**DECOMPOSITION OF A FUZZY FUNCTION BY ONE- DIMENSIONAL FUZZY MULTIRESOLUTION ANALYSIS**

**Jean-louis Akakatshi Ossako<sup>1\*</sup>, Rebecca Walo Omana <sup>2</sup>, Richard Bopili Mbotia<sup>3</sup>, Antoine Kitombole Tshovu <sup>4</sup>**

<sup>1,2,4</sup> Departement of Mathematics and Computer Science, Faculty of Science and Technology, University of Kinshasa, Kinshasa, D.R.Congo

<sup>3</sup>Department of Physics, Faculty of Science and Technology, University of Kinshasa, Kinshasa, D.R.Congo

Corresponding author: \*<jlakakatshi@gmail.com>

Received Date: \*date

Accepted Date: \*date

Published Date: \*date

**HIGHLIGHTS**

  - Demonstration of the existence of fuzzy multi-resolution analyzes for the decomposition of a fuzzy signal

  - Obtaining the fuzzy spaces containing the details of the fuzzy signal by the existence of a fuzzy wavelet

  - Construction of a fuzzy wavelet

  - obtaining a fuzzy orthonormal basis of *F* on which to decompose a fuzzy signal

**ABSTRACT**

*Signal compression and data compression are techniques for storing and transmitting signals using fewer bits as possible for encoding a complete signal. A good signal compression scheme requires a good signal decomposition scheme. The decomposition of the signal can be done as follows: The signal is split into a low-resolution part, described by a smaller number of samples than the original signal, and a signal difference, which describes the difference between the low-resolution signal and the real coded signal. Our paper deals with the proofs of these properties in a fuzzy environment. The proof of one- dimensional multiresolution analysis is given. The concept of fuzzy wavelets is introduced and as a byproduct a special fuzzy space of details of a signal is given and an orthonormal basis of Fdecomposing the fuzzy signal is obtained.*

***Keywords:** Fuzzy image, fuzzy multiresolution analyzes, fuzzy basis functions, fuzzy basis*

*Riesz, fuzzy orthonormal basis.*

# INTRODUCTION

The one-dimensional multiresolution analysis of is an appropriate tool for wavelet study it, allows in particular, the construction of an orthonormal bases (Mallat, 1999; Meyer, 1987; Daubechies, 1992; Mehra, 2018).

The multiresolution analysis of a sequence of nested and closed subspaces satisfying the following properties:

1)  .

2)  
3)  
4)  .

5)  
Moreover, there exist *θ ϵ* such that {*θ*(*t* - *n*)}*<sub>nϵZ</sub>* is a Riesz basis of *V<sub>0</sub>* .

A function *f ϵ*is approximated at any level j of this analysis, and the approximation in*V<sub>j</sub>* is twice finer than in *V<sub>j - 1</sub>* for every j = - ∞, ….., +∞.

**Problematic**

This multiresolution analysis defines *f* in using an orthonormal basis, as a sum of details.

The paper deals with this analysis in a fuzzy environment.

**Methodology**

Our methodological scheme follows the following steps:

  - Fuzzy multi-resolution analysis ;

  - Detail spaces and wavelets ;

  - Construction of the fuzzy wavelet ;

  - Fuzzy orthonormal bases of F

**Interest of the subject**

The interest of our work is that it takes into account the fuzzy environment in the signal

decomposition by one-dimensional multiresolution analysis in wavelet theory.

**Results obtained:**

The main result is **multiresolution analysis and fuzzy orthonormal bases of**

F

Consider an interval \[a, b\] as a fuzzy universe set.

The fuzzy partition of this universe is given by the fuzzy subsets of the universe \[a, b\] which admit the properties given in the following definition:

**Definition** 1.1 (Perfilieva, 2006; Ohlan, 2021; Bloch, 2015 ; Sussner, 2016)

Consider x<sub>1</sub> ˂ ......... ˂ x<sub>n</sub> fixed nodes such that x<sub>0</sub> = a and x<sub>n+1</sub> = b with n ≥ 2.

Then the fuzzy sets *A<sub>1</sub>,......., A<sub>n</sub>* , of membership functions *A<sub>1</sub>* (*x*) ,........., *A<sub>n</sub>* (*x*) defined on \[a, b\], form a fuzzy partition of \[a, b\] if they satisfy the following conditions for

k = 1,.........,n :

1)  *A<sub>k</sub>*<sub> </sub> : \[a, b\] \[0, 1\], *A<sub>k</sub>* (*x<sub>k</sub>* ) = 1 ;

2)  *A<sub>k</sub>* (*x*) = 0 if *x* ( *x<sub>k-1</sub>*, *x<sub>k+1</sub>* ) ;

3)  *A<sub>k</sub>* is continuous;

4)  *A<sub>k</sub>* , for k = 2,........,n , increases strictly on \[x<sub>k-1</sub> , x<sub>k</sub> \] and decreases strictly on

\[x<sub>k</sub> , x<sub>k+1</sub> \] for k = 1,.........., n - 1.

5)  For all x \[a, b\],

And the membership functions that can be identified with the sets *A<sub>1</sub>,.........,A<sub>n</sub>* are called fuzzy basis functions.

**Fuzzy multi-resolution analysis**

Let *f* : \[0, 1\] → F (*R*) a fuzzy function and *K* (*R*) be the set of closed intervals of R

Then α-cuts of .

**Theorem** 1.2

There is a sequence of fuzzy sets forming a multi-resolution analysis of

F

**Proof**

Consider a sequence inF and ꓯ αϵ \[0, 1\], let

the α – level sets of *V<sub>j</sub>* .

We have: .

Assume that this sequence of closed intervals is nested and verifies the following properties:

1)  
2\) F

3\)

4\)

5\)

∀ αϵ \[0, 1\], we shown in lemma 1.4 the existence of a Riesz basis {*θ<sup>α</sup>* (*t* - *n*)}*<sub>nϵ Z</sub>* .

Note that j stands for resolution and represents the level of analysis of the function<sub> </sub> ; the

approximation in of is twice fine as in but half good as that in .

Define F (1.1)

Then for *vϵ V<sub>j</sub>* , we have :

Therefore, and *v ϵ* .

The choice of v being arbitrary, we have :

1')

2') By definition, if , then : and by 2),

.

3') Similarly, if ,, then : and by 3),

.

Note that :

(i) .

(ii) .

5') From (ii), we have :

\= and

Hence, = .

Since we have = .

4') forms decreasing nested intervals when that is so we have :

.

To complete the proof of theorem 1.2, we need to show the existence of a Riesz basis for and therefore, by (1.1) a Riesz basis for .

This is done in lemma 1.4

**Definition** 1.3 (Mallat, 1999 ; Le Cadet, 2004)

A family of vectors is a Riesz basis of H if it is linearly independent and there exist A ˃ 0 and B ˃ 0 such that for any H, we can find *a*\[*n*\] with

satisfactory .

Note that this energy equivalence ensures that the development of on is numerically stable.

The following theorem, inspired by (Mallat, 1999), gives a necessary and sufficient condition for to be a Riesz basis of .

**Lemma** 1.4

A family , α \[0, 1\], is a Riesz basis of if and only if

0 ˂ A and 0 ˂ B such that , (1.2)

**Proof**

1)  By definition, is a Riesz basis of *V<sub>0</sub><sup>α</sup>* if ꓯ f ϵ *V<sub>0</sub><sup>α</sup>* ,

and there exist A ˃ 0 and B ˃ 0 such that

(1.3)

The Fourier transform of f is where *w* ϵ \[- π, π\].

By the Parseval identity, we have :

and

Using the periodicity of , we have :

And by (1.3), we have : ꓯ w ϵ \[-π, π\] :

Hence

Similarly, we have : which implies

(2i) Conversely, if f verifies (1.2) then is a Riesz basis of *V<sub>0</sub><sup>α</sup>* if and only if ꓯ *f* ϵ *V<sub>0</sub><sup>α</sup>* and for any sequence (*a*(*n*))*<sub>nϵZ</sub>* ⊂ l<sup>2</sup> , we have :

Suppose that for one of these sequences, (1.2) is not verified.

Then ꓯ *w* ϵ\[-π, π \],, with support in \[-π, π\], such that

Let us first assume that for these *w* ϵ \[-π, π \], we have

So

, that is

Assume also that for these *w* ϵ \[-π, π \], we have :

So

, that is

By this double contradiction, the reciprocal is well verified.

**Detail spaces and wavelets**

**Definition** 1.5 (Beg, 2013; Cheng, 2015 ; Huang, 2016) 

Let *A<sub>k</sub>* be a fuzzy basis function and let *δ<sub>k</sub>* (*x*) be an other basis function satisfying all the conditions given in Definition 1.1

Then there exists *p ϵ N* with *p* ˃ 1 such that *δ<sub>k</sub> (x) = A<sub>k</sub> <sup>p</sup> (x)* (1.4)

where *A<sub>k</sub> <sup>p</sup> (x) = A<sub>k</sub> (x) .................A<sub>k</sub> (x)* (*p* times), and *δ<sub>k</sub> (x)* is called the fuzzy delta function.

This implies that (1.5)

and (1.6)

**Definition** 1.6 (Beg, 2013; Cheng, 2015 ; Huang, 2016)

Let *A<sub>k</sub> (x)* ( for k = 0,........., n ) be fuzzy basis functions.

{*A<sub>k</sub>* (*x*)} are orthogonal fuzzy if (1.7) where *ε*(*x*) is a function such that (1.8)

where *α* is an arbitrary positive real number close to 0.

**Definition <span class="underline"> </span>**1.7 (Beg, 2013)

Consider a fuzzy basis function *A*(*x*) centered on the first node, that is k = 0.

We define a displacement operator (*R<sub>k</sub>* ) as follows:

*A<sub>k</sub>* (*x*) = *R<sub>k</sub>* *A*(*x*) (1.9)

**Definition** 1.8 (Beg, 2013)

The fuzzy scalar product is defined by : (1.10)

where (1.11)

is an ordinary product.

Furthermore, the sum of any 2 terms in (1.10) is calculated as follows:

**Definition** 1.9 (Beg, 2013)

Let *A<sub>k</sub>* (*x*) = *R<sub>k</sub>* *A* (*x*) (for k = 0,.........,n) be fuzzy basis functions satisfying the equations (1.6) and (1.7).

Then {*A<sub>k</sub>* (*x*)} are fuzzy orthogonal. This implies : (1.13)

where is a scalar product.

as , we can approximate as follows: (1.14)

From this approximation, it is possible to orthogonalize the basis {*θ* (*t - n*)}*<sub>nϵZ</sub>* of *V<sub>0</sub>* , and obtain an orthonormal basis {*Ф*(*t - n*)}*<sub>nϵZ</sub>* of *V<sub>0</sub>* .

Thus, as {*Ф*(*t - n*)}*<sub>nϵZ</sub>* is an orthonormal basis of *V<sub>0</sub>* , the properties (2') and (3') of fuzzy multiresolution analysis allow us to deduce that form a fuzzy orthonormal basis of *V<sub>j</sub>* for any j ϵ Z.

While these bases are suitable for approximation problems, they do not a priori have properties that facilitate the detection of singularities in an image; on the other hand, the details that are lost when going from a resolution j to a coarser resolution j – 1, are high-frequency components of the image.

Let *W<sub>j -1</sub>* be the fuzzy space containing these details.

In the following, we define the direct sum between two fuzzy sets by using α-cuts.

Let P<sub>K</sub> (R) be the set of compact and convex subsets of R.

It is known that ꓯ u ϵ F , the α - cut \[u\]<sup>α</sup> ϵ P<sub>K</sub> (R), 0 ≤ α ≤ 1.

For every 0 ≤ α ≤ 1 and for every u, v ϵ F, we define uv using α-cuts \[uv\]<sup>α</sup> as follows:

**Lemma** 1.10 (Lakshmikantham, 2003; De Barros, 2017; Gomes, 2015 ; Mazandarani, 2021).

Let u and v ϵ F, then ꓯ α ϵ \[0, 1\] : \[uv \]<sup>α</sup> = \[u\]<sup>α</sup> + \[v\]<sup>α</sup> .

We can define the direct sum between two fuzzy sets using α-cuts by :

**Definition** 1.11 (Cognet, 2000; Grifone, 2019)

where :

As , there is a subset W<sub>j</sub> such that

We define this relationship using the α-cuts by :

**Definition <span class="underline"> </span>**1.12

with

The second condition implies orthogonality.

Now we present fuzzy orthonormal bases of these detail spaces; they will have interesting properties for the detection of singularities in an image, and in particular for the compression problem.

According to the definition of a fuzzy multiresolution analysis, we have :

Since *Ф(t) ϵ V<sub>0</sub>* , we have *Ф(t) ϵ V<sub>1</sub>* ; hence, there exists a sequence (*h<sub>k</sub>* )*<sub>kϵZ</sub>* such that :

Given *Ф*, this relation allows to construct *h<sub>k</sub>* (via its transfer function m<sub>0</sub> (w), given in equation (1.16)).

On the other hand,

If *Ψ*(*t*) is a function of *W<sub>0</sub>* , there exists a sequence (*g<sub>k</sub>* )*<sub>kϵZ</sub>* such that :

This relationship and the previous one are called fuzzy two-scale relationships.

These two relations allow us to construct a fuzzy wavelet *Ψ* such that

{*Ψ*(*t - n*)}*<sub>nϵZ</sub>* be a fuzzy orthonormal basis of *W<sub>0</sub>* .

By compressing or expanding *Ψ*, we then construct fuzzy orthonormal bases of the other detail spaces:

**Construction of Ψ**

**Definition** 1.13 (Kumwimba, 2016; Feng, 2001; Hesamian, 2022; Chachi, 2018)

Let F.

We define the operator : Fx F by the equation

> for all α ϵ \[0, 1\] (1.15)

Thus, the two filters *g* = (*g<sub>n</sub>* )*<sub>nϵZ</sub>* and *h* = (*h<sub>n</sub>* )*<sub>nϵZ</sub>* that appear in the two-scale relations are expressed in terms of *Ф* and *Ψ*: it is sufficient to do the scalar product above between each of the two relations and and to note is orthonormal to obtain :

;

Applying the Fourier transform to each of the scaling relationships, we obtain

(Meyer, 1987; Daubechies, 1992) the equations :

(1.16)

(1.17)

where

are the transfer functions of the filters and .

Let us look for a function *Ф* that is a smoothing kernel that is and reapply (1.16) to , then to , and so on.

Finally, we obtain: .

This makes it possible to express *Ф* as a function of h in the case where the starting data of the problem is the filter h.

Knowing *m<sub>1</sub> (w)*, the expression of the function *Ψ* in the case where the starting point of the problem is the filter g can be deduced by equation (1.17).

**Fuzzy orthonormal bases of** F

**Theorem <span class="underline"> </span>**1.14

Letbe a fuzzy multiresolution analysis of F

If *Ψ* is a fuzzy wavelet constructed according to the above procedure, then this wavelet provides a fuzzy orthonormal basis of F

**Proof**

To do this, it is sufficient to use definition 1.12 on *V<sub>j</sub>*, then on *V<sub>j -1</sub>* , ... up to a certain level L to obtain :

By properties 4') and 5') of the fuzzy multiresolution analysis : F that is: the space F is decomposed as an orthogonal sum of detail spaces at all resolutions.

Consider a fuzzy function f of F .

The previous formula allows us to decompose it on the fuzzy orthonormal bases defined on the spaces

(*W<sub>j</sub>* )*<sub>jϵZ</sub>* :

with the coefficients (*d<sub>j,k</sub>* )*<sub>kϵZ</sub>* corresponding to the wavelet coefficients of *f* at resolution j

Thus, {*Ψ<sub>jk</sub>* (*t*)}<sub>*jϵZ*, *kϵZ*</sub> defines a fuzzy orthonormal basis of F on which *f* is decomposed into a sum of finer and finer details as j increases.

Note, again by properties 4') and 5') of the fuzzy multiresolution analysis, that we also have:

F

F is then decomposed as follows :

is the projection of f onto an approximation space *V<sub>L</sub>* , contains all the details that were lost when approximating *f* onto *V<sub>L</sub>* .

**Restriction to the bounded interval \[0, 1\]**: periodic fuzzy wavelet bases

**Theorem** 1.15

Consider a fuzzy multiresolution analysis of F

Given a fuzzy wavelet *Ψ*, this wavelet allows us to obtain a fuzzy orthogonal basis of

F

**Proof**

In fact, since in this case the signals we manipulate are in practice of bounded support: we must define fuzzy wavelet bases on a bounded interval \[0, 1\].

To define a fuzzy wavelet basis on \[0, 1\], we start from a basis of

F

The fuzzy wavelets *Ψ<sub>jn</sub>* (*t*) spanning t = 0 or t = 1 will have to be adapted.

The simplest method is to periodise the wavelets *Ψ<sub>jn</sub>* and the function *f*.

To do this, we define :

are periodic, of period 1.

If the support of *Ψ<sub>jn</sub>* lies in \[0, 1\], (and even if the support of the fuzzy wavelet *Ψ* is not compact, on a small scale, will tend to ): the behaviour of the fuzzy inner wavelets is not affected.

is defined in the same way by periodising the fuzzy scale functions.

This gives that for all J ≥ 0, the family

is a fuzzy orthonormal basis of

F

The spaces of fuzzy approximations and the spaces of fuzzy details are of finite dimensional spaces.

In other words, since , at resolution j there are only *2<sup>j</sup>* different fuzzy wavelets.

The same applies to fuzzy scale functions.

Thus, is in fact finite-dimensional:

Specifically, is of dimension *2<sup>j</sup>* .

In particular, V<sub>0</sub> , the coarsest fuzzy approximation space, is of dimension 1: it is the set of constants on

\[0, 1\].

We also have dim = *2<sup>j</sup>* .

This periodisation method has the advantage of being simple, but it can generate large wavelet coefficients at the edges, if the function f is not itself periodic.

Note, however, that when periodic boundary conditions are used, the notations can be abbreviated by writing *V<sub>j</sub>* rather than , *Ψ<sub>jk</sub>* instead of ,..........

**Discussion**

Our results, in particular the definition and the proof of a one-dimensional fuzzy multiresolution analysis, constitute our major and original contribution.

It allowed us to perform the decomposition of a fuzzy signal.

**CONCLUSION**

A good signal compression scheme requires a good signal decomposition scheme.

The decomposition of the signal can be done as follows:

The signal is subdivided into a low-resolution part, which can be described by a smaller number of bits than the original signal, and a signal difference, which describes the difference between the low-resolution signal and the real coded signal.

However, a major problem arises: "how to decompose this signal (function), in a fuzzy environment (containing inaccuracies), onto a fuzzy orthonormal basis of the space F

We have seen that, for a fuzzy signal, this decomposition can be obtained by one-dimensional fuzzy multiresolution analysis via the use of α-cuts.

This fuzzy multiresolution analysis allowed the definition of the detail spaces as well as the constructions of a fuzzy wavelet and a fuzzy orthonormal basis of the space F on which the signal is decomposed.

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