**The orthogonal fuzzy wavelet transform: a fast algorithm for the decomposition and reconstruction in fuzzy wavelets of a one and two dimensional fuzzy signal**
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**HIGHLIGHTS**
- Recall of some definitions and main results obtained on the existence of fuzzy multiresolution analysis in the one-dimensional case, for the decomposition of a fuzzy image.
- Presentation of a fast construction algorithm for fuzzy signal analysis and synthesis based on fuzzy multiresolution analysis.
- Extension of the algorithm to 2 dimensions, by the two-dimensional fuzzy multiresolution analysis.
**ABSTRACT**
*The decomposition of an image can be done in the following way: The image is split into a low-resolution part, which can be described by a smaller number of samples than the original image, and a signal difference, which describes the difference between the low-resolution image and the real coded image. Therefore, this low-resolution image is also decomposed into a low-resolution image and a difference image, making more efficient coding possible. This decomposition is repeated several times, so that a hierarchical image decomposition is created. Thus, the low-resolution image is only half the size of the original image. This reduced image is enlarged to the size of the original image. The result is a detailed image that is the same size as the original image. Our problem is: "Can we build algorithms allowing the decomposition and reconstruction of a signal in a fuzzy environment? We will answer in the affirmative.*
*This construction is first made possible by one-dimensional fuzzy multiresolution analysis, which will later be extended to two dimensions. In the first part, we recall some definitions and main results obtained on the existence of fuzzy multiresolution analysis in the one-dimensional case, for the decomposition of a fuzzy image. The second part, based on this multi-resolution analysis, presents a fast construction algorithm for the analysis and synthesis of a fuzzy signal. Finally, the third part is nothing else than an extension of this algorithm to 2 dimensions, by the fuzzy multiresolution bidimensional analysis.*
***Keywords:** fuzzy image, fuzzy basis function, Riesz basis, multi – analysis fuzzy resolution, fuzzy orthonormal basis, orthogonal transform in fuzzy wavelets.*
# INTRODUCTION
A good image compression scheme requires a good image decomposition scheme.
The decomposition of the image can be done as follows:
The image is split into a low-resolution part, which can be described by a smaller number of samples than the original image, and a difference image, which describes the difference between the low-resolution image and the real coded image.
Thus, wavelet decomposition reduces the size of the image at each step relative to a given resolution; each of the details of these images being taken into account by the wavelet coefficients, which allow a reconstructed signal to be obtained for its best transmission.
**Problematic**
The hierarchical decomposition as described above reduces the size of the image from one resolution to another; however, the problem is that it is insufficient because it does not take into account the parameters of the image in an environment containing inaccuracies.
**Methodology**
Our methodological scheme includes the following steps:
- Reminder of the definitions and main results on monodimensional fuzzy multi-resolution analysis;
- One-dimensional fuzzy wavelet decomposition and reconstruction algorithm ;
- Generalization of the algorithm to 2 dimensions.
**Interest of the subject**
The transmission of an image, like any natural phenomenon, involves a large number of uncertainties.
A natural approach to controlling these uncertainties is to consider decomposition and reconstruction in a fuzzy environment.
The interest of this work concerns the taking into account of the fuzzy environment in the construction, by the uni and bidimensional multiresolution analysis, of a wavelet decomposition and reconstruction algorithm
**Results obtained**
**1. Reminder of the definitions and main results on fuzzy multiresolution analysis (FMA) unidimensional**
**Definition** 1.1 (Perfilieva. I., 2006; Ohlan, R., et al., 2021; Bloch, I., 2015 ; Sussner, P., 2016)
Consider x1 ˂ ......... ˂ xn fixed nodes such that x0 = a and xn+1 = b with n ≥ 2.
Then the fuzzy sets A1 ,......., An , of membership functions A1 (x) ,........., An (x) defined on \[a, b\], form a fuzzy partition of \[a, b\] if they satisfy the following conditions for k = 1,.........,n :
1) Ak : \[a, b\] \[0, 1\], Ak (xk ) = 1 ;
2) Ak (x) = 0 if x ( xk-1 , xk+1 ) ;
3) Atk is continuous;
4) Ak , for k = 2,........ .n , grows strictly over \[xk-1 , xk \] and decreases strictly over
\[xk , xk+1 \] for k = 1,.........., n - 1.
5) For all x \[a, b\],
And the membership functions that can be identified with the sets A1 ,.........,An are called fuzzy basis functions.
**Theorem** 1.2
There is a sequence of fuzzy sets forming a multi-resolution analysis of
F
**Proof**
Consider a sequence in F 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 ꓯ αϵ\[0, 1\], ∃θ ϵ F such that {θα (t -n)}nϵ Z is a Riesz basis of .
Note that j stands for resolution and represents the level of analysis of the function ; 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 Vj , 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 is - to -
say we have :
Hence, by reasoning similar to 5') using (i).
Therefore, we obtain:
To complete the construction, let's prove that is a Riesz basis of , to deduce that generates 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
ⱻ 0 ˂ A and 0 ˂ B such that , (1)
**Proof**
1) By definition, is a Riesz basis of if ꓯ f ϵ V 0α ,
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 if and only if ꓯ f ϵ V 0α and for any sequence (a(n))nϵZ a(n) ϵ l2 , we have :
Suppose that for one of these sequences, (1) is not verified.
Then ꓯ w ϵ\[-π, π \], ⱻ whose support is in \[-π, π\] such that
Let us first assume that for these w ϵ \[-π, π \], we have
So
Assume also that for these w ϵ \[-π, π \], we have :
So
By this double contradiction, the reciprocal is well verified.
**Lemma** 1.5 (Lakshmikantham. V., et al., 2003; De Barros, L. C., et al., 2017; Gomes, L. T., et., 2015 ; Mazandarani, M., et al., 2021).
Consider u and v F(the set of all fuzzy numbers of R).
So .
**Definition** 1.6 (Cognet. M., 2000; Grifone, J. 2019)
is defined by the α-cuts by :
as .
**Definition **1.7
Given Wj – 1 the set of details of Vj , such that
with .
**Definition** 1.8 (Kumwimba. D., 2016; Feng, Y., et al., 2001; Hesamian, G., et al., 2022; Chachi, J., 2018)
Let and F
We define the operator : F F by the equation
(3)
**Theorem** 1.9
Consider a FMRA of F,…………. ...........
If is a fuzzy wavelet constructed for this AMRF, then this wavelet provides a fuzzy orthonormal basis of F
Note, however, that the signals we handle in practice are of bounded support: in this case we must define fuzzy wavelet bases on a bounded interval, in this case the interval \[0, 1\].
**Theorem** 1.10
Consider an AMRF of F
Given a fuzzy wavelet of this AMRF, this wavelet allows to obtain a fuzzy orthonormal basis of F
In addition to the fact that the signals and images manipulated in practice are of bounded media, they are necessarily of finite resolution.
Suppose that the 1D blurred signal to be analyzed is defined on \[0, 1\] and comprises N = 2J points: it is represented by a vector
For the sake of simplicity, let us consider the case of periodic boundary conditions; this means that each of the coefficients characterizing the projection of f(signal) into Vj and Wj is periodic with period 2j : thus,
cj,k = cj,k+2j and dj,k = dj,k+2j .
Performing an orthogonal fuzzy wavelet transform of the signal f will consist in decomposing it in and thus in finding the coefficients of its projection on V0 on the one hand, and on each of the Wj , j = 0,........., J - 1 on the other hand.
And f belongs to leads to: .
**ALGORITHM**
Our algorithm, inspired by (Le Cadet, O., 2004), proceeds in 2 steps: analysis and synthesis.
**Analysis**:
The analysis step consists in finding, from these 2J coefficients , representing f on , the 2J coefficients and representing f on .
Let's exploit the fact that .
Decomposing on , we have :
(4)
Noting (Wu, J., 2002, Diamond, P., and Kloeden, P., 2000 and Butnariu, D., 1989) that a fuzzy function θ is strongly measurable if and only if and are measurable , and taking the change of variable 2j – 1 t1 - k = t, we can write:
for j fixed at 1
(5)
Therefore, (4) implies that :
(6)
And by calculating the scalar product of f with the vectors of each member of this equality, we have :
Similarly, by decomposing onto , we have :
As in (5), the change of variable 2j – 1 t1 - k = t proves that :
(7)
And so (8)
Taking the scalar product of f with each member of (8), we get :
The equalities given by cj - 1,k and dj - 1,k can also be expressed in terms of circular convolution of period 2 j
.
**Convolution - Decimation**
From the above, we obtain the following result:
(9)
(10)
with
This result can be interpreted as follows:
h is a low-pass filter, which will smooth the coordinates keeping the low frequencies and g is a high-pass filter, which will select the details, the high frequencies of the signal.
The fuzzy wavelet transform is thus obtained as an iteration of two operations: the data (initially, the vector ), are convolved by the filters h and g; of the result of these two convolutions, only those of even indices are kept, thus one out of two is eliminated. This is the decimation stage.
The resulting vector is used as a new starting point, and the vector is stored.
and are of size while is of size .
Figure 1 summarises this procedure.
**Figure 1**: Analysis: filter bench
**SUMMARY**:
The synthesis is the opposite step of the analysis: from the wavelet coefficients, and thus from the vector data , we want to find
.
Since Wj – 1 is the orthogonal complement of Vj – 1 in Vj , the union of their bases
and is a fuzzy orthonormal basis of Vj .
And therefore, any can be decomposed into this base:
.
Using (5) and (7) in the latter equality, we obtain :
.
Taking the scalar product of f with each member of this equality, we find :
.
This can also be written in vector form as :
(11)
In other words, at each step, we double the size of cj and dj (on sampling) by interposing zeros between the coefficients of consecutively even indices, then we convolve them with the filters h and g, and add the two terms.
Figure 2 illustrates this procedure
**Figure 2**: Synthesis: filter bench
**Generalization to two dimensions**
The general definition of a fuzzy multiresolution analysis of F is similar to that given in the one-dimensional case.
It is sufficient to consider fuzzy functions defined on IR2 and no longer on IR.
**Tensor product in** F
Tensor products, in both the classical and fuzzy cases, are used to extend one-dimensional signal spaces to multi-dimensional signal spaces.
**Definition** 1.11 (Mallat, S., 1999)
A tensor product between vectors of two Hilbert spaces satisfies the following properties:
- Linearity :
Ȼ, (12)
- Distributivity :
(13)
This tensor product defines a new Hilbert space which contains all vectors of the form where x1 ϵ H1 and x2 ϵ H2 , as well as linear combinations of such vectors.
**Two-dimensional fuzzy multi-resolution analysis**
We use the tensor product to define a fuzzy multiresolution analysis of F
Consider a sequence of nested fuzzy spaces defining a one-dimensional fuzzy multiresolution analysis of F
**Theorem** 1.12
{ϑJ = }jϵ Z defines a fuzzy multiresolution analysis of F
**Note** 1.13
Before proving this theorem, let us first note that: .
Therefore, ϑj =
Thus we can write: ϑj = ϑ j-1 W 1j-1 W 2j-1 W3j-1 where
j-1 ϑ=
W1j-1 =
W2j-1 =
W3j-1 =
Each fuzzy approximation space ϑj is therefore decomposed into a coarser fuzzy approximation space ϑj-1 and 3 fuzzy detail spaces.
This decomposition is repeated several times on each lower resolution fuzzy approximation space, so that a hierarchical decomposition of nested fuzzy spaces {ϑj }jϵZ can be constructed.
**Proof**
Since A, B fuzzy sets, we can define the product \[A. B\] using α-cuts as follows:
.
On the other hand, if ,
If we also consider , we have :
Hence, taking , we also have: .
For example, .
Therefore, posing H and H1 by respectively, we have :
.
If.
And let ϑj = . It is clear that if then the first property below holds:
1\) ϑ j ϑj +1 .
2\) Let F F
If ϵ ϑαj ϵ ϑαj+1.
ϵ ϑj ϵ ϑj+1.
3\)
> If
ϵ ϑ0 α ϵ ϑ0 α that isϵ ϑ0 ϵ ϑ0 .
Note that :
(i) \[ ϑj \]α = ϑJ α
(ii) \[ ϑi \]α = ϑ j α
5\)
Starting from (ii), we have :
Firstly,
ϑj α =ϑj α
and on the other hand,
ϑj α =ϑj α .
Hence
ϑj =ϑj .
And since ϑj ϑj +1, we deduce thatϑj =
4\) Since . Hence ϑj α form decreasing nested intervals when that is ϑα-(j+1) ϑα-j .
We deduce that: ϑjα = {0}.
Hence, ϑjα = {0} = ϑj α = ϑ jα by reasoning similar to 5) using (i).
Therefore, we obtain: ϑj = ϑj = .
To complete the construction, we state the following theorem:
**Theorem** 1.14
Let θj and Фj be two bases of the fuzzy space Vj , then
is a Rieszbasis of ϑj = .
**Proof**
Indeed, and being two bases of Vj , is a base of ϑj that is - to say is a base ϑj α .
We deduce, by Theorem 2.4, that is a Riesz basis of ϑj α = if and only if ⱻ 0 ˂ C and 0 ˂ D such that ,
.
Hence, is a Riesz basis of ϑj if and only if ⱻ 0 ˂ C and 0 ˂ D such that ,
.
Therefore, { ϑj = }j ϵ Z generates a fuzzy multiresolution analysis of
F
Thus, the fast fuzzy wavelet transform algorithm presented in one dimension can be extended to two dimensions.
Consider for all scales 2j and for all n = (n1 , n2 ):
and for 1 ≤ k ≤ 3 where
(fuzzy orthonormal basis of ϑj )
and
(fuzzy orthonormal basis of F )
with ,
,
For any pair of one-dimensional filters y \[m\] and z \[m\], we can write yz \[n\] = y \[n1 \]. z\[n2 \], and .
Consider h\[m\] and g\[m\] as two conjugate filters associated with the fuzzy wavelet .
**ALGORITHM**
**Decomposition**
Wavelet coefficients at scale 2j are calculated from cj by convolution and two-dimensional separable subsampling.
The decomposition formulas are obtained by applying the one-dimensional convolution formulas given in (9) and (10) to the separable two-dimensional fuzzy wavelets and scaling functions for n = (n1 , n2 ):
(14)
(15)
(16)
(17)
This result can be interpreted as follows:
A separable two-dimensional convolution can be factored into the one-dimensional convolutions along the rows and columns of the blurred image.
As the factorisation is illustrated in Figure 3, these 4 convolution equations are obtained with only 6 one-dimensional convolution groups.
The lines of are first convolved with and and sub-sampled by 2.
Then, the columns of these two produced images are convoluted respectively
with and and subsampled to give the 4 subsampled images , ,
We can conclude as follows:
The fuzzy wavelet transform of the image cL gives 3J + 1 fuzzy sub-images
\[ cL – J , { dj 1 , dj 2 , dj 3 }L - J≤ j ˂ L \], (18)
calculated by iteration of the recursion relations (14) - (17) for L - J ≤ j ˂ L; J being the number of octaves (frequency band) considered for this decomposition.
**Figure 3**: Decomposition of cJ into 6 convolution groups and one-dimensional sub-samples along the rows and columns of the blurred image.
**RECONSTRUCTION**
Let ỹ \[n\] = ỹ \[n1 , n2 \] denote the fuzzy image of size twice that of y\[n\], obtained by inserting a row of zeros and a column of zeros between rows and even columns consecutively.
The approximation cJ is reconstructed from the coarse scale approximation cJ -1 and the wavelet coefficients dJ-1 k with separable two-dimensional convolutions derived from the one-dimensional reconstruction formula in (11).
The result is :
Therefore, the image cL is reconstructed by the wavelet representation (19) for
L - J ≤ j ˂ L .
The 4 separable convolutions in (19) can also be factorised into 6 groups of one-dimensional convolutions along the rows and columns, as shown in Figure 4.
**Figure 4**: Reconstruction of cJ by inserting the zeros between the rows and columns of cJ -1 and
dJ -1 k , and filtering the result.
**Discussion**
From the above, we can therefore state that the results obtained, in particular by defining the one and two-dimensional fuzzy multi-resolution analyses, constitute, without doubt, our major contribution, which is in fact a complement to the construction of algorithms for the decomposition and reconstruction of a one and two-dimensional signal as presented in the research work carried out so far, 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**
The decomposition of an image can be obtained as follows:
The image is subdivided into a low-resolution part, which can be described by a smaller number of bits than the original image, and a signal difference, which describes the difference between the low-resolution image and the real coded image.
The low-resolution image is in turn decomposed into another low-resolution image and a difference image, making more efficient coding possible.
This decomposition is repeated several times, so that a hierarchical image decomposition is created.
Thus, the low-resolution image is only half the size of the original image.
This reduced image is enlarged to the size of the original image.
The result is a detailed image that is the same size as the original image.
As with conventional decomposition, fuzzy wavelet decomposition reduces the size of the fuzzy image at each stage; each of the fuzzy details in these images being taken into account by the fuzzy wavelet coefficients which allow a good reconstructed signal to be obtained for better transmission.
In this work, we have described, in the case of the orthogonal fuzzy wavelet transform, the algorithms for the decomposition and reconstruction in fuzzy wavelets of a one- and two-dimensional signal by the associated fuzzy multiresolution analyses.
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