=Paper=
{{Paper
|id=Vol-2577/paper12
|storemode=property
|title=Rational Wavelet Transform with Reducible Rational Dilation Factor
|pdfUrl=https://ceur-ws.org/Vol-2577/paper12.pdf
|volume=Vol-2577
|authors=Oleg Chertov,Volodymyr Malchykov
|dblpUrl=https://dblp.org/rec/conf/its2/ChertovM19
}}
==Rational Wavelet Transform with Reducible Rational Dilation Factor==
https://ceur-ws.org/Vol-2577/paper12.pdf
146
Rational Wavelet Transform with Reducible Rational
Dilation Factor
© Oleg Chertov1[0000-0003-0087-1028] and © Volodymyr Malchykov1[0000-0002-1710-9171]
1 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv,
Ukraine
mavr2k@gmail.com
Abstract. Wavelet analysis is very effectively used in analysis of different
types of data. Mostly often dyadic wavelet transforms are used. But non-dyadic
wavelet transforms allow more accurate determination of data features. Com-
monly used value of dilation factor for rational wavelet transforms is an irre-
ducible fraction. In this paper we will show on an example that for reducible
fraction as a dilation factor perfect reconstruction condition is satisfied.
Keywords: Wavelet Transform, Non-Dyadic Wavelet, Dilation Factor.
1 Introduction
1.1 Wavelet Transform
Wavelet transform (WT) is a very powerful tool for analyzing the data of different
nature. Unlike Fourier analysis WT gives as a result time-frequency representation of
source signal.
Wavelet analysis has shown its efficiency in various areas, such as image and vid-
eo processing, data compression and noise reduction, solving of partial differential
equations, speech recognition, processing of Electroencephalography (EEG), Elec-
tromyography (EMG), Electrocardiography (ECG) signals, etc. [1-4]
1.2 Dyadic and Non-dyadic Wavelet Transform
Discrete wavelet transform is characterized by its dilation factor. It was shown [5]
that as a value of dilation factor any real number greater than one can be taken.
If dilation factor equals 2 the discrete WT is called dyadic, otherwise non-dyadic.
Usually dyadic WT is used due to its simple and effective implementation. But in the
case when signal singularities will be located in adjoined frequency intervals the re-
sults of dyadic WT will not be eligible. Thus, it will be better to select frequency in-
tervals in a way that they would cover such singularities. For example, non-dyadic
WT are more suitable for flexible decompositions of the data [6], non-dyadic scale
ratios [7], non-dyadic frequency divisions [8], or constructing non-tensor-based multi-
D wavelets with coset sum method [9].
Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
147
Various approaches to non-dyadic WT were proposed by different authors.
Bratteli and Jorgensen [10] proposed a method based on the construction of the set
mapping to Kunzh’s algebra representation. In their approach task of non-dyadic WT
filters choice reduces to unitary matrix construction. As a dilation factor only a natu-
ral number greater than one can be selected.
Pollock and Cascio [8] proposed a method for construction of packet WT with a
possibility of a dilation factor choice through each level of decomposition. They gen-
eralized dyadic WT packet technique, where each frequency range at each step of
wavelet decomposition is divided into two intervals. Main idea was to divide range
into p intervals, where p is an arbitrary prime number.
There can be cases where integer dilation factor is not enough. Auscher [11] gave
the formal definition of a rational multiresolution analysis and specified the method of
corresponding orthogonal wavelet bases construction. Using his ideas, Baussard,
Nicolier and Truchetet [12] developed a fast pyramidal algorithm for WT coefficients
construction in a case of a rational dilation factor. Thus, they gave a generalization of
the Malla algorithm for dyadic case.
Also, there are some works that deal with using of irrational dilation factors. One
of the first was Feauveau [13], where the value of dilation factor was √2.
1.3 Rational Wavelet Transform
The most general and effective approach for non-dyadic WT is rational multiresolu-
tion analysis proposed in [11] and [12].
In this approach dilation factor N is equal to a rational number p/q. At each level of
wavelet decomposition, we get one approximation component and (p – q) detail com-
ponents. Rational multiresolution analysis filters construction is carried out in the
frequency domain. Wavelet decomposition pyramidal algorithm with a rational dila-
tion factor is also transferred to the frequency domain.
2 Problem Formulation
The main advantage of non-dyadic wavelet transform is that it can provide more pre-
cise separation of signal components. Very often the irreducible dilation factor is used
for such decomposition. Commonly its value is taken 3/2.
The purpose of this work is to show that as value of the dilation factor for rational
wavelet transform a reducible fraction 6/4 can be taken and this will allow getting the
more precise detection of signal singularities. Also, authors will show that perfect
reconstruction condition is satisfied for this case.
148
3 Problem Solution
3.1 Conditions for Building Filters
In order to build filters for rational wavelet transform with dilation factor 6/4 we will
use the approach described in [5] for case 3/2.
We have function
6
𝜑 ∈ 𝑉 𝑉 = 𝑆𝑝𝑎𝑛 𝜑 ∙ −𝑛
4
so, that
𝜑 𝑥 = ∑ ℎ 𝜑 𝑥−𝑛 (1)
6 6 6 6
𝜑 𝑥 − 4𝑙 = ℎ 𝜑 𝑥 − 4𝑙 − 𝑛 = ℎ 𝜑 𝑥 − 6𝑙 − 𝑛 =
4 4 4 4
6 6
= ℎ 𝜑 𝑥−𝑘
4 4
This set of functions has to be orthonormal
𝛿 = 〈𝜑 𝑥 , 𝜑 𝑥 − 4𝑙 〉 = 𝜑 𝑥 ∙ 𝜑 𝑥 − 4𝑙 𝑑𝑥 =
ℝ
6 6 6 6
= ℎ 𝜑 𝑥−𝑛 ℎ 𝜑 𝑥−𝑘 𝑑𝑥
4 4 4 4
ℝ
6 6 6
= ℎ ℎ 𝜑 𝑥 − 𝑛 ∙ 𝜑 𝑥 − 𝑘 𝑑𝑥 = ℎ ℎ
4 4 4
ℝ
So, orthonormality of the 𝜑 ∙ −4𝑙 implies the condition
ℎ ℎ =𝛿
On the other hand, 𝜑 𝑥 − 1 is also in V0, and therefore can also be written as
𝜑 𝑥−1 = ∑ ℎ 𝜑 𝑥−𝑛 (2)
149
6 6 6 6
𝜑 𝑥 − 4𝑙 − 1 = ℎ 𝜑 𝑥 − 4𝑙 − 𝑛 = ℎ 𝜑 𝑥 − 6𝑙 − 𝑛 =
4 4 4 4
6 6
= ℎ 𝜑 𝑥−𝑘
4 4
Set of functions 𝜑 ∙ −1 also has to be orthonormal.
𝛿 = 〈𝜑 𝑥 − 1 , 𝜑 𝑥 − 4𝑙 − 1 〉 = 𝜑 𝑥 − 1 ∙ 𝜑 𝑥 − 4𝑙 − 1 𝑑𝑥 =
ℝ
6 6 6 6
= ℎ 𝜑 𝑥−𝑛 ℎ 𝜑 𝑥−𝑘 𝑑𝑥
4 4 4 4
ℝ
6 6 6
= ℎ ℎ 𝜑 𝑥 − 𝑛 ∙ 𝜑 𝑥 − 𝑘 𝑑𝑥 = ℎ ℎ
4 4 4
ℝ
Also, it has to be orthogonal to the previous set of functions.
0 = 〈𝜑 𝑥 − 1 , 𝜑 𝑥 − 4𝑙 〉 = 𝜑 𝑥 − 1 ∙ 𝜑 𝑥 − 4𝑙 𝑑𝑥 =
ℝ
6 6 6 6
= ℎ 𝜑 𝑥−𝑛 ℎ 𝜑 𝑥−𝑘 𝑑𝑥
4 4 4 4
ℝ
6 6 6
= ℎ ℎ 𝜑 𝑥 − 𝑛 ∙ 𝜑 𝑥 − 𝑘 𝑑𝑥 = ℎ ℎ
4 4 4
ℝ
So, orthonormality of the 𝜑 . −4𝑙 − 1 and its orthogonality of the 𝜑 . −4𝑙 − 1 with
the respect to 𝜑 . −4𝑙 give us such two conditions
ℎ ℎ =𝛿
ℎ ℎ =0
Repeating the similar operations for functions 𝜑 𝑥 − 2 and 𝜑 𝑥 − 3
150
𝜑 𝑥−2 = ∑ ℎ 𝜑 𝑥−𝑛
(3)
𝜑 𝑥−3 = ∑ ℎ 𝜑 𝑥−𝑛
and taking into account orthogonality and orthonormality properties will bring us to
the conditions
ℎ ℎ =𝛿
ℎ ℎ =0
ℎ ℎ =0
ℎ ℎ =𝛿
ℎ ℎ =
ℎ ℎ =
ℎ ℎ =
Now, we will introduce auxiliary functions. Let’s denote by 𝜑 𝜔 the Fourier trans-
form of function 𝜑 𝑥 . Then, applying the Fourier transform to (1), we will get
6 4 4 4 4
𝜑 𝜔 = ℎ 𝜑 𝜔 𝑒 = ℎ 𝑒 ∙𝜑 𝜔
4 6 6 6 6
We can rewrite last expression as
𝜑 𝜔 =𝑚 𝜔 ∙𝜑 𝜔 (4)
where 𝑚 𝜔 is defined as
4
𝑚 𝜔 ≡ ℎ 𝑒
6
In the same way we can define the function 𝑚 𝜔 . Applying the Fourier transform
to (2) will give us
151
6 4 4 4 4
𝜑 𝜔 𝑒 = ℎ 𝜑 𝜔 𝑒 = ℎ 𝑒 ∙𝜑 𝜔
4 6 6 6 6
𝜑 𝜔 𝑒 =𝑚 𝜔 ∙𝜑 𝜔 (5)
where
4
𝑚 𝜔 ≡ ℎ 𝑒
6
After applying the same operations to functions (3) we will get
𝜑 𝜔 𝑒 =𝑚 𝜔 ∙𝜑 𝜔 (6)
𝜑 𝜔 𝑒 =𝑚 𝜔 ∙𝜑 𝜔 (7)
where
4
𝑚 𝜔 ≡ ℎ 𝑒
6
4
𝑚 𝜔 ≡ ℎ 𝑒
6
Similar to [5] we can also define two wavelet functions
6 6
𝜓 𝑥 = 𝑔 𝜑 𝑥−𝑛
4 4
6 6
𝜓 𝑥 = 𝑔 𝜑 𝑥−𝑛
4 4
Properties of their orthonormality, mutual orthogonality and orthogonality with previ-
ously defined functions ϕ give us the next conditions
𝑔 𝑔 =𝛿 𝑔 𝑔 =𝛿
𝑔 ℎ =0 𝑔 ℎ =0
152
𝑔 ℎ =0 𝑔 ℎ =0
𝑔 ℎ =0 𝑔 ℎ =0
𝑔 ℎ =0 𝑔 ℎ =0
𝑔 𝑔 =0
Also, we get another two necessary functions 𝑚 and 𝑚
4
𝑚 𝜔 = 𝑔 𝑒
6
4
𝑚 𝜔 = 𝑔 𝑒
6
3.2 Perfect Reconstruction Conditions
According to [5], received conditions for filters coefficients are equivalent to the uni-
tarity of matrix
𝑚 𝜔 𝑚 𝜔 𝑚 𝜔 𝑚 𝜔 𝑚 𝜔 𝑚 𝜔
⎛𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ ⎞
⎜ ⎟
⎜𝑚 𝜔 + 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ ⎟
⎜ ⎟
⎜𝑚 𝜔 + 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ ⎟
⎜ ⎟
⎜𝑚 𝜔 + 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ ⎟
⎝𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ 𝑚 𝜔+ ⎠
(8)
Li [14] proved that perfect reconstruction condition is satisfied for the case of the
irreducible dilation factor and showed two examples of wavelet basis.
In [10] the extension of Littlewood-Paley wavelet basis to the rational case was in-
troduced.
Both the scale function and wavelet functions in this case are defined in Fourier
domain by next formulas:
153
𝑞∙𝜋 𝑞∙𝜋
1 𝑓𝑜𝑟 − ≤𝜔<
|𝜑 𝑁𝜔 | = 𝑝 𝑝
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
𝑞+𝑚−1 ∙𝜋 𝑞+𝑚 ∙𝜋
𝜓 𝑁𝜔 = 1 𝑖𝑓 ≤ |𝜔| <
𝑝 𝑝
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
where 𝑁 = and 𝑚 = 1, … , 𝑝 − 𝑞.
We will consider this wavelet basis for the case of reducible fraction 6/4 and check
that perfect reconstruction condition will be satisfied.
If 𝑁 = then scale function can be taken as
6 4𝜋
𝜑 𝜔 = 1 𝑖𝑓 |𝜔| ≤ 6
4
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
or
1 𝑖𝑓 |𝜔| ≤ π
𝜑 𝜔 =
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
Wavelet functions can be defined as
6 4𝜋 5𝜋
𝜓 𝜔 = 1 𝑖𝑓 ≤ |𝜔| <
4 6 6
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
6 5𝜋
𝜓 𝜔 = 1 𝑖𝑓 ≤ |𝜔| < π
4 6
0 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒
According to the (4) – (7) we can write
6
𝜔 𝜑
𝑚 𝜔 = 4
𝜑 𝜔
6
𝜔𝜑
𝑚 𝜔 = 4 ∙𝑒 ∙
𝜑 𝜔
6
𝜔𝜑
𝑚 𝜔 = 4 ∙𝑒 ∙
𝜑 𝜔
6
𝜔𝜑
𝑚 𝜔 = 4 ∙𝑒 ∙
𝜑 𝜔
154
Similarly, we get
6
𝜓 𝜔
m 𝜔 = 4
𝜑 𝜔
6
𝜓 𝜔
m 𝜔 = 4
𝜑 𝜔
After substituting the above expressions into the matrix (8) direct check allows us to
verify that the condition of perfect reconstruction, given in [14], will be also satisfied
for the case of reducible fraction 6/4.
3.3 Experimental Results
In order to illustrate that using of the reducible dilation factor can lead to more precise
identifying of signal singularities we will take a look at the example which is based on
model data.
This mode data is the sum of three sinusoidal signals (see Fig. 1).
Fig. 1. Model signal
Fourier spectrum of this model signal contains three peaks in frequency domain, that
fall into the following frequency ranges (see Fig.2 ):
• first signal (S1) – from 2π/3 to 5π/6
155
• second signal (S2) – from 0 to 2π/3
• third signal (S3) – from 5π/6 to π
Fig. 2. Fourier spectrum of signal
When using the dilation factor 3/2 frequency range is divided into two parts – from 0
to 2π/3 for approximation component and from 2π/3 to π for detail component. Due
to this rational wavelet transform will separate signal S2 (see Fig.3. Approximation
component), but signal S1 and S3 will be mixed (see Fig.3. Detail component).
156
Fig. 3. Decomposition of model signal with dilation factor 3/2
But, if we take dilation factor 6/4 then frequency range is divided into three parts –
again from 0 to 2π/3 for approximation component, but now from 2π/3 to 5π/6 for the
first detail component and from 5π/6 to π for the second detail component.
Thus, in this case all three source signals will be separated (see Fig. 4).
So, for this model signal the use of dilation factor 6/4 is preferable, since it allows
separation of all three components from the original signal.
157
Fig. 4. Decomposition of model signal with dilation factor 6/4
4 Conclusions
Authors proposed to use reducible rational dilation factor 6/4 except most often used
3/2. It was shown that for this value of dilation factor perfect reconstruction condi-
tions are satisfied. An example shows that using dilation factor 6/4 allows more pre-
cise separation of signal singularities.
Further researches have to be done in order to generalize the results to the case of
arbitrary reducible dilation factor.
158
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