**Algorithms for the Extension two Dimensional Fuzzy Signal via Decomposition and Reconstruction in Fuzzy Wavelets**
**Jean-louis Akakatshi Ossako1\*, Rebecca Walo Omana 2, Richard Bopili Mbotia3, Antoine Kitombole Tshovu 4**
1,2,4 Departement of Mathematics and Computer Science, Faculty of Science and Technology, University of Kinshasa, Kinshasa, D.R.Congo
3Department of Physics, Faculty of Science and Technology, University of Kinshasa, Kinshasa, D.R.Congo
Corresponding author: \*
Received Date: \*date
Accepted Date: \*date
Published Date: \*date
**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 two-dimensional 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 one-dimensional 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 one dimensional**
**Definition** 1.1 (Perfilieva, 2006; Ohlan, 2021; Bloch, 2015; Sussner, 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) *Ak* 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 α ϵ \[0, 1\], ∃*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, , that is *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 that is
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, 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 .
**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 V0α* ,
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 V0α* 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\[-π, π \], ∃ 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, 2003; De Barros, 2017; Gomes, 2015 ; Mazandarani, 2021).
Consider u and v F(the set of all fuzzy numbers of R).
So .
**Definition** 1.6 (Cognet, 2000; Grifone, 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, 2016; Feng, 2001; Hesamian, 2022; Chachi, 2018)
Let and F
We define the operator : F F by the equation
(3)
**Theorem** 1.9
Consider a fuzzy multiresolution analysis of F,….. ...........
If is a fuzzy wavelet associated to this fuzzy multiresolution analysis, then it generates a fuzzy orthonormal basis of F
In practice signals have bounded support : therefore we must define fuzzy wavelet bases on a bounded interval \[0, 1\]. In this case, an equivalent result is :
**Theorem** 1.10
Consider a fuzzy multiresolution analysis of F
Let be the wavelet associated to this fuzzy multiresolution analysis, then it generates a fuzzy
orthonormal basis of F
Since in pratice, signals and images are considered of bounded support, they must be of finite resolution.
Suppose that the 1D signal to be analyzed is defined on \[0, 1\] and contains N = 2J points: then it can be 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 : then
cj,k = cj,k+2j and dj,k = dj,k+2j .
Performing an orthogonal fuzzy wavelet transform of the signal f consist in decomposing it in and hence 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.
Therefore *f* belongs to means to: .
**ALGORITHM**
Our algorithm, inspired by (Le Cadet, 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 .
Note the fact that .
Decomposing on , we have :
(4)
According to (Wu, 2002; Diamond, 2000; Butnariu, 1989) 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)
Calculating the scalar product of *f* with the vectors of each member of this equality, we have :
Similarly, 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 2j
.
**Convolution - Decimation**
From the discussion 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 and give the high frequencies of the signal.
The fuzzy wavelet transform is obtained as an iteration of two operations: the data (initially, the vector ), are convolved by the filters h and g. The result of these two convolutions is that, only those of even index are kept, the other one is eliminated. This is the decimation.
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
**SYNTHESIS**:
The synthesis is the opposite step of the analysis: from the wavelet coefficients, and thus from the vector data , we obtain
.
Indeed *Wj – 1* is the orthogonal complement of *Vj – 1* in *Vj*, the union of their bases
and is a fuzzy orthonormal basis of *Vj* .
Therefore, any can be decomposed into this base by:
.
Using (5) and (7) in this 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 index, 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 R2 and no longer on R.
For this purpose, we need the concept of fuzzy tensor product.
**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
Let A and B be fuzzy sets.
We define the product \[A. B\] using α – cuts as follows:
.
Note that ,
**Two-dimensional fuzzy multi-resolution analysis**
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
**Proof**
Since A, B fuzzy sets, we can define the product \[A. B\] using α-cuts as follows:
.
Since ,
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*.
Therefore : ϑjα = {0} and ϑjα = {0} = ϑj α = ϑ jα by reasoning similarly to 5) and using (i).
Hence : ϑj = ϑj = .
To complete the construction of the fuzzy multiresolution analysis, we prove the following theorem:
**Theorem** 1.13
Let *θj* and *Фj * be two bases of the fuzzy space *Vj* , then
is a Riesz basis of ϑj = .
**Proof**
Indeed, and being two bases of Vj, is a base of ϑj that is is a base of ϑj α.
By Theorem 1.4, is a Riesz basis of ϑj α = if and only if ∃ 0 ˂ C and 0 ˂ D such that ,
.
Hence, is a Riesz basis of ϑj if and only if ∃ 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*):
(12)
(13)
(14)
(15)
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 fuzzy 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
\[ *cL – J* , {*dj 1* , *dj 2* , *dj 3* }L- J≤ j ˂ L \], (16)
calculated by iteration of the recursion relations (12) - (15) 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 two, 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-1k* 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 (17) 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**
Our results, mainly the definition and the proof of two – dimensional fuzzy multiresolution analysis as well as the construction of decomposition et renconstruction algorithms, constitute our major and original contribution.
**CONCLUSION**
The paper deals with new algoritm for decomposition and reconstruction of a fuzzy one or two dimensional fuzzy signal in fuzzy wavelets.
In general case, decomposition and reconstruction of image follow a suitable procedure.
For the decomposition:
The image is divided 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.
Repeating this decomposition several times, we obtain a hierarchical image.
The resulting low-resolution image has only half the size of the original image.
The reduced image is then enlarged to the size of the original image.
The result is a detailed image that has the same size as the original image.
Similarly, fuzzy wavelet decomposition reduces the size of the fuzzy image at each stage and each fuzzy detail in these images is taken into account by the fuzzy wavelet coefficients which allow a good reconstruction of signal for better transmission.
In this work, we have described, using orthonormal wavelet transform , algorithms for the decomposition and reconstruction in fuzzy wavelets for a one and two-dimensional signal by associated fuzzy multiresolution analyses.
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