvec{t}}(x,y)> 0$ follows

  $0=g(1)\le d(x,y)=g({\varvec{t}}(x,y))< g(0).$ For the fuzzy T-metric, in the broader sense, it holds that $0\le d(x,y)\le g(0).$

• 2. From the strict monotonicity of the function g and its injectivity, it follows that $g(a)=0\Leftrightarrow a=1$, and due to condition 2 of the fuzzy metric, it holds that:

  $$\begin{aligned} d(x,y)=g({\varvec{t}}(x,y))=0\Leftrightarrow {\varvec{t}}(x,y)=1 \Leftrightarrow x=y. \end{aligned}$$

• 3. ${\varvec{t}}(x,y)={\varvec{t}}(y,x)\Rightarrow$

  $d(x,y)=g({\varvec{t}}(x,y))=g({\varvec{t}}(y,x))=d(y,x).$

• 4. From condition 4 of fuzzy T-metric ${\varvec{t}}$ and theorems on the representation of the t-norm T follows

  $$\begin{aligned} T({\varvec{t}}(x,y),{\varvec{t}}(y,z))= & {} g^{(-1)}(g({\varvec{t}}(x,y))+g({\varvec{t}}(y,z))\\\le & {} {\varvec{t}}(x,z), \end{aligned}$$

  and since g is a decreasing generator, it follows that

  $$\begin{aligned} g(g^{(-1)}(g({\varvec{t}}(x,y))+g({\varvec{t}}(y,z)))\ge g({\varvec{t}}(x,z)). \end{aligned}$$

  For $g({\varvec{t}}(x,y))+g({\varvec{t}}(y,z))\in [0,g(0)]$, it holds that $g({\varvec{t}}(x,y))$$+$$g({\varvec{t}}(y,z))$$\ge g({\varvec{t}}(x,z)),$ and for d the inequality of the triangle is true. In case that $g({\varvec{t}}(x,y))+g({\varvec{t}}(y,z))>g(0)$ holds, and from $0< {\varvec{t}}(x,z)$ it follows that $g(0)> g({\varvec{t}}(x,z)),$ and the inequality of the triangle is valid. $\square$

Theorem 6

If ${\varvec{s}}_1{:}\,X\times X\times (0,+\,\infty ) \rightarrow [0,1)$ and ${\varvec{s}}_2{:}\,X\times X\times (0,+\,\infty ) \rightarrow [0,1)$ are fuzzy S-metrics, with respect to the strict triangular conorm S, then the mapping ${\sigma }({\varvec{s}}_1,{\varvec{s}}_2){:}\,X\times X\times (0,+\,\infty ) \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} {\sigma }({\varvec{s}}_1,{\varvec{s}}_2)(x,y,\alpha )= S\left( {\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha )\right) \end{aligned}$$

is also a fuzzy S-metric with respect to the conorm S. If S is not a strict t-conorm, then ${{{\sigma }}}({\varvec{s}}_1,{\varvec{s}}_2)$ is a fuzzy S-metric in a broader sense.

Proof

• 1. From property 1 for ${\varvec{s}}_1$ and ${\varvec{s}}_2$, it follows that

  ${\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha ) \in [0,1]$, because S is a t-conorm, it follows that

  $$\begin{aligned} S({\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha ))\in [0,1]. \end{aligned}$$

  If S is a strict t-conorm, then from

  ${\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha )\in [0,1)$ and from Remark 8, it follows that

  $$\begin{aligned} S({\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha )) \in [0,1). \end{aligned}$$

• 2. From the property $S(a_1,a_2)=0\Leftrightarrow a_1=a_2=0,$ given in Remark 7, it follows that

  $$\begin{aligned}&S({\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha ))=0\\&\quad \Leftrightarrow {\varvec{s}}_1(x,y,\alpha )= {\varvec{s}}_2(x,y,\alpha )=0 \Leftrightarrow x=y. \end{aligned}$$

• 3. The triangular conorm S is well defined; therefore, it holds:

  $$\begin{aligned}&{\varvec{s}}_1(x,y,\alpha )={\varvec{s}}_1(y,x,\alpha )\wedge {\varvec{s}}_2(x,y,\alpha )={\varvec{s}}_2(y,x,\alpha )\\&\quad \Rightarrow \;S({\varvec{s}}_1(x,y,\alpha ),{\varvec{s}}_2(x,y,\alpha ))\\&\quad =S({\varvec{s}}_1(y,x,\alpha ),{\varvec{s}}_2(y,x,\alpha )). \end{aligned}$$

• 4. For simplicity, the next labels are introduced:

  $$\begin{aligned}&a_1={\varvec{s}}_1(x,y,\alpha ),a_2={\varvec{s}}_2(x,y,\alpha ), b_1={\varvec{s}}_1(y,z,\beta ),\\&b_2 ={\varvec{s}}_2(y,z,\beta ), c_1={\varvec{s}}_1(x,z,\alpha +\beta ),\\&c_2={\varvec{s}}_2(x,z,\alpha +\beta ). \end{aligned}$$

  From axiom 4 of the metric for ${\varvec{s}}_1$ and ${\varvec{s}}_2$, it holds that

  $$\begin{aligned} S(a_1,b_1)\le c_1,\quad S(a_2,b_2)\le c_2, \end{aligned}$$

  thus from monotonicity follows:

  $$\begin{aligned} S(S(a_1,b_1),S(a_2,b_2))\le S(c_1,c_2). \end{aligned}$$

  (1)

  From the associativity and commutativity of S, it follows

  $$\begin{aligned}&S(S(a_1,b_1),S(a_2,b_2))=S(a_1,S(b_1,S(a_2,b_2)))\\&\quad =S(a_1,S(S(b_1,a_2),b_2))=S(a_1,S(S(a_2,b_1),b_2))\\&\quad =S(a_1,S(a_2,S(b_1,b_2)))=S(S(a_1,a_2),S(b_1,b_2)), \end{aligned}$$

  since (1) is valid it follows that

  $$\begin{aligned} S(S(a_1,a_2),S(b_1,b_2))\le S(c_1,c_2). \end{aligned}$$

• 5. The mappings ${\varvec{s}}_1(x,y,{}_{-})$ and ${\varvec{s}}_2(x,y,{}_{-})$ and t-conorm S are continuous; thus, $\sigma ({\varvec{s}}_1(x,y,{}_{-}),{\varvec{s}}_2(x,y,{}_{-}))$ is also a continuous function. $\square$

Theorem 7

If ${\varvec{t}}_1{:}\,X\times X\times (0,+\,\infty ) \rightarrow (0,1]$ and ${\varvec{t}}_2{:}\,X\times X\times (0,+\,\infty ) \rightarrow (0,1]$ are fuzzy T-metrics with respect to the strict triangular norm T, then a mapping $\tau ({\varvec{t}}_1,{\varvec{t}}_2){:}\,X\times X\times (0,+\,\infty ) \rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \tau ({\varvec{t}}_1,{\varvec{t}}_2)(x,y,\alpha )= T\left( {\varvec{t}}_1(x,y,\alpha ),{\varvec{t}}_2(x,y,\alpha )\right) \end{aligned}$$

is a fuzzy T-metric with respect to the norm T.

If T is not a strict t-norm, then $\mathbf{\tau }({\varvec{t}}_1,{\varvec{t}}_2)$ is a fuzzy T-metric in a broader sense.

Theorem 8

If ${\varvec{t}}_i{:}\,X_i\times X_i\rightarrow (0,1]$ are fuzzy T-metrics with respect to a strict triangular norm T, then ${\varvec{t}}{:}\,X^2\rightarrow [0,1],$$X=X_1\times \dots \times X_n$ given by

$$\begin{aligned}&{\varvec{t}}(x,y)=T^{n-1}({\varvec{t}}_1 (x_1,y_1),{\varvec{t}}_2 (x_2,y_2),\ldots ,{\varvec{t}}_n (x_n,y_n)),\\&x=(x_1,\ldots ,x_n),\;y=(y_1,\ldots ,y_n), \end{aligned}$$

is a fuzzy T-metric with respect to the triangular norm T. If T is not a strict t-norm, then ${\varvec{t}}$ is a fuzzy T-metric in the broader sense.

Proof

• 1. ${\varvec{t}}_i(x_i,y_i)\in (0,1]$

  $\Rightarrow {\varvec{t}}(x,y)=T^{n-1}({\varvec{t}}_1(x_1,y_1),\ldots ,{\varvec{t}}_n(x_n,y_n))\in [0,1].$ If T is a strict triangular norm, then $T^{n-1}$ is also a strict monotone function, then from ${\varvec{t}}_i(x_i,y_i)\in (0,1],\;$ based on Remark 4, follows

  $$\begin{aligned} {\varvec{t}}(x,y)=T^{n-1}({\varvec{t}}_1(x_1,y_1),\ldots ,{\varvec{t}}_n(x_n,y_n))\in (0,1]. \end{aligned}$$

• 2. From Remark 3 follows:

  $$\begin{aligned}&{\varvec{t}}(x,y)=T^{n-1}({\varvec{t}}_1(x_1,y_1),\ldots ,{\varvec{t}}_n(x_n,y_n))=1\\&\quad \Leftrightarrow (\forall i\in \{1,\dots ,n\})\;{\varvec{t}}_i(x_i,y_i)=1\\&\quad \Leftrightarrow (\forall i\in \{1,\dots ,n\})\;x_i=y_i \Leftrightarrow x=y. \end{aligned}$$

• 3. ${\varvec{t}}(x,y)=T^{n-1}({\varvec{t}}_1(x_1,y_1),...,{\varvec{t}}_n(x_n,y_n))$

  $\quad =T^{n-1}({\varvec{t}}_1(y_1,x_1),...,{\varvec{t}}_n(y_n,x_n)) ={\varvec{t}}(y,x).$

• 4. From axiom 4 of the fuzzy T-metric ${\varvec{t}}_i$

  follows $T({\varvec{t}}_i(x_i,y_i),{\varvec{t}}_i(y_i,z_i))\le {\varvec{t}}_i(x_i,z_i),\; i\in \{1,\ldots ,n\},$ so it follows

  $$\begin{aligned}&T({\varvec{t}}(x,y),{\varvec{t}}(y,z))=T\left( T^{n-1}({\varvec{t}}_1(x_1,y_1),\ldots ,{\varvec{t}}_n(x_n,y_n)),\right. \\&\quad \left. T^{n-1}\left( {\varvec{t}}_1(y_1,z_1),\ldots ,{\varvec{t}}_n(y_n,z_n)\right) \right) \\&\quad =T^{2n-1}\left( {\varvec{t}}_1(x_1,y_1),\ldots ,{\varvec{t}}_n(x_n,y_n),\right. \\&\left. \quad {\varvec{t}}_1(y_1,z_1),\ldots ,{\varvec{t}}_n(y_n,z_n)\right) \\&\quad =T^{2n-1}\left( {\varvec{t}}_1(x_1,y_1),{\varvec{t}}_1(y_1,z_1),\ldots ,\right. \\&\quad {\varvec{t}}_n(x_n,y_n),{\varvec{t}}_n(y_n,z_n))\\&\quad =T^{n-1}\left( T({\varvec{t}}_1(x_1,y_1),{\varvec{t}}_1(y_1,z_1)),\ldots ,\right. \\&\quad \left. T\left( {\varvec{t}}_n(x_n,y_n),{\varvec{t}}_n(y_n,z_n)\right) \right) \\&\quad \le T^{n-1}\left( {\varvec{t}}_1(x_1,z_1),\ldots ,{\varvec{t}}_n(x_n,z_n)\right) ={\varvec{t}}(x,z). \end{aligned}$$

• 5. Since fuzzy T-metric is a constant function on the parameter $\alpha$ (and therefore a continuous function) and T is a continuous t-norm, the composition of these mappings is also a continuous function.

$\square$

Theorem 9

If ${\varvec{s}}_i{:}\,X_i\times X_i\rightarrow [0,1)$ are fuzzy S-metrics with respect to the strict triangular conorm S, then ${\varvec{s}}{:}\,X^2\rightarrow [0,1],$$X=X_1\times \dots \times X_n$ defined with

$$\begin{aligned}&{\varvec{s}}(x,y)=S^{n-1}({\varvec{s}}_1 (x_1,y_1),{\varvec{s}}_2 (x_2,y_2),\ldots ,{\varvec{s}}_n (x_n,y_n)),\\&x=(x_1,\ldots ,x_n),\;y=(y_1,\ldots ,y_n), \end{aligned}$$

is the fuzzy S-metric with respect to the triangular conorm S. If S is not a strict triangular conorm, then ${\varvec{s}}$ is a fuzzy S-metric in a broader sense.

Proof

• 1. ${\varvec{s}}_i(x_i,y_i)\in [0,1)$

  $\Rightarrow {\varvec{s}}(x,y)=S^{n-1}({\varvec{s}}_1(x_1,y_1),\ldots ,{\varvec{s}}_n(x_n,y_n))\in [0,1].$ If S is a strict triangular conorm, then $S^{n-1}$ is a strict monotone function, and from ${\varvec{s}}_i(x_i,y_i)\in [0,1),\;$ it follows that

  ${\varvec{s}}(x,y)=S^{n-1}({\varvec{s}}_1(x_1,y_1),\ldots ,{\varvec{s}}_n(x_n,y_n))\in [0,1).$

• 2. ${\varvec{s}}(x,y)=S^{n-1}({\varvec{s}}_1(x_1,y_1),\ldots ,{\varvec{s}}_n(x_n,y_n))=0$

  $\Leftrightarrow \;(\forall i\in \{1,\dots ,n\})\;{\varvec{s}}_i(x_i,y_i)=0$

  $\Leftrightarrow \; (\forall i\in \{1,\dots ,n\})\;x_i=y_i$

  $\Leftrightarrow \; x=y.$

• 3. ${\varvec{s}}(x,y)=S^{n-1}({\varvec{s}}_1(x_1,y_1),\ldots ,{\varvec{s}}_n(x_n,y_n))$

  $=S^{n-1}({\varvec{s}}_1(y_1,x_1),\ldots ,{\varvec{s}}_n(y_n,x_n))$

  $={\varvec{s}}(y,x).$

• 4. From axiom 4 of the fuzzy S-metric ${\varvec{s}}_i$, it follows

  $S({\varvec{s}}_i(x_i,y_i),{\varvec{s}}_i(y_i,z_i))\ge {\varvec{s}}_i(x_i,z_i),\; i\in \{1,\ldots ,n\}, i.e.,$

  $S({\varvec{s}}(x,y),{\varvec{s}}(y,z))$

  $=S(S^{n-1}({\varvec{s}}_1(x_1,y_1),\ldots ,{\varvec{s}}_n(x_n,y_n)),$

  $S^{n-1}({\varvec{s}}_1(y_1,z_1),\ldots ,{\varvec{s}}_n(y_n,z_n)))$

  $=S^{2n-1}({\varvec{s}}_1(x_1,y_1),\ldots ,{\varvec{s}}_n(x_n,y_n),$

  ${\varvec{s}}_1(y_1,z_1),\ldots ,{\varvec{s}}_n(y_n,z_n))$

  $=S^{2n-1}({\varvec{s}}_1(x_1,y_1),{\varvec{s}}_1(y_1,z_1),\ldots ,$

  ${\varvec{s}}_n(x_n,y_n),{\varvec{s}}_n(y_n,z_n))$

  $=S^{n-1}(S({\varvec{s}}_1(x_1,y_1),{\varvec{s}}_1(y_1,z_1)),$

  $\ldots ,S({\varvec{s}}_n(x_n,y_n),{\varvec{s}}_n(y_n,z_n)))$

  $\ge S^{n-1}({\varvec{s}}_1(x_1,z_1),\ldots ,{\varvec{s}}_n(x_n,z_n))$

  $={\varvec{s}}(x,z).$

• 5. Since the fuzzy S-metric is a constant function on the parameter $\alpha$ (and therefore a continuous function) and S is a continuous triangular conorm, the composition of these mappings is also a continuous function. $\square$

Example 5

If ${\varvec{t}}_i{:}\,X_i\times X_i\rightarrow (0,1]$ are fuzzy T-metrics with respect to the product, then ${\varvec{t}}{:}\,X^2\rightarrow (0,1], X=X_1\times \dots \times X_n$ defined with

$$\begin{aligned}&{\varvec{t}}(x,y)=\prod _{i=1}^{n}{\varvec{t}}_i (x_i,y_i),\;\; x=(x_1,\ldots ,x_n),\\&y=(y_1,\ldots ,y_n), \end{aligned}$$

is the fuzzy T-metric with respect to the product.

4 Filtering images by using fuzzy metrics

In this section, filtering color images will be considered, with color components red, green, blue (RGB). When filtering an image, a window is used, most often labeled with W, whose size is $n\times n$, where n is an odd number. Pixels in that window are labeled with $({\varvec{i}},{\varvec{F}}_{{\varvec{i}}}),$ where ${\varvec{i}}=(i_1,i_2)\in I\times I,$$I=\{0,1,\ldots ,n-1\}$, ${\varvec{i}}$ is a vector with spatial coordinates of pixel $i_1,\;i_2$ (points from the screen with integer coordinates), ${\varvec{F}}_{{\varvec{i}}}$ is a three-dimensional vector, whose first coordinate represents quantity of red color, second coordinate is a quantity of green color, and third represents quantity of blue color, i.e., ${\varvec{F}}_{{\varvec{i}}}=({\varvec{F}}_{{\varvec{i}}}^R,{\varvec{F}}_{{\varvec{i}}}^G,{\varvec{F}}_{{\varvec{i}}}^B)$ or ${\varvec{F}}_{{\varvec{i}}}=({\varvec{F}}_{{\varvec{i}}}^1,{\varvec{F}}_{{\varvec{i}}}^2,{\varvec{F}}_{{\varvec{i}}}^3).$

The essence of image filtering is to replace the pixel which represents noise by pixel without noise, which can be achieved by replacing a middle pixel in window W with pixel that represents the other pixels from the window W in the best possible way, i.e., by a pixel which is the most similar in color and spatial distance to all the other pixels in W.

It is of great importance to choose a good criterion for selecting such a pixel without noise, which will replace the pixel with noise in a given window W, because the choice of pixels affects the image quality, i.e., affects the degree of the removed noise.

The most important part of choosing that criterion will be a good selection of fuzzy T-metric ${\varvec{c}}$. On the set of all pixels in the given window W, a order relation will be induced by using fuzzy T-metric ${\varvec{c}}$. This order relation will be used to compare pixels $({\varvec{i}},{\varvec{F}}_{{\varvec{i}}})$ (“position”, “color”) of the image and to choose a pixel that differs the least from all the other pixels in the window, i.e., which is the most similar to all other pixels in W (regarding color and distance). The middle pixel in the given window W will be replaced by the pixel found using the algorithm. The algorithm is applied on each sliding window.

In the algorithm for filtering the image, the mapping ${\varvec{c}}{:}\,W\times W\rightarrow \mathbb {R}$ will be used, defined on the window $W=\{({\varvec{i}},{\varvec{F}}_{{\varvec{i}}})| {\varvec{i}}\in I\times I\}$, defined with

$$\begin{aligned} {{\varvec{c}}}=T({{\varvec{\tau }}},{{\varvec{t}}}), \end{aligned}$$

(2)

where ${{\varvec{\tau }}}$ and ${{\varvec{t}}}$ are fuzzy T-metrics with respect to a triangular norm T. It follows from Theorem 8 that ${{\varvec{c}}}$ is a fuzzy T-metric.

Fuzzy T-metric ${\varvec{\tau }}$ is defined with

$$\begin{aligned}&{\varvec{\tau }}\left( {\varvec{F}}_{{\varvec{i}}},\; {\varvec{F}}_{{\varvec{j}}}\right) \\&\quad =T^3\left( {\varvec{\tau }}_1\left( {\varvec{F}}^1_{{\varvec{i}}},\; {\varvec{F}}^1_{{\varvec{j}}}\right) , {\varvec{\tau }}_2\left( {\varvec{F}}^2_{{\varvec{i}}},\; {\varvec{F}}^2_{{\varvec{j}}}\right) , {\varvec{\tau }}_3\left( {\varvec{F}}^3_{{\varvec{i}}},\; {\varvec{F}}^3_{{\varvec{j}}}\right) \right) , \end{aligned}$$

and it is used to measure the similarity between corresponding colors (equality of quantity of colors) between two pixels ${\varvec{F}}_{{\varvec{i}}}$ and ${\varvec{F}}_{{\varvec{j}}}$, i.e., similarity of kth color ($k=1,2,3$) is measured by fuzzy T-metric ${\varvec{\tau }}_k$. It follows from Theorem 8 that ${\varvec{\tau }}$ is a fuzzy T-metric.

Spatial distance of pixels ${\varvec{i}}$ and ${\varvec{j}}$ is measured with fuzzy T-metric ${\varvec{t}}.$ In the fuzzy T-metric ${\varvec{t}}$, there is a parameter t that affects the sensitivity of fuzzy T-metric ${\varvec{t}}$.

In paper (Valentin et al. 2011), a special case of fuzzy T-metric ${\varvec{c}}$ is used. Instead of arbitrary t-norm, the usual multiplication operation is used, i.e., fuzzy T-metric ${\varvec{c}}$ is defined with:

$$\begin{aligned} {\varvec{c}}\left( {\varvec{F}}_{{\varvec{i}}},\;{\varvec{F}}_{{\varvec{j}}}\right) = {\varvec{\tau }}\left( {\varvec{F}}_{{\varvec{i}}},\; {\varvec{F}}_{{\varvec{j}}}\right) \cdot {\varvec{t}}\left( {\varvec{i}},\; {\varvec{j}}\right) . \end{aligned}$$

(3)

Fuzzy T-metrics ${\varvec{\tau }}_k,$$k=1,2,3$ are like in Example 1,

$$\begin{aligned} {\varvec{\tau }}\left( {\varvec{F}}_{{\varvec{i}}},\; {\varvec{F}}_{{\varvec{j}}}\right) ={\varvec{\tau }}_1\left( {\varvec{F}}^1_{{\varvec{i}}},\; {\varvec{F}}^1_{{\varvec{j}}}\right) \cdot {\varvec{\tau }}_2\left( {\varvec{F}}^2_{{\varvec{i}}},\; {\varvec{F}}^2_{{\varvec{j}}}\right) \cdot {\varvec{\tau }}_3\left( {\varvec{F}}^3_{{\varvec{i}}},\; {\varvec{F}}^3_{{\varvec{j}}}\right) . \end{aligned}$$

Fuzzy T-metric ${\varvec{t}}$ is like in Example 4, where d is the Euclidean metric. By defining fuzzy T-metric ${\varvec{c}},$ it is proposed one more method for filtering color image, which provides rather good results because it takes into account simultaneously the criterion of similarity of colors and spatial distance.

5 Application

In this section will be given the concrete examples of application of fuzzy metrics in color (RGB) image filtering, which provides better quality of the filtered image compared to filtering by median filter. Image is determined by the position of pixels in image, ordered pairs which represent the coordinates of pixels and three-dimensional vector which is assigned to every pixel. Each of the coordinates of the three-dimensional vector represents quantity of color that is assigned to that pixel, respectively red, green, blue. Filtering is done using sliding window W, whose size is $n\times n$, where n is an odd number. Two different fuzzy metrics are used for defining one fuzzy metric applying t-norm, the usual multiplication.

The value of the middle pixel in given window W will be determined by the values of all the other pixels inside the window, on which unique metric will be applied by using method that will be explained below. Image that will be processed is labeled with J. A pixel that is currently in process of calculating fuzzy metric c is labeled with $J_t$, and the coordinates of three-dimensional vector of colors are labeled with: $J_t=(J_t^R,J_t^G,J_t^B).$ Fuzzy T-metric which is used to measure similarity in colors among pixels is marked with ${\varvec{\tau }}$. It is defined in the following way:

$$\begin{aligned} {\varvec{\tau }}(J_i,J_j)=\prod _{l=1}^3\frac{\frac{J_i^l+J_j^l}{2}+K}{\max \{J_i^l,J_j^l\}+K}. \end{aligned}$$

(4)

Fuzzy T-metric that considers spatial distance between pixels is marked with ${\varvec{t}}$. It is defined in the following way:

$$\begin{aligned} {\varvec{s}}(J_i,J_j,t)=\frac{t}{t+|i_1-j_1|+|i_2-j_2|}. \end{aligned}$$

(5)

The metric used for the comparison of the quality of images is UIQI, which is defined in Wang and Bovik (2002). Metric of quality UIQI is based on the fact that every image distortion is observed as a combination of three factors: loss of correlation, luminance distortion and contrast distortion. Since filtered images are color images (RGB), image quality UIQI index is calculated for each of them. Thus, we get that instead of one value, the image quality is represented by three-dimensional vector. The value of that metric for each color is in the interval from − 1 to 1. The closer the metric quality of image is to one, the better is the quality of image. Therefore, if the index of quality of each color is closer to one, the quality of the processed image is better. For calculating the image quality UIQI, the sliding window is used, that slides through the whole image from pixel to pixel from top to the bottom of the image. For each of these windows, UIQI is calculated by formula given in Wang and Bovik (2002), and in the end the values assigned to each of these windows, during calculation, are summed and divided by the number of windows. In the following examples of image quality UIQI, the chosen size of window is 8 (Fig. 1).

Fig. 1

Baboon, $256\times 256$

The filtered image given below is contaminated with $10\%$ salt and pepper noise. Values of metric of image quality UIQI for each color of image with $10\%$ of noise are equal to: [0.4639, 0.4737, 0.5047] (Fig. 2)

Fig. 2

Baboon, $256\times 256$, 10 % salt and pepper

It is concluded that for $t\in [2.6,3.0]$ and $K\in [640,896],$ the metric ${\varvec{c}}$ which is defined in this paper gives better quality of image with respect to quality of image filtered by median filter, where the quality of image is compared with the metric for image quality UIQI. In this paper, image filtered by fuzzy metrics ${\varvec{c}}$ with parameters $t=2.6$, $K=768$ and window size 3 is represented. The values of metric for the image quality UIQI for each color for the filtered image by applying the method proposed in this paper are equal to: [0.5257, 0.5702, 0.5662] (Fig. 3)

Fig. 3

Baboon, image filtered by the algorithm for filtering, presented in the paper, window size is 3, $K=768$, $t=2.6$

The values of metric of image quality UIQI for each color for filtered image by median filter with window size three are equal to: [0.5033, 0.5649, 0.5447] (Fig. 4)

Fig. 4

Baboon, filtered by median filter, window size is 3

Comparing the index of metric for image quality UIQI for corresponding colors (respectively, red, green, blue), it can be concluded that all indices of images filtered by the method proposed in this paper are greater than the corresponding indices of images filtered by median filter. As those indices are closer to one, it can be concluded that the image quality is better.

For additional literature reading about fuzzy filtering, authors recommend the following list: Astola et al. (1990), Morillas et al. (2005a, b) and Smolka et al. (2001).

6 Conclusion

The fuzzy T-metric and fuzzy S-metric defined in the manner presented in the paper provide a wide range of possibilities for further application in image processing. Depending on the type of problem, i.e., of the expected performance, the appropriate t-norm or t-conorm can be chosen to contribute to achieve desired performance. In the paper, a type of fuzzy T-metric was applied when constructing an image filtering algorithm. We concluded that for parameters in a certain range, using the UIQI image quality metric, the quality of the image filtered by the operation from this work is better than the image quality obtained by filtering with the median filter. We plan to use the T-metric in the future for filtering the image, replacing it with the appropriate fuzzy S-metric, and selecting the appropriate parameter domains for obtaining the appropriate filtering results. It is also possible to observe the fuzzy metric and aggregation operators constructed by the fuzzy metric that are applied in image filtering or image segmentation as in the article (Nedović et al. 2017).