Novel design of a fractional wavelet and its application to image denoising

Bulletin of Electrical Engineering and Informatics
Vol. 9, No. 1, February 2020,pp. 129~140
ISSN: 2302-9285,DOI: 10.11591/eei.v9i1.1548  129
Journal homepage: http://beei.org
Novel design of a fractional wavelet and its application to
image denoising
Lanani Abderrahim1, Meghriche Salama2,Djouambi Abdelbaki3
1
Department of Electronics, University of Batna 2, Mostefa Ben Boulaid, Batna, Algeria
2
Department of Pharmaceutical Engineering, University of Constantine 3, Salah Boubnider, Constantine, Algeria
3
Department of Electrical Engineering, Larbi Ben M’hidi University, Oum El Bouaghi, Algeria
Article Info ABSTRACT
Article history:
Received Feb 23, 2019
Revised Jun 2, 2019
Accepted Sep 26, 2019
This paper proposes a new wavelet family based on fractional calculus.
The Haar wavelet is extended to fractional order by a generalization
of the associated conventional low-pass filter using the fractional delay
operator 𝑍−𝐷
. The high-pass fractional filter is then designed by a simple
modulation of the low-pass filter. In contrast, the scaling and wavelet
functions are constructed using the cascade Daubechies algorithm.
The regularity and orthogonality of the obtained wavelet basis are ensured by
a good choice of the fractional filter coefficients. An application example
is presented to illustrate the effectiveness of the proposed method. Thanks to
the flexibility of the fractional filters, the proposed approach provides better
performance in term of image denoising.
Keywords:
Fractionaldelay
Fractionalfilters
Image denoising
The Haarwavelet
This is an open access article under the CC BY-SA license.
Corresponding Author:
LananiAbderrahim,
Department of Electronics,
University of Batna 2,Mostefa Ben Boulaid, Batna,Algeria.
Email: lanani_arahim@yahoo.fr
1. INTRODUCTION
The main purpose of signal processing is to describe signals related to the real world for the purpose
of processing, identifying, compressing, understanding or transmitting. In this context, wavelets have always
played a very important role, and we no longer count the applications that use them. It is a tool allows
analyzing and locating the discontinuities of a signal along one or two dimensions and at different scales.
This feature is used to solve many problems,such as image compression and restoration [1].
Since their introduction, the wavelets have been very rapidly developed and have attracted
the attention of many researchers. Solid mathematical foundations (basis) were quickly put into place for
the notions of an orthogonal basis [2], multiresolution analysis [3-5] and wavelets with compact support [6],
giving birth to several real, complex and fractional wavelet families [1]. However, identification of an
adequate analyzing wavelet remains an important problem. No single wavelet has been adapted to all cases.
For some applications,it is possible that no suitable wavelet exists amongthe known wavelets. In such cases,
it is necessary to try to build a new wavelet adapted to the specific problem to be addressed.
Thus, many studies have appeared whose purpose is to build wavelets which have more elegant
mathematical properties and are more efficient. Among the wavelets which are currently of considerable
interest those based on fractional calculus. The main advantage of having a fractional order is flexibility,
which allows adjustments in transform parameters such as regularity and localization of the base
functions [5]. These advantages can be exploited to improve several approaches and ameliorate problems
such as denoising by thresholding wavelet coefficients [1]. The idea of fractional wavelets was first proposed

 ISSN: 2302-9285
Bulletin of Electr Eng & Inf, Vol. 9, No. 1, February 2020 : 129 – 140
130
by Mendlovic [7] in the context of optical signal processing, where they were defined as a cascade
of the classical wavelet transform and the fractional Fourier transform (FRFT) [8]. Shortly thereafter,
Huang and Suter [9] proposed a fractional version of the wave packet transform (FRWPT), based on
combining FRFT and the wave packet transform (WPT). Due to a lack of physical interpretation
and the complexity of the calculation, the FRWPT unfortunately did not receive much attention.
However, several years later, the idea was extended in the context of spline wavelets to non-integer
degrees [10]. Quincunx wavelets have also been generalized to non-integer orders with a construction based
on fractional Quincunx filters, which are generated through the diamond McClellan transform [11].
Several fractional wavelet models have recently been proposed and developed, with a wide
range of applications[12˗16].
The aim of the present paper is to introduce the new fractional wavelet (NFRW), which we consider
a generalization of the Haar wavelet [1]. However, this idea can also be extended to generalize other wavelet
families. The construction generally begins with the choice of a low-pass digital filter with the orthogonality
property checked in Fourier or direct space; then it is generalized through the fractional delay (FD) Z-D
[17]
while ensuring correct properties of orthogonality, compact support and regularity. The high-pass filter can
then be built from the low-pass filter, and the associated scaling function and wavelet can be deduced.
To illustrate the potential of the proposed wavelet, we present a denoising application on images. The rest of
this paper is organized as follows. Section 2 briefly defines the discrete wavelet transform (DWT) and
introduces the relation between multiresolution analysis and filter banks. Section 3 proposes the NFRW.
Section 4 examines and simulates an experimental application of the NFRW. Finally, section 5 presents
concluding remarks.
2. DISCRETE WAVELET TRANSFORM AND FILTER BANKS
The DWT is a multiresolution/multifrequency representation. This tool separates the data, functions
or operators in frequency components according to a resolution adapted to the scale. The multiresolution
analysis developed by Mallat [5] connects the DWT and filter banks in an elegant way, with the DWT
considered as a process of decomposing the signal into approximations and details. The original signal 𝑥
̃[n]
passes through two complementary low-pass and high-pass filters ℎ
̃[n] and 𝑔
̃[n], followed by a factor 2
subsampling, and emerges as two signals: the approxima tion signal A and the detail signal D,
respectively [5]. Figure 1 illustrates this principle. The reconstruction is performed by low-pass and high-pass
synthesis filters ℎ
̅[n] and 𝑔̅[n], preceded by a factor2 upsampling, as illustrated in Figure 2.
Figure 1. One-dimensional
wavelet decomposition
Figure 2. One-dimensional wavelet reconstruction
The filter bank ensures a perfect reconstruction if and only if [18]
(1)
Where 𝐻
̃, 𝐺
̃, 𝐻
̅ and 𝐺̅ are, respectively, the transfer functions of the low-pass and high-pass analysis
and the synthesis filters. In the case of an orthogonal basis, the high-pass filter is a modulated version of
the low-pass filter; however, the analysisand synthesis filters are identical up to a centralsymmetry.

Bulletin of Electr Eng & Inf ISSN: 2302-9285 
Novel design of a fractional wavelet and its application to image denoising (Lanani Abderrahim)
131
Moreover, the orthogonal wavelet transform is generally characterized by a scaling function 𝝋 and
wavelet 𝝍 (for more details see [19]), where the scaling function can be expressed in terms of the low-pass
filter by the equation
𝜑(𝑥) = ∑ ℎ
̃[𝑛]√2𝜑(2𝑥 − 𝑛).
𝑛 (2)
In contrast, the wavelet function decomposes at the lower scale based on the scaling functions and can be
expressed in terms of the high-pass filter by the equation
𝜓(𝑥) = ∑ 𝑔
̃[𝑛]√2
𝑛 𝜑(2𝑥 − 𝑛). (3)
3. PROPOSED DISCRETE FRACTIONAL WAVELET STRUCTURE
From a structural point of view, the DWT is identical to an iterated filter bank, which gives
it a multiresolution character, so we can use a fractional filter to implement a fractional discrete wavelet as
shown in Figure 3; indeed, based on fractional operators, we can build fractional filters which lead to
wavelets with interesting properties (precision, flexibility, regularity, etc.).
In this paper, we have chosen the Haar filter as the starting point for our construction; this is an
orthogonal FIR filter which has a linear phase, one vanishing moment and a finite support. It is generalized
using a fractional delay (FD) 𝑍−𝐷
, which is simulated by a filter which we call the “fractional delay filter”
(FDF). We can then build the high-pass fractional filter from the low-pass fractional filter and deduce
the associated scaling function and wavelet by the cascade algorithm [6]. A perfect reconstruction is ensured
by fractionalsynthesisfilters, built in accordance with the conditions of (1).
Figure 3. One-dimensional fractionalwavelet decomposition and reconstruction
3.1. Fractional filter design
The transfer function of the low-pass filter associated with the Haar analysis is described by
the equation [20]
𝐻
̃(𝑧) =
(1+𝑧−1)
√2
(4)
We propose a generalization of this function to a fractional order by exploiting the fractional operator Z-D
with D considered to be a real number:
𝐻𝑓
̃(𝑧) =
(1+𝑧−𝐷)
√2
(5)
This function can be written in a more general way as follows:
𝐻𝑓
̃(𝑧) = 𝐴 + 𝐵 ∙ 𝑍−𝐷
(6)
where 𝐷 ∈ ℝ is the filter order, and 𝐴 ∈ ℝ and 𝐵 ∈ ℝ are the coefficients. The orthogonality and regularity
of the scaling function and wavelet are ensured by a good choice of the coefficients A and B. In the classical
case, D=1, 𝐴 = 1 √2
⁄ and 𝐵 = 1 √2
⁄ .

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132
In contrast, it should be noted that the fractional order digital delay 𝑍−𝐷
is represented by an
irrational function in which the impulse response is infinitely long. This is similar to a non-causal filter for
any finite shift in time, which makes its implementation an unrealizable task [17] and implies a very limited
implementation for the transfer function of the fractional filter 𝐻𝑓
̃ . Therefore, it is necessary to use
approximation techniques to obtain a realizable implementation of the FD and, obviously, a rational
representation for the transfer function of the fractional filter 𝐻𝑓
̃ . The majority of the design techniques
which have been proposed in the literature to approximate the FD focus exclusively on finite impulse
response (FIR) and infinite impulse response filters (IIR) [17]. In this paper, we focus on the FIR filter due to
its easy implementation and good approximation.
3.1.1. Fractional delay approximation using FIR filters
The transferfunction of the FIR filter approximatingthe FD 𝑍−𝐷
is given by [17]:
𝐻(𝑍) = ∑ ℎ[𝑛]𝑍−𝑛
𝑁
𝑛=0 , (7)
where N is the order of the filter and ℎ[n], n=0,1,...N, are the real coefficients which form the impulse
response of the FIR filter.
Several techniques have been developed to identify the coefficients h[n], such as windowing
methods, maximally flat FIR approximation (Lagrange interpolation), FIR filter design with a smooth
transition band function,the minimax or Chebyshev method,and the stochastic method. More deta ils a bout
these can be found in [17]. In the present paper, we choose the method based on Lagrange interpolation,
due to the ease of calculating filter coefficients from a closed form equation and also because the filter
obtained by this method has a completely flat magnitude frequency response at low frequencies.
Indeed, the Lagrange interpolation is one of the simplest and most efficient methods for designing an FIR
filter approximatingthe FD 𝑍−𝐷
. The FDF coefficients are given by [17]:
ℎ[𝑛] = ∏ 𝐷−𝑘
𝑛−𝑘
𝑁
𝑘=0
𝑘≠𝑛
𝑓𝑜𝑟 𝑛 = 0,1,2, … 𝑁 (8)
where N is the order of the filter and D is the desired delay,
𝑁−1
2
≤ 𝐷 ≤
𝑁+1
2
𝑓𝑜𝑟 𝑜𝑑𝑑 𝑁
such that (9)
𝑁
2
− 1 ≤ 𝐷 ≤
𝑁
2
+ 1 𝑓𝑜𝑟 𝑒𝑣𝑒𝑛 𝑁.
Using (7), we draw up Table 1, which groups the coefficients for Lagrange FD filters of order
N=1, 2 and 3.
Table 1. The coefficients of the Lagrange fractionaldelay filters of order N=1, 2 and 3
h[0] h[1] h[2] h[3]
N=1 1-D D
N=2 (D-1)(D-2)/2 -D(D-2) D(D-1)/2
N=3 -(D-1)(D-2)(D-3)/6 D(D-2)(D-3)/2 -D(D-1)(D-3)/2 D(D-1)(D-2)/6
3.1.2. Fractional filter bank approximation
Following the approximation of the FD 𝑍−𝐷
by an FIR filter, the transfer function of the fractional
filter 𝐻𝑓
̃ (6) is represented by the following function:
𝐻𝑓
̃(𝑧) ≈ 𝐴 + 𝐵 ∙ (∑ ℎ[𝑛]𝑍−𝑛
).
𝑁
𝑛=0 (10)
Where h(n) and N are respectively the coefficients and the order of the FIR filter approximating
the FD 𝑍−𝐷
, and A and B are the coefficients of the fractional filter which constitutes the wavelet. In order to
ensure the orthogonality and regularity of the scaling function and the wavelet, the coefficients A and B are
chosen according to the following conditions [2-3], [21]:

133
∀ 𝜔 ∈ ℝ, |𝐻𝑓
̃(𝜔)|
2
+ |𝐻𝑓
̃(𝜔 + 𝜋)|
2
= 2, (11)
𝐻𝑓
̃(0) = √2, (12)
|𝐻𝑓
̃(𝜋)| = 0. (13)
Condition (11) is a constraint corresponding to the orthonormality of the scaling function,and it also
means that the filter 𝐻𝑓
̃ is a conjugate mirror filter [22]. Condition (12) is a simple normalization, and
condition (13) ensures the regularity of the scaling functions and wavelets. There exist only a few solutions
for identifying A and B while respecting these three conditions. In this paper, we calculate A and B while
ensuring that conditions (12) and (13) hold; however, condition (11) is provided only approximately, i.e.:
|𝐻𝑓
̃(𝜔)|
2
+ |𝐻𝑓
̃(𝜔 + 𝜋)|
2
≈ 2, (14)
|𝐻𝑓
̃(𝜔)|
2
+ |𝐻𝑓
̃(𝜔 + 𝜋)|
2
− 2 = 𝜀. (15)
The coefficients A and B of the fractionalfilter 𝐻𝑓
̃ are given in Table 2 for N=1, 2 and 3.
The error 𝜀 is very small and does not have dramatic effects on our construction, while the fractional
filter 𝐻𝑓
̃ generates regular orthogonal bases for 0.5 < 𝐷 < 1.7, which implies a limitation of the FD order to
the interval [0.5, 1.7]. Beyond this interval, the accuracy of the calculation can cause problems,
and the orthogonality of the obtained bases is lost.
Table 2. Fractionalfilter coefficients for different values of N
A B
N=1 √2 −
√2
1 − (ℎ[0]− ℎ[1])
√2
1 − (ℎ[0]− ℎ[1])
N=2 √2 −
√2
1 − (ℎ[0]− ℎ[1] + ℎ[2])
√2
1 − (ℎ[0]− ℎ[1]+ ℎ[2])
N=3 √2 −
√2
1 − (ℎ[0]− ℎ[1]+ ℎ[2] − ℎ[3])
√2
1 − (ℎ[0]− ℎ[1]+ ℎ[2] − ℎ[3])
According to the above table, the fractional filter 𝐻𝑓
̃ is represented by a finite number of non-zero
coefficients, i.e. it is a filter with compact support, where the number of coefficients is directly related to
the FD filter order (length of filter 𝐻𝑓
̃=FD filter order + 1). The regularity of the scaling function
and the wavelet are ensured as long asthe zero of 𝐻𝑓
̃(𝜔) at 𝜔 = 𝜋 has an order of at least 1 [21].
On the other hand, as indicated above (in section 2), the high-pass fractional filter is a modulated
version of the low-pass filter, where the transfer function is defined by the property of quadrature mirror
filters [23]; its coefficient is given by the following expression:
𝑔𝑓
̃[𝑛] = (−1)𝑁−𝑛
ℎ𝑓
̃[𝑁 − 𝑛]. (16)
However, the synthesis filters (H and G) are constructed according to (1), and their coefficients
are defined asfollows:
ℎ
̅𝑓
[𝑛] = ℎ𝑓
̃[𝑁 − 𝑛], (17)
𝑔̅𝑓
[𝑛] = 𝑔𝑓
̃[𝑁 − 𝑛]. (18)
The frequency responses of the analysis and synthesis filters (𝐻
̃, 𝐺
̃, 𝐻
̅ and 𝐺̅) are adjusted by
the variation of the parameter D, as shown in Figure 4. It can be seen that the generalization of these filters
through the fractional operator Z-D
leads to more flexible filters with better precision, allowing a continuous
adjustment of the key parametersof the filters.

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134
(a) (b)
(c) (d)
Figure 4. Frequency responses of analysis and synthesis filters for N=2 and D=0.5 1.5: (a) low-pass analysis
filter, (b) high-pass analysisfilter, (c) low-pass synthesis filter, (d) high-pass synthesis filter
3.2. Fractional scaling function and fractional wavelet
The construction of the scaling and wavelet functions is ensured by the cascade algorithm [6].
They are respectively represented in terms of the low-pass and high-pass filters ℎ
̃𝑓 and 𝑔
̃𝑓 by
the following expressions:
𝜑𝑓
(𝑥) = ∑ ℎ
̃𝑓
𝑛
[𝑛]√2𝜑𝑓
(2𝑥 − 𝑛), (19)
𝜓𝑓
(𝑥) = ∑ 𝑔
̃𝑓
𝑛
[𝑛]√2𝜑𝑓
(2𝑥 − 𝑛). (20)
Figure 5 presents the scaling and wavelet functions for different values of D: 0.5, 0.8, 1, 1.1, 1.3 and 1.5.
It can be seen that these functions gain in regularity with the increase of the parameter D, until the value
D=1.7, where the orthogonality of the obtained base is completely lost.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
Normalized Frequency
Magnitude
D=0.5
D=1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
Magnitude
D=1.5
D=0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
Magnitude
D=0.5
D=1.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
Magnitude
D=1.5
D=0.5

135
Figure 5. The fractionalscaling and wavelet functions fordifferent values of D
4. APPLICATION AND RESULTS
In this section, to illustrate the potential of the proposed wavelet, some results of using the proposed
fractional wavelet in image denoising algorithm are presented. The denoising algorithms, which threshold
wavelet coefficients, are based on an estimation of the noise variance. This estimation is obtained according
to a statistical calculation performed on the wavelet coefficients at the first scale. The threshold computed
from this estimation is then applied at each scale to denoise the image (for more details see [24]).
In the context of dyadic analysis, an adaptive threshold selection using Stein's unbiased risk estimate
(SURE) principle and hard thresholding [25], the denoising algorithm is applied to a set of 8-bit gray scale
images: Lena,boat,cameraman and building as shown in Figure 6, corrupted by simulated additive Gaussia n
0 1 2 3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Scaling function for D = 0.5
Time
Magnitude
0 1 2 3
-3
-2
-1
0
1
2
3
4
5
Wavelet function for D = 0.5
Time
Magnitude
0 1 2 3
-1
-0.5
0
0.5
1
1.5
2
Time
Magnitude
0 1 2 3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Time
Magnitude
0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Scaling function for D = 1
Time
Magnitude
0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Wavelet function for D = 1
Time
Magnitude
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Scaling function for D =1.1
Time
Magnitude
0 1 2 3 4
-1.5
-1
-0.5
0
0.5
1
Wavelet function for D =1.1
Time
Magnitude
0 1 2 3 4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time
Magnitude
0 1 2 3 4
-1.5
-1
-0.5
0
0.5
1
Time
Magnitude
0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Magnitude
0 1 2 3
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time
Magnitude

 ISSN: 2302-9285
136
white noise with standard deviations of 10, 20 and 30. The wavelet basis proposed in this paper (generalized
Haar)is exploited for different valuesof D, where the ordinary case is D=1.
Figure 6. Input images (Lena, boat,cameraman and building)
To evaluate and measure the quality of the restoration, the peak signal-to-noise ratio (PSNR) is computed
(with MATLAB® coding). It is defined for a B-bit gray scale level by the equation
𝑃𝑆𝑁𝑅 = 10𝑙𝑜𝑔10 (
(2𝐵−1)
2
𝑀𝑆𝐸
). (21)
where MSE is the mean square error between the initial image (without noise) and the denoised image.
The PSNR results after denoising are reported in Table 3, for different values of the parameter D.
Figure 7 presents the PSNR variationsof the Lena,building, cameraman and boat image denoising with three
different noise levels using the proposed wavelet basis with different values of D.
Table 3. PSNR (dB) results of denoising images with Gaussian noise with standard deviations σ=10, 20 and
30, using several values of D.
PSNR denoisedimage (dB)
Input
image
σ/PSNR
noised
image
D=0.5 D=0.6 D=0.7 D=0.8 D=0.9 D=1 D=1.1 D=1.2 D=1.3 D=1.4
D=1.5
D=1.6
Lena
512x5
12
σ=10/ 20.00 21.89 22.16 22.86 23.44 24.15 24.95 26.23 27.36 27.81 28.13
28.43
27.65
σ=20/ 13.98 16.58 17.00 17.60 18.25 18.93 18.93 21.10 22.29 22.88 23.43
23.86
22.89
σ=30/ 10.46 13.21 13.74 14.45 15.22 15.98 15.98 17.77 19.00 19.51 20.18
20.67
19.56
Buildi
ng
512x5
12
σ=10/20.00 18.64 18.84 19.33 20.08 21.28 22.60 23.36 23.69 23.83 23.71
23.81
23.58
σ=20/13.98 15.55 16.22 16.73 17.29 17.99 18.72 19.82 17.14 21.04 21.35
21.59
21.01
σ=30/10.46 13.13 13.62 13.62 14.87 15.51 15.85 17.14 18.20 18.61 19.11
19.43
18.63
Came
raman
256x2
56
σ=10/20.02 18.29 18.81 19.65 20.20 20.92 23.18 24.15 24.59 24.61 24.36
24.46
21.30
σ=20/13.97 15.81 16.28 16.91 17.40 18.07 18.07 20.12 21.05 21.32 21.66
21.94
18.74
σ=30/10.45
13.46 13.54 14.11 14.89 15.52 15.52 17.28 18.44 18.71 19.31
19.63
18.74
Boat
512x5
12
σ=10/20.00
20.87 21.64 21.79 22.55 23.06 24.22 25.29 26.10 26.39 26.51
26.71
26.18
σ=20/13.98 16.35 16.79 17.44 17.94 18.64 19.44 20.77 21.88 22.31 22.82
23.18
22.30
σ=30/10.46 13.04 13.55 14.24 15.04 15.81 16.21 17.60 18.83 19.30 19.92
20.36
19.33

137
Figure 7. Presents the PSNR variations of the Lena, building, cameraman and boat image denoising
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
12
14
16
18
20
22
24
26
28
30
lena 512x512
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
30
lena 512x512
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
12
14
16
18
20
22
24
building 512x512
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
building 512x512
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
12
14
16
18
20
22
24
26
Cameraman 256x256
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
Cameraman 256x256
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
12
14
16
18
20
22
24
26
28
Boat 512x512
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
5
10
15
20
25
30
Boat 512x512
Fractional order D
PSNR
(dB)
Noise 10%
Noise 20%
Noise 30%

 ISSN: 2302-9285
138
As can be seen, the best value of the PSNR for all images with different noise densities corresponds
to a fractionalvalue of the parameterD, which indicates that the fractionalanalysis leads to an improvemen t
in the denoising result due to the flexibility of the fractional filters which constitute the proposed wavelet
basis. Moreover, it can be seen in the denoising results for the images Lena,boat,cameraman and building as
shown in Figure 8 that the use of fractional calculation through the FD 𝑧−𝐷
offers considerable performance
improvement in terms of visual quality, especially because the noise is eliminated with significant
preservation of edge sharpness.
Noisy image D=0.8 D=1
D=1.1 D=1.3 D=1.5
D=1.1 D=1.3 D=1.5
Figure 8. Comparison of the visual quality of the images lena, boat,cameraman and building
for different values of D

139
D=1.1 D=1.3 D=1.5
D=1.1 D=1.3 D=1.5
Figure 8. Comparison of the visual quality of the images lena, boat,cameraman and building
for different values of D (continue)
5. CONCLUSION
In this paper, we have introduced a generalization of the Haar wavelet using fractional operators,
where the construction is based on exploiting the FD 𝑍−𝐷
, approximated by an FIR filter. Variation of
the parameter D provides greater flexibility for the fractional filters which constitute the wavelet, leading to
more precise decomposition (in terms of detail and approximation). To illustrate the effectiveness of
the proposed fractional wavelet (generalized Haar), we have presented the results of using a denoising
application on images. The results are very promising, and this idea can be expanded to generalize other
families of wavelets.

 ISSN: 2302-9285
140
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