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. Commonly used value of dilation factor for rational wavelet transforms is an irreducible fraction. In this paper we will show on an example that for reducible fraction as a dilation factor perfect reconstruction condition is satisfied.
1.1
Introduction
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
video processing, data compression and noise reduction, solving of partial differential
equations, speech recognition, processing of Electroencephalography (EEG),
Electromyography (EMG), Electrocardiography (ECG) signals, etc. [
Dyadic and Non-dyadic Wavelet Transform
Discrete wavelet transform is characterized by its dilation factor. It was shown [
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
results of dyadic WT will not be eligible. Thus, it will be better to select frequency
intervals in a way that they would cover such singularities. For example, non-dyadic
WT are more suitable for flexible decompositions of the data [
Bratteli and Jorgensen [
Pollock and Cascio [
There can be cases where integer dilation factor is not enough. Auscher [
Also, there are some works that deal with using of irrational dilation factors. One
of the first was Feauveau [
Rational Wavelet Transform
The most general and effective approach for non-dyadic WT is rational
multiresolution analysis proposed in [
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 components. Rational multiresolution analysis filters construction is carried out in the frequency domain. Wavelet decomposition pyramidal algorithm with a rational dilation 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 precise 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.
Problem Solution
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 [
We have function − 4 = − 4 − = − 6 −
= 6 4 so, that So, orthonormality of the ∙ −4 implies the condition ℎ ℎ = On the other hand, − 1 is also in V0, and therefore can also be written as − 1 =
= 6 4 = 6 4
ℎ Set of functions ∙ −1 also has to be orthonormal. 6 4 ℎ − 6 − = 6 4 6 4 6 4 ℎ ℎ 6 6 4 =
ℎ Also, it has to be orthogonal to the previous set of functions.
0 = 〈 − 1 , − 4 〉 = − 1 ∙ − 4 = = ℎ = 6 4 = 6 4 the respect to . −4 give us such two conditions Repeating the similar operations for functions − 2 and − 3 and taking into account orthogonality and orthonormality properties will bring us to the conditions ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ ℎ
=
≡
ℎ
=
Now, we will introduce auxiliary functions. Let’s denote by
form of function . Then, applying the Fourier transform to (1), we will get
the Fourier
trans
=
6
4
ℎ
4
6
ℎ
∙
4
6
We can rewrite last expression as
where
is defined as
(4)
to (2) will give us
In the same way we can define the function
. Applying the Fourier transform
After applying the same operations to functions (3) we will get
where
Similar to [
Also, we get another two necessary functions and
+ + + + + (8) + + + + + ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 3.2
Perfect Reconstruction Conditions
According to [
⎛
⎜
⎜
⎜
⎝
⎜
⎜
⎜
+
+
+
+
+
troduced.
domain by next formulas:
Li [
In [
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). 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 • second signal (S2) – from 0 to 2π/3 • third signal (S3) – from 5π/6 to π 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). 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. 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 conditions are satisfied. An example shows that using dilation factor 6/4 allows more precise separation of signal singularities.
Further researches have to be done in order to generalize the results to the case of arbitrary reducible dilation factor.