CN107992711B - Optimization design method of M-channel oversampling modulation diagram filter bank - Google Patents

Optimization design method of M-channel oversampling modulation diagram filter bank Download PDF

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CN107992711B
CN107992711B CN201810049810.4A CN201810049810A CN107992711B CN 107992711 B CN107992711 B CN 107992711B CN 201810049810 A CN201810049810 A CN 201810049810A CN 107992711 B CN107992711 B CN 107992711B
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蒋俊正
刘松辽
欧阳缮
孙希延
谢跃雷
杨玉琳
曹想
杨杰
杨圣
赵海兵
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Guilin University of Electronic Technology
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Abstract

The invention discloses an optimal design method of an M-channel oversampling modulation diagram filter bank, which resolves the design problem of the diagram filter bank into the design of four prototype filters, realizes the overall design of the diagram filter bank through the modulation of the four prototype filters, fully considers the design complexity of the prototype filters, and designs the prototype filters under the condition of considering the complete reconstruction condition and the frequency spectrum characteristic of the diagram filter bank, thereby improving the overall performance of the diagram filter bank. The invention provides a simple and effective solution for reducing the design complexity and reconstruction errors of the image filter bank and realizing the recovery and reconstruction of signals.

Description

Optimization design method of M-channel oversampling modulation diagram filter bank
Technical Field
The invention relates to the technical field of multi-rate signal processing, in particular to an optimal design method of an M-channel oversampling modulation diagram filter bank.
Background
The graph signal processing is an important part of the signal processing, and has attracted attention in recent years. In order to process irregular signals and network data, researchers have introduced graph signal processing, and regular signals can also be modeled as graph signals for processing, so graph signal processing is more advantageous than conventional signal processing. The image signal processing has very important application significance in the aspects of network, denoising, smart city, signal abnormity detection and recovery, semi-supervised classification, image processing, big data processing and the like. Compared with the traditional filter bank, the advantage of the graph filter bank is that the graph filter bank can be designed by transforming the eigenvector and eigenvalue of the Laplace matrix of the graph into the frequency domain, and the frequency domain can be designed by adopting a normalized real frequency domain, so that the graph filter bank is relatively easy to realize.
The image filter bank is mainly divided into two channels and an M channel according to the number of channels. In order to meet the requirements of practical application, the design of an M-channel map filter bank is paid attention and researched by more scholars, the M-channel map filter bank has more sub-band division and has multi-resolution analysis characteristics, and more refined signal components can be selectively reserved or extracted according to needs, so that the M-channel map filter bank has more advantages in processing large-scale signals. The existing design method of the M-channel image filter bank mainly considers the spectrum selectivity and the reconstruction error of the image filter bank, does not consider the design complexity, and the design complexity mainly depends on the design of a prototype filter. The number M of prototype filters designed by the conventional method increases linearly with the number of channels, and the design of the prototype filters becomes more complicated when the number of channels is larger. Therefore, a graph filter bank design method that sufficiently considers the design complexity of the prototype filter and the spectral characteristics of the graph filter bank, the complete reconstruction condition, is yet to be proposed.
Disclosure of Invention
The invention aims to solve the problem that the design of a prototype filter of the existing M-channel map filter bank is relatively complex, and provides an optimal design method of an M-channel oversampling modulation map filter bank.
In order to solve the problems, the invention is realized by the following technical scheme:
the optimal design method of the M-channel oversampling modulation diagram filter bank specifically comprises the following steps:
step 1, according to the characteristic that the low-pass and band-pass filters in the graph filter bank have different frequency spectrum shapes, the design of the M-channel graph filter bank is reduced to the design of 0 th and 1 st analysis filters and 0 th and 1 st synthesis filters, while the 2 nd to M-1 st analysis filters are modulated by the 0 th and 1 st analysis filters, and the 2 nd to M-1 st synthesis filters are modulated by the 0 th and 1 st synthesis filters; wherein the modulation formula is:
Figure BDA0001552028200000021
Figure BDA0001552028200000022
in the formula, hi(. h) is the ith analysis filter, gi(·) is the ith synthesis filter, λ is the graph frequency, i ═ 0,1, …, M-1, k ═ 2,3, …, M-2; m is the number of channels, M >2 and is an even number;
step 2, under the zero point constraint condition, taking the passband ripple energy and the stopband energy of the two analysis filters as objective functions, and solving the optimization problem of the analysis filter which enables the objective function to be minimum to obtain the coefficient of the 0 th analysis filter and the coefficient of the 1 st analysis filter, so as to obtain the 0 th analysis filter and the 1 st analysis filter;
step 3, with the solved analysis filter as a known condition and the stop band energies of the two synthesis filters as an objective function under a complete reconstruction constraint condition, solving an optimization problem of the synthesis filter with the minimum objective function to obtain a 0 th synthesis filter coefficient and a 1 st synthesis filter coefficient, and further obtain a 0 th synthesis filter and a 1 st synthesis filter;
and 4, substituting the 0 th and 1 st analysis filters obtained in the step 2 and the 0 th and 1 st synthesis filters obtained in the step 3 into the modulation formula in the step 1 to obtain the 2 nd to M-1 st analysis filters and the 2 nd to M-1 st synthesis filters, thereby obtaining the whole M-channel modulation diagram filter bank.
In the step 3, the optimization problem of the analysis filter is as follows:
Figure BDA0001552028200000023
Figure BDA0001552028200000024
wherein h ═ h0；h1]，h0For the 0 th analysis filter coefficient vector, h1For the 1 st analysis filter coefficient vector; ep(h) Is the sum of the passband ripple energies of the 0 th and 1 st analysis filters;
Es(h) is the sum of the stop band energies of the 0 th and 1 st analysis filters; alpha is a weight factor;
Figure BDA0001552028200000025
for the frequency vector of the 1 st analysis filter, Lh1For the 1 st analysis filter coefficient vector h1Length of (d); upper labelTIndicating transposition.
In the step 4, the optimization problem of the synthesis filter is as follows:
Figure BDA0001552028200000026
s.t.|E(λk′)|≤εr；g1(0)＝0
wherein g ═ g0；g1]，g0Is the 0 th synthesis filter coefficient vector, g 11 st synthesis filter coefficient vector; es(g0) Is the stop band energy of the 0 th synthesize filter; es(g1) The stopband energy of the 1 st synthesize filter; alpha is a weight factor; e (-) is the reconstruction error, λk′For the k' th frequency discrete point,
Figure BDA0001552028200000027
k' is 0,1, …, N +1 is the number of frequency discrete points given; epsilonrFor a given reconstruction error margin; g1(0) Is the value of the 1 st synthesize filter at frequency zero.
In the above steps 3 and 4, the analysis filter and the synthesis filter are solved by using a convex programming solving tool cvx.
Compared with the prior art, the design problem of the graph filter bank is solved into the design of four prototype filters, the overall design of the graph filter bank is realized through the modulation of the four prototype filters, the design complexity of the prototype filters is fully considered, and the design of the prototype filters is carried out under the condition of considering the complete reconstruction condition and the frequency spectrum characteristic of the graph filter bank, so that the overall performance of the graph filter bank is improved. The invention provides a simple and effective solution for reducing the design complexity and reconstruction errors of the image filter bank and realizing the recovery and reconstruction of signals.
Drawings
Fig. 1 is a basic structure of an M-channel oversampled picture filter bank.
Fig. 2 is a modulation spectrum diagram of an M-channel oversampled map filter bank.
Fig. 3 is the magnitude response of the resulting graph filter after optimization in example 1 of the present invention.
Fig. 4 shows the magnitude response of the resulting filter after optimization in example 2 of the present invention.
Detailed Description
For ease of understanding, the present invention is further described in detail below by taking a six-channel (M ═ 6) filter bank as an example.
Vertex domain analysis filter H for M-channel map filter bankiAnd synthesis filter GiRespectively expressed as:
Figure BDA0001552028200000031
in the formula, HiAnalyze the filter for the ith vertex domain, GiFor the ith vertex domain synthesis filter, hi(lambda) is the value of the ith spectral domain analysis filter at the characteristic root lambda, gi(λ) is the value of the ith spectral domain synthesis filter at the characteristic root λ, λ is the characteristic root of the laplacian matrix of the graph G, σ (G) is the characteristic space formed by all the characteristic roots of the laplacian matrix of the graph G, PλIs the orthogonal projection matrix of the eigenspace and i is the order of the subband filter.
According to the characteristic that the frequency spectrum shapes of a low-pass filter and a band-pass filter in a graph filter bank are different, a modulation formula for modulating the graph filter bank by an M channel is designed as follows:
Figure BDA0001552028200000032
Figure BDA0001552028200000033
in the case of a normal channel M (M is an even number), the full reconstruction condition can be expressed as:
Figure BDA0001552028200000034
wherein λ ∈ [0,2 ]. The complete reconstruction condition of the M-channel modulation map filter bank obtained by substituting modulation equations (2) and (3) into equation (4) is:
Figure BDA0001552028200000041
the analysis filter and the synthesis filter of the six-channel map filter bank are respectively represented as:
Figure BDA0001552028200000042
in the formula, Lhi,LgiRespectively representing analysis filters hiAnd a synthesis filter giLength of (d).
Figure BDA0001552028200000043
Notation and representation of the graph frequency analogous to a conventional filter bank, lambdapd0,λsd0Represents h0,g0Of the passband and stopband, lambdapd1,λpd2,λsd1,λsd2Represents h1,g1Pass band and stop band cut-off frequencies. The passband ripple of the low pass prototype filter can be measured by equation (8):
Figure BDA0001552028200000044
when i is 0, a is 0, b is λpd0When formula (8) is Ep(h0) I.e. h0The energy of the passband ripple of (a); when i is 1, a is lambdapd1,b＝λpd2When formula (8) is Ep(h1) I.e. h1The energy of the passband ripple. The stopband attenuation is determined by the stopband energy:
Figure BDA0001552028200000045
Figure BDA0001552028200000046
in a six-channel modulation diagram filter bank, the signal is filtered by a prototype filter h0(λ),h1(lambda) and g0(λ),g1The (lambda) modulation results in a further filter,
Figure BDA0001552028200000051
the full reconstruction condition of the six-channel modulation map filter bank can be expressed as:
Figure BDA0001552028200000052
the reconstruction error function of the six-channel modulation diagram filter bank obtained from the complete reconstruction condition is:
Figure BDA0001552028200000053
maximum reconstruction error is defined as Emax＝Maxλ|E(λ)|。
Based on the above analysis, the spectrum selectivity, the reconstruction error and the design complexity of the graph filter bank are fully considered, the design problem of the modulation graph filter bank can be summarized as the design of four prototype filters, the design of the prototype filters is further summarized as the band-constrained optimization problem, and the prototype filters are designed in two steps:
fig. 1 shows a basic structure of an M-channel oversampled filter bank, fig. 2 shows a spectrogram of an M-channel modulation diagram filter bank, and based on the above spectral characteristics, the method for optimally designing a prototype filter of the M-channel modulation diagram filter bank includes the following steps:
the first step is as follows: and designing an analysis prototype filter, taking the passband ripple and the stopband energy of the two analysis prototype filters as objective functions, and solving the analysis prototype filter which enables the passband distortion and the stopband energy to be minimum under the constraint condition of zero, wherein the optimization problem is a convex programming problem and can be effectively solved.
Figure BDA0001552028200000054
Constraint of zero point
Figure BDA0001552028200000055
α and β are weights, and α ═ β is usually taken. For easy solution, remember
h＝[h0；h1]；h0＝[I0,0]h＝B0h；h1＝[0,I1]h＝B1h；(14)
Wherein, B0Is of size Lh0×(Lh0+Lh1) Matrix of (A), B1Is of size Lh1×(Lh0+Lh1) Matrix of (I)0Is Lh0×Lh0Identity matrix of (1)1Is Lh1×L h10 is an all-zero matrix. The constraint solving problem can be simplified to equation (15):
Figure BDA0001552028200000056
the second step is that: and taking the solved analysis prototype filter as a known condition, and under the condition of complete reconstruction constraint, taking the stop band energy of the synthesis prototype filter bank as an objective function to solve the synthesis prototype filter in consideration of maximizing the stop band attenuation of the synthesis filter.
Figure BDA0001552028200000057
Wherein
Figure BDA0001552028200000061
k′＝0,1,…,N，εrIs the reconstruction error tolerance, N +1 is the number of discrete points, and many examples show that N-100 guarantees the reconstruction error accuracy. For the convenience of calculation, the substitution shown in equation (17) is made:
g＝[g0；g1]；g0＝[I0,0]g＝C0g；g1＝[0,I1]g＝C1g； (17)
wherein, C0Is of size Lg0×(Lg0+Lg1) Moment ofArray, C1Is of size Lg1×(Lg0+Lg1) Matrix of (I)0Is Lg0×Lg0Identity matrix of (1)1Is Lg1×L g10 is an all-zero matrix. The above problem can be equivalently written as:
Figure BDA0001552028200000062
wherein b isk′2, k' is 0,1, …, N, note:
Figure BDA0001552028200000063
vector aT(λk′) Is represented as follows:
aT(λk′)＝d(λk′)+d(2-λk′),k′＝0,…,N (19)
the optimization problems are all convex programming problems, the convex programming solving tool cvx can be used for effectively solving the problems, and the solved graph filter is a real-valued function of lambda.
The above design process can be generalized to any M (M >2 and even) channel map filter bank. The generalized optimal design method of the M-channel oversampling modulation diagram filter bank based on convex optimization comprises the following steps:
step 1, according to the characteristic that the frequency spectrum shapes of low-pass filters and band-pass filters in a graph filter bank are different, the design of an M-channel graph filter bank is reduced into the design of four prototype filters, and other filters are modulated by the four prototype filters. The modulation formula of the M-channel modulation chart filter bank is designed as follows:
Figure BDA0001552028200000064
Figure BDA0001552028200000065
in the formula hi(λ), i ═ 0,1,2,3, …, M-1 is the ith analysis filter, gi(λ) is the ith synthesis filter, λ is the eigenroot of the Laplace matrix of graph G.
In step 2, in the case of a normal channel M (M is an even number), the complete reconstruction condition can be expressed as:
Figure BDA0001552028200000066
wherein λ ∈ [0,2 ]. The complete reconstruction condition of the M-channel modulation diagram filter bank obtained by substituting the modulation formula in the step 1 into the above formula is as follows:
Figure BDA0001552028200000071
the reconstruction error function from the full reconstruction condition is:
Figure BDA0001552028200000072
maximum reconstruction error is defined as Emax＝Maxλ|E(λ)|。
And step 3, according to the characteristics of the modulation diagram filter bank, the design problem of the modulation diagram filter bank is summarized into the design of four prototype filters.
And 3.1, designing two analysis filters, taking the passband ripple and the stopband energy of the two analysis filters as objective functions, and solving the analysis filters which enable the passband distortion and the stopband energy to be minimum under the zero constraint condition, wherein the optimization problem is a convex programming problem and can be effectively solved. The constraint solving problem can be simplified as:
Figure BDA0001552028200000073
Figure BDA0001552028200000074
in the formula, Ep(h) Analyzing the passband ripple energy of the filter; es(h) Analyzing the stop band energy of the filter; alpha is a weight factor; h is an analysis filter bank, h ═ h0；h1]；h0For the 0 th analysis filter coefficient vector, h1For the 1 st analysis filter coefficient vector,
Figure BDA0001552028200000075
for the frequency vector of the 1 st analysis filter, Lh1For the 1 st analysis filter coefficient vector h1Length of (d).
And 3.2, designing two comprehensive filters. And taking the solved analysis filter as a known condition, and under the constraint condition of complete reconstruction, considering that the stop band attenuation of the synthesis filter is maximized, and taking the stop band energy of the two synthesis filters as an objective function to solve the synthesis filter.
Figure BDA0001552028200000076
s.t.|E(λk′)|≤εr；g1(0)＝0；
k′＝0,1,…,N；
In the formula, Es(g0) Is a synthesis filter g0The stop band energy of (a); es(g1) Is a synthesis filter g1The stop band energy of (a); g is the synthesis filter bank, g ═ g0；g1]，g0Is the 0 th synthesis filter coefficient vector, g 11 st synthesis filter coefficient vector;
Figure BDA0001552028200000077
E(λk′) Is the reconstruction error at the k' th frequency discrete point; epsilonrTo reconstruct the error tolerance; a isT(λk′) Is the response vector at the k' th frequency discrete point; g1(0) Is the 1 st synthesize filter g1At frequency zeroTaking values; n +1 is the number of frequency dispersion points, and many examples show that N100 guarantees reconstruction error accuracy.
Because the optimization problems of solving the analysis filter and the synthesis filter are both convex programming problems, the analysis filter and the synthesis filter can be effectively solved by adopting a convex programming solving tool cvx.
And 4, modulating the analysis filter and the synthesis filter obtained in the step 3 by the modulation mode in the step 1 to obtain the M-channel modulation diagram filter bank.
The performance of the present invention is illustrated by the following specific simulation examples.
Simulation example 1:
designing a graph filter bank, wherein the parameters are as follows:
Lh0＝12,Lh1＝12,Lg0＝11,Lg1＝11,λpd0＝0.3,λpd1＝0.35,λpd2＝0.55,λsd0＝0.55,λsd1＝0.1,λsd2＝1.0,α＝0.1,εr＝10-10(ii) a The obtained amplitude response of the graph filter bank is shown in fig. 3, and PR in the graph represents the reconstruction error of the graph filter bank corresponding to different values of λ. The maximum reconstruction error and the orthogonality value obtained by simulation calculation are respectively Emax＝3.31×10-10Table 1 shows the reconstruction performance and boundary ratio of the graph filter bank designed by the method of the present invention and the graph filter bank designed by the prior method 1 (near-orthogonal M-channel oversampling) and the prior method 3 (a new method for designing an M-channel biorthogonal oversampling graph filter bank) under the same graph filter bank length and operation environment.
TABLE 1
Figure BDA0001552028200000081
RBIs a boundary ratio[14]，
Figure BDA0001552028200000082
Taking R in simulationBIs subjected toRatio, RBA value equal to 1 indicates that the filter bank is fully reconstructed, when the reconstruction characteristic of the filter bank is better. The comparison shows that the reconstruction error of the algorithm is obviously lower than that of the existing method 1 and the existing method 3, the spectral characteristics of the graph filter bank designed by the method are obviously better than those of the existing method 1 and the existing method 3, and the reconstruction error and the spectral selectivity are important indexes for measuring the performance of the graph filter bank, so that the overall performance of the modulation graph filter bank designed by the method is better.
Simulation example 2:
designing a graph filter bank, wherein the parameters are as follows:
Lh0＝8,Lh1＝8,Lg0＝7,Lg1＝7,λpd0＝0.3,λpd1＝0.35,λpd2＝0.55,λsd0＝0.55,λλsd1＝0.1,λsd2＝1,α＝0.1,εr＝10-13(ii) a The resulting modulation map filter bank amplitude response is shown in fig. 4. Table 2 shows the reconstruction performance and orthogonality comparison of the graph filter bank designed by the method of the present invention with the graph filter bank designed by the existing method 2 (M-channel oversampled graph filter bank) and the existing method 3 (a new method for designing an M-channel biorthogonal oversampled graph filter bank).
TABLE 2
Figure BDA0001552028200000083
A closer to 1 orthogonality Θ indicates a better orthogonality of the filter bank. The comparison shows that the reconstruction error of the graph filter bank designed by the method is obviously smaller than that of the existing method 2 and the existing method 3, the orthogonality is better than that of the existing method 2, the method is equivalent to that of the existing method 3, and the overall spectrum characteristic is better than that of the existing method 2 and the existing method 3, so that the overall performance of the graph filter bank designed by the method is better.
It should be noted that, although the above-mentioned embodiments of the present invention are illustrative, the present invention is not limited thereto, and thus the present invention is not limited to the above-mentioned embodiments. Other embodiments, which can be made by those skilled in the art in light of the teachings of the present invention, are considered to be within the scope of the present invention without departing from its principles.

Claims (2)

  1. The optimal design method of the M-channel oversampling modulation diagram filter bank is characterized by comprising the following steps:
    step 1, according to the characteristic that the low-pass and band-pass filters in the graph filter bank have different frequency spectrum shapes, the design of the M-channel graph filter bank is reduced to the design of 0 th and 1 st analysis filters and 0 th and 1 st synthesis filters, while the 2 nd to M-1 st analysis filters are modulated by the 0 th and 1 st analysis filters, and the 2 nd to M-1 st synthesis filters are modulated by the 0 th and 1 st synthesis filters; wherein the modulation formula is:
    Figure FDA0002774435280000011
    hM-1(λ)＝h0(2-λ)
    Figure FDA0002774435280000012
    gM-1(λ)＝g0(2-λ)
    in the formula, hM-k(. h) is the M-k th analysis filter, hM-1(. h) is the M-1 th analysis filter, h1(. 1) is the analysis filter, h0(. 0) is the 0 th analysis filter; gM-k(. h) is the M-k th synthesis filter, gM-1(. h) is the M-1 th synthesis filter, g1(. 1) is a synthesis filter, g0(. 0) is the 0 th synthesis filter; λ is graph frequency, k is 2,3, …, M-2; m is the number of channels, M >2 and is an even number;
    step 2, under the zero point constraint condition, taking the passband ripple energy and the stopband energy of the two analysis filters as objective functions, and solving the optimization problem of the analysis filter which enables the objective function to be minimum to obtain the coefficient of the 0 th analysis filter and the coefficient of the 1 st analysis filter, so as to obtain the 0 th analysis filter and the 1 st analysis filter; the optimization problem of the analysis filter is as follows:
    Figure FDA0002774435280000013
    Figure FDA0002774435280000014
    wherein h ═ h0；h1]，h0For the 0 th analysis filter coefficient vector, h1For the 1 st analysis filter coefficient vector; ep(h) Is the sum of the passband ripple energies of the 0 th and 1 st analysis filters; es(h) Is the sum of the stop band energies of the 0 th and 1 st analysis filters; alpha is a weight factor;
    Figure FDA0002774435280000015
    for the frequency vector of the 1 st analysis filter, Lh1For the 1 st analysis filter coefficient vector h1Length of (d); upper labelTRepresenting a transpose;
    step 3, with the solved analysis filter as a known condition and the stop band energies of the two synthesis filters as an objective function under a complete reconstruction constraint condition, solving an optimization problem of the synthesis filter with the minimum objective function to obtain a 0 th synthesis filter coefficient and a 1 st synthesis filter coefficient, and further obtain a 0 th synthesis filter and a 1 st synthesis filter; the optimization problem of the synthesis filter is as follows:
    Figure FDA0002774435280000016
    s.t.|E(λk′)|≤εr；g1(0)＝0
    in the formula, g0Is the 0 th synthesis filter coefficient vector, g11 st synthesis filter coefficient vector;Es(g0) Is the stop band energy of the 0 th synthesize filter; es(g1) The stopband energy of the 1 st synthesize filter; alpha is a weight factor; e (-) is the reconstruction error, λk′For the k' th frequency discrete point,
    Figure FDA0002774435280000021
    k' is 0,1, …, N +1 is the number of frequency discrete points given; epsilonrFor a given reconstruction error margin; g1(0) Is the value of the 1 st synthesis filter at the frequency zero point;
    and 4, substituting the 0 th and 1 st analysis filters obtained in the step 2 and the 0 th and 1 st synthesis filters obtained in the step 3 into the modulation formula in the step 1 to obtain the 2 nd to M-1 st analysis filters and the 2 nd to M-1 st synthesis filters, thereby obtaining the whole M-channel modulation diagram filter bank.
  2. 2. The method for optimally designing an M-channel oversampled modulation map filter bank as claimed in claim 1, wherein in steps 2 and 3, the analysis filter and the synthesis filter are solved using a convex programming solver cvx.
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