Datasets of different nature can be effectively analyzed with the help of wavelet analysis. There are two types of discrete wavelet transforms - dyadic and non-dyadic. The latter one allows for more accurate detection and separation of features that are present in analyzed data. A lot of methods use the irreducible fractions as a dilation factor for rational wavelet transform. In this paper the general case of reducible rational dilation factor will be considered. The procedure for building filters will be shown as well as the perfect reconstruction condition. Also, an approach for selecting the best reducible rational dilation factor will be proposed.
Introduction Usually, the irreducible dilation factor is used in rational wavelet transforms. But in some cases, using the reducible dilation factor can improve the quality of wavelet analysis.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). condition for this case. portional fractions will be introduced.
The conditions for building filters for the dilation factor that equals 6/4 were shown
earlier [
Conditions for Filters Coefficients Let’s take arbitrary reducible fraction as a dilation factor of rational wavelet trans , ∙ , ∙ , ∈ ℕ , 2 form where fraction is irreducible.
We have function that can be represented as Let’s define set of functions ℎ ∈
ℎ
This set of functions has to be orthonormal, then 〈 , 〉
∙ ℎ
ℎ ℎ So, orthonormality of the set of functions ∙ implies the condition Let’s take functions 1 , 2 , … , 2 , 1 from the same V0. Due to this fact we can write each of these functions as , , ℎ ℎ
ℎ
ℎ ℎ ℎ This gives us q-1 conditions for filter coefficients: Also, all sets of functions have be mutually orthogonal ℎ ∙ ℎ , ∙ , ℎ
ℎ ̃ ℝ ℎ ℎ
̃ ℎ
0 , ̃〉
̃
ℎ ℎ where This gives us next conditions: Finally, we have next conditions for low-pass filter coefficients: , ̃ 0 …
1, ̃ ℎ ∙ ℎ ̃ 0, , ̃ 0 …
1, ̃ ⎧ ℎ ⎪ ⎨ ℎ ⎪ ⎩
∙ ℎ ∙ ℎ ̃ , , ∙ ∙ ∙ where functions
are defined according to Next, we define p-q wavelet functions ∙ ∙ ̃ ∙
∙ ∑ ℎ ∙ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ℎ ∙ ∙
, (1) (2) From the orthonormality of these functions and their orthogonality to the functions we get next conditions: ∙ ℎ ̃ ,
1 … p 0, , ̃
1 … p , ̃ 0, 1 … p , ̃ functions.
1 … ∑ ∙ ,
1 … ∙ ,
1 … are the Fourier transform of the corresponding wavelet
Perfect Reconstruction Condition ⎝ ⎪ ⎩ ℎ . . . . . . . . . . . . . . . .
: ⎞ ⎟ ⎟ ⎟ ⎠ 2 2 , , is a complex conjugate of the transposition of where arguments , 0 …
1 are defined according to the formula: After substituting expressions (1) and (2), multiplying sums and grouping of similar elements we get (3) where
is a unit matrix of dimension p. In [
For this purpose, we will calculate elements of matrix A. Diagonal elements of this matrix can be written as . . . . . . . . . . . . ∙ ∙ ∙ ℎ ∙ ∙ 2 2 ∙ ∙ ∗ ∙ ∙ . . ∙ ∙ ∙ ∙ ∙ Let’s take a look at the last multiplier. After substituting n-k by m it can be written as 1 ⋯ After denoting ≡
we can write this expression as If does not equal to one, then we can multiply last expression by 1- and get 1 1 ρ ⋯ 1 1 Due to the fact that
1 we get that in this case
⎧ ℎ ⎪ ⎪⎨ ⎩ 1 1 ρ
0
1 ∙ ℎ ∙ ∙ ∙ , If equals one, i.e. for the values of m that are multiplies of the numerator, then from the definition of the function it immediately follows that in this case So, now we can write the values of the diagonal elements of matrix A as that, after taking into account conditions for filters coefficients, gives us ,
So, matrix A satisfies condition (3). This means that it can be looked as a condition for the perfect reconstruction in the case of reducible dilation factor of rational wavelet 3.3
Dilation Factor Selecting Criterion In order to select optimal value from the set of reducible fractions that all are the multipliers of the same irreducible rational number authors propose to use entropy-based criterion.
Entropy is usually understood as a measure of uncertainty or unpredictability of
information [
∑
∙
,
decomposition. In their previous work [
Calculation of entropy for the rational wavelet decomposition at the level N of the signal is based on relative energy. First, it is necessary to find the energy Ej at the level j of the decomposition coefficients. lated where dj,k are the detailed coefficients at this level and Nj is the whole number of the
Next, the relative energy Pj that shows the distribution of energy by levels is calcuAnd at last entropy is calculated according to the expression 3.4
Experimental Results We will illustrate the process of selecting the reducible dilation factor by the decomposition of signal of the spontaneous electromagnetic emission of the Earth.
The whole set of data was measured by “Yugneftegazgeologiya” company during the experiments on the September, 2012 in Bazeliyivka, Ukraine. measurement stations at the “zero” point.
Fourier spectrum of this signal is shown at the Fig. 2.
Due to the Fourier spectrum and nature of the measured signal a decision to analyze it by rational wavelet transform with the value of 5/3 for dilation factor was made first. Two other values – 10/6 and 15/9 – for the dilation factor were also considered.
In order to select the most optimal value entropy-based criterion was used. Results are shown in Table 1. The minimum value of entropy is bolded. Entropy for dyadic wavelet decomposition is also included for comparison.
Entropy 0.1071
Orange vertical lines bound the frequency band of Fourier spectrum that corresponds to the detailed components on third level of wavelet decomposition.
Dark red vertical line separates the frequency intervals for the detailed components of decomposition with dilation factor 5/3. Green vertical lines show the spectrum bands for the detailed components of rational wavelet decomposition with dilation factor 10/6.
It can be easily seen that in the second case singularities of the signal are better separated from each other.
Further fragmentation of frequency band into smaller parts leads to the fact that some singularities will be placed at the boundaries of intervals, and, so, will not be separated. Increasing of wavelet entropy in Table 1 proofs this. 4
Conclusions Authors have proposed to use arbitrary reducible rational fraction as a dilation factor for the rational wavelet transform. It has been shown that perfect reconstruction condition for such values of dilation factor is satisfied.
Criterion for selecting the optimal dilation factor from the set of reducible fractions that all are the multipliers of the same irreducible rational number has been introduced. Authors have proposed to use entropy-based criterion.
Process of selecting the optimal value for the dilation factor of rational wavelet transform is demonstrated on the signal of the spontaneous electromagnetic emission of the Earth.