**DECOMPOSITION OF A FUZZY FUNCTION BY FUZZY MULTIRESOLUTION ANALYZES**

**ONE-DIMENSIONAL**

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**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 techniques and, in general, data compression techniques are techniques for storing or transmitting signals using fewer bits than are 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, which can be 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 problem is posed in these terms: "can we, in a fuzzy environment, decompose a function (a signal) on a fuzzy orthonormal basis of the space*

*F ?". We will answer in the affirmative. This decomposition is made possible by the definitions of, among others, one-dimensional fuzzy multiresolution analyzes, detail spaces and fuzzy wavelets, as well as by the constructions of a fuzzy wavelet and the fuzzy orthonormal bases of F In the first part, the existence of fuzzy multiresolution analyzes (FMA) for the decomposition of a fuzzy signal (signal containing uncertainties) via the use of α-cuts is demonstrated.*

*The second part allows us to obtain the fuzzy spaces containing the details of the fuzzy signal, by the existence of a fuzzy wavelet. The third part shows how, in a practical way, to build this fuzzy wavelet.*

*Finally, the fourth part specifies the obtaining of a fuzzy orthonormal basis of F on which we can decompose this fuzzy signal.*

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

*Riesz, fuzzy orthonormal basis.*

# INTRODUCTION

The one-dimensional multiresolution analyzes of is an appropriate tool for wavelet analysis allowing, in particular, to construct orthonormal bases of this space (Mallat, S., 1999; Meyer, Y. et al., 1987 and Daubechies, I., 1992; Mehra, M., 2018).

It consists of a sequence of nested closed subspaces , in which a function is approximated at each analyzes (resolution) level j, and which verify the following properties:

1)  .

2)  
3)  
4)  .

5)  
with the existence of θ ϵ such that {θ(t - n)}<sub>nϵZ</sub> is a Riesz basis of V<sub>0</sub> .

By this analysis, the approximation in V<sub>j</sub> of f is twice as fine as that in V<sub>j - 1</sub> but twice as bad as that in

V<sub>j + 1</sub>.

**Issue**

The decomposition made by the above analysis allows us to obtain f on an orthonormal basis of L<sup>2</sup> (R) as a sum of finer and finer details as j increases; however, the problem with this analysis is that it is insufficient because it does not take into account the parameters of the function in an environment containing inaccuracies

**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 this work is to take into account the fuzzy environment in the signal decomposition by one-dimensional multiresolution analysis in wavelet theory.

**Results obtained:**

1\. **Multiresolution and fuzzy orthonormal bases of** F

> In this section, we will show how to obtain, in the one-dimensional case, the following a fuzzy multi-resolution analysis of F

We begin by giving some preliminary notions.

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. I., 2006; Ohlan, R., et al., 2021; Bloch, I., 2015 ; Sussner, P., 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> : \[a, b\] \[0, 1\], 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)  At<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 (FMA)**

F

This implies that the α-cuts of (the set of all closed intervals of R).

**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

We have: .

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

1)  
2\) F

3\)

4\)

5\)

It follows that , it can be shown that there exists {θ<sup>α</sup> (t -n)}<sub>nϵ Z</sub> which is a Riesz basis of (see Theorem 2.4).

Note that j stands for resolution and represents the level of analysis of the function<sub> </sub> ; the approximation in of is twice as fine as that in but half as good as that in .

Note that we can define F

If vϵ V<sub>j</sub> , we have :

Thus, , i.e. v ϵ .

Hence, the choice of v being arbitrary, we have :

1')

2') By definition, if f(t) is such that , we have: .

Hence by 2), .

3') Similarly, if f(t) is such that , , we have: .

Hence by 3), .

Note that :

(i) .

(ii) .

5') From (ii), we have :

On the one hand, =

and on the other hand,

Hence, = .

And since we deduce that = .

4') Since forms decreasing nested intervals when i.e.

we have :

Hence, by reasoning similar to 5') using (i).

Therefore, we obtain:

To complete the construction, let us prove that is a Riesz basis of , to deduce that the family such that is a Riesz basis of and that defines a multiresolution analysis of F

To do this, let us first define a Riesz basis of a space denoted H (Hilbert space).

**Definition** 1.3 (Mallat, S., 1999 and Le Cadet, O., 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, S., 1999), gives a necessary and sufficient condition for to be a Riesz basis of .

**Theorem** 1.4

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

0 ˂ A and 0 ˂ B such that , (1)

**Proof**

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

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

(2)

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

By the Parseval identity, we have :

and

By exploiting the periodicity of , we have :

Using (2), we have: ꓯ w ϵ \[-π, π\] given:

Hence

Similarly, we have : which implies

(2i) Conversely, if f verifies (1) 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> a(n) ϵ l<sup>2</sup> , we have :

Suppose that for one of these sequences, (1) is not verified.

Then ꓯ w ϵ\[-π, π \], whose support is 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, I. et al., 2013; Cheng, R., et al., 2015 ; Huang, W., et al., 2016) 

Let A<sub>k</sub> be a fuzzy basis function.

Consider also δ<sub>k</sub> (x) a basis function satisfying all the conditions given in Definition 2.1

Then there exists p ϵ IN with p ˃ 1 such that δ<sub>k</sub> (x) = A<sub>k</sub> <sup>p</sup> (x) (3)

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

This implies that (4)

and (5)

**Definition** 1.6 (Beg, I. et al., 2013; Cheng, R., et al., 2015 ; Huang, W., et al., 2016)

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

{A<sub>k</sub> (x)} are orthogonal fuzzy if (6) where ε(x) is a function such that (7)

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

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

Consider a fuzzy basis function A(x) centred on the first node, i.e. k = 0.

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

A<sub>k</sub> (x) = R<sub>k</sub> A(x) (8)

**Definition** 1.8 (Beg, I. et al., 2013)

The fuzzy scalar product is defined by : (9)

where (10)

is an ordinary product.

Furthermore, the summation of any 2 terms in (9) is calculated as follows:

**Definition** 1.9 (Beg. I., et al., 2013)

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

(5 - 6).

Then { A<sub>k</sub> (x) } are fuzzy orthogonal. This implies : (12)

where is a scalar product.

as , we can approximate as follows: (13)

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

Thus, since {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 and "smoother" resolution j - 1 are high-frequency components of the image; let us call W<sub>j -1</sub> the blurred space containing these details.

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

Let P<sub>K</sub> (R) denote 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.

Now, for every 0 ≤ α ≤ 1 and for every u, v ϵ F , we can define u v by the α-cuts

\[u v \]<sup>α</sup> as follows:

**Lemma** 1.10 (Lakshmikantham. V., et al., 2003; De Barros, L. C., et al., 2017; Gomes, L. T., et., 2015 ; Mazandarani, M., et al., 2021).

Let u and v ϵ F , then ꓯ α ϵ \[0, 1\] : \[u v \]<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. M., 2000; Grifone, J. 2019)

w = u v is defined by the α-cuts by :

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

In this case, W<sub>j</sub> must be an orthogonal supplement.

We define this relationship using the α-cuts by :

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

Given W<sub>j-1</sub> the set of details of V<sub>j</sub> , such that

with

The second condition implies orthogonality.

We now need to present fuzzy orthonormal bases of these detail spaces; they will have interesting properties for the detection of singularities in an image, and thus in particular for the compression problem.

According to the definition of an AMRF, we have :

Since Ф(t) ϵ V<sub>0</sub> , Ф(t) also belongs to V<sub>1</sub> ; this means that there exists a sequence h = (h<sub>k</sub> )<sub>kϵZ</sub> such that :

Knowing Ф, this relation allows to construct h<sub>k</sub> (via its transfer function m<sub>0</sub> (w), given in equation (15) below).

On the other hand,

If Ψ(t) is a function of W<sub>0</sub> , there exists a sequence g = (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. D., 2016; Feng, Y., et al., 2001; Hesamian, G., et al., 2022; Chachi, J., 2018)

Let F .

We define the operator : F x F by the equation

> for all α ϵ \[0, 1\] (14)

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, Y., et al., 1987; Daubechies, I., 1992) the equations :

(15)

(16)

where

are the transfer functions of the filters and .

Let us look for a function Ф that is a smoothing kernel i.e. and reapply (15) 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 (16).

**Fuzzy orthonormal bases of** F

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

Letbe a FMRA 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 AMRF: 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)}jϵZ<sub>, 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 AMRF, 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 an AMRF of F

Given a fuzzy wavelet Ψ constructed according to the procedure in 2.3, this wavelet allows us to obtain a fuzzy orthogonal basis of F

**Proof**

In fact, the signals that we manipulate in practice are of bounded support: in this case, we must define fuzzy wavelet bases on a bounded interval (here, we will place ourselves on the interval \[0, 1\]).

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

F

The fuzzy wavelets Ψ<sub>jn</sub> (t) that span 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 Vj<sup>per</sup> and the spaces of fuzzy details W <sub>j</sub><sup>per</sup> are this time finite dimensional spaces.

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

The same applies to fuzzy scale functions.

Thus, is in fact finite-dimensional:

Specifically, V <sub>j</sub><sup>per</sup> 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 W <sub>j</sub><sup>per</sup> = 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 V <sub>j</sub><sup>per</sup> , Ψ<sub>jk</sub> instead of Ψ <sub>jk</sub><sup>per</sup> ,..........

**Discussion**

From the above, we can therefore state that the results obtained, in particular by the definition of a one-dimensional fuzzy multiresolution analysis, constitute, without any doubt, our major contribution, which is in fact a complement on the decomposition of a signal by the one-dimensional multiresolution analysis as presented in the research works carried out until now to our knowledge (Mallat, S. (1999), Daubecies, I. (1992), Antoine, J.P., et al. (2008), Meyer, Y., et al. (1987)), in the deterministic case not taking into account the imprecision parameter, on the wavelet theory.

**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 (FMA) via the use of α-cuts.

This AMRF 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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