*2.6. Fuzzy Transform*

In Section 2.2, we defined the fuzzy partition and its units, basic functions. Below, we proceed by the establishment of fuzzy transform. The closeness space (*<sup>X</sup>*, *w*) serves as a ground on which F-transform is established as its key element, a fuzzy partition, is selected on condition (3) that allows the creation of (*<sup>X</sup>*, *w*) based on (8) in Section 2.5.

**Definition 9** (Direct F-Transform [4])**.** *Let X be a set, Y* ⊆ *X* ⊂ [*y*0, *yl*+<sup>1</sup>] ⊂ R*,* 0 < |*Y*| = *l* ≤ |*X*| = *L* < +<sup>∞</sup>*,* P = {*At* : *X* → [0, 1] | *t* ∈ *Y*} *be a fuzzy partition of X with nodes in Y, s.t. y*0 < *x*1 < *x*2 < ... < *xL* < *yl*+1*, and satisfying* (3)*, i.e.,* ∀*t*,*<sup>s</sup>* ∈ *Y*: *At*(*s*) = *As*(*t*)*. Let dtt be a node degree given by* (7) *and u*: *X* → R *be a function where* ∀*i* = 1, ... , *L*: *u*(*xi*) = *ui. Then the* direct discrete F-transform *of u with respect to* P *is a vector <sup>F</sup>*[*u*] ∈ R*<sup>l</sup> of* F-transform components *defined by*

$$\forall t \in \mathcal{Y} \colon F[\boldsymbol{u}]\_t = \frac{\sum\_{i=1}^{L} u\_i A\_t(\mathbf{x}\_i)}{\sum\_{i=1}^{L} A\_t(\mathbf{x}\_i)} = \frac{\sum\_{i=1}^{L} u\_i A\_t(\mathbf{x}\_i)}{d\_{tt}}.\tag{10}$$

**Remark 4.** *Note that in [4], the direct F-transform is defined only with respect to a fuzzy partition (without closeness) and in that case, the requirement that* P *satisfies* (3) *is omitted. This paper, however, assumes a closeness background. That is why for every node t* ∈ *Y, the value*

$$d\_{tt} = \sum\_{i=1}^{L} w\_{ti} \, \, \, \, \, \, \, \, \, \tag{11}$$

*is also called a node degree which fully agrees with* (7)*. For the corresponding space with closeness* (*<sup>X</sup>*, *w*) *where the closeness values are denoted according to* (1)*, we equivalently define the direct F-transform of u as a vector <sup>F</sup>*[*u*] ∈ R*<sup>l</sup> of F-transform components defined by*

$$\forall t \in \mathcal{Y} \colon F[u]\_t = \frac{\sum\_{i=1}^{L} u\_i w\_{ti}}{\sum\_{i=1}^{L} w\_{ti}} = \frac{\sum\_{i=1}^{L} u\_i w\_{ti}}{d\_{tt}} \,. \tag{12}$$

*which fully agrees with* (10)*.*

Let *F* be a real vector-valued operator that acts on the space of all real functions on the universe *X* and gives us a real vector in R*<sup>l</sup>* given by (10) that can be uniquely identified with a function on the set of nodes *Y*, an element of a functional space on *Y*, s.t. ∀*t* ∈ *Y*: *<sup>F</sup>*[*u*](*t*) = *<sup>F</sup>*[*u*]*<sup>t</sup>*. So,

$$F\colon u \mapsto F[u]\_\prime \text{ where } u: X \to \mathbb{R} \text{ and } F[u]: Y \to \mathbb{R}\text{ }\dots$$

Then Definition 9 describes F-transform as a vector of F-transform components, i.e., as an image of the operator *F*.

**Lemma 3** (F-Transform Identity)**.** *Let X be a set, Y* ⊆ *X,* 0 < |*Y*| = *l* ≤ | *X*| = *L* < + <sup>∞</sup>*,* P = {*At* : *X* → [0, 1] | *t* ∈ *Y*} *be a fuzzy partition of X with nodes Y satisfying the assumptions of Lemma 2 and* (*<sup>X</sup>*, *w*) *be a corresponding space with closeness where W* ∈ R*l*×*<sup>L</sup> is the closeness matrix given by* (8) *and* (9)*, and D* ∈ R*l*×*<sup>l</sup> be a diagonal scaling matrix with diagonal elements* (*dtt*)*<sup>t</sup>*∈*<sup>Y</sup> given by* (7) *and* (11)*. Let <sup>F</sup>*[*u*] ∈ R*<sup>l</sup> be the vector of F-transform components of the function u*: *X* → R *with respect to* P *and u* ∈ R*<sup>L</sup> be the vector form of the function u*: *X* → R *given by* ∀*i* = 1, . . . , *L*: *ui* = *<sup>u</sup>*(*xi*)*. Then the following holds:*

$$DF[u] = \mathcal{W}u.\tag{13}$$

*Equation* (13) *is called the* F-transform identity*.*

Lemma 3 characterizes the operator *F* in a matrix form utilizing the fact that both functional spaces contain discrete functions; hence, they can be represented as vectors (matrices of finite order):

$$F\colon \mathbb{R}^L \to \mathbb{R}^l,\ u \mapsto F[\!u\!\!u\!\/] = D^{-1}\mathcal{W}\!u\!\!u\!\/.$$

There is a historical reason for representing F-transform as a mapping.

**Definition 10** (Inverse F-Transform)**.** *Let X be a set, Y* ⊆ *X,* 0 < |*Y*| = *l* ≤ | *X*| = *L* < + <sup>∞</sup>*,* P = {*At* : *X* → [0, 1] | *t* ∈ *Y*} *be a fuzzy partition of X with nodes Y satisfying the assumptions of Lemma 2 and* (*<sup>X</sup>*, *w*) *be a corresponding space with closeness where W* ∈ R*l*×*<sup>L</sup> is the closeness*

*matrix given by* (8) *and* (9)*. Let W<sup>t</sup> be its t-th row and <sup>F</sup>*[*u*] *be the direct F-transform of a function u*: *X* → R *in accordance with Definition 9 and* (12)*. Then the inverse F-transform of the function u is the vector*

$$\hat{F}[\boldsymbol{\mu}] = \mathcal{W}^{\top}F[\boldsymbol{\mu}] = \sum\_{t \in \mathcal{Y}} F[\boldsymbol{\mu}]\_t \mathcal{W}^{t^{\top}} \in \mathbb{R}^L.$$

Definition 10 agrees with the standard case given in [4] where it is shown that the inverse F-transform approximates the original function defined on the fuzzy partitioned space (the function is sparsely represented by the direct F-transform). Note that it is called "inverse" as it transforms the vector of direct F-transform components back to the original space. It is not an inverse mapping, though.
