CN107239623B - Optimal design method of M-channel oversampling image filter bank based on convex optimization - Google Patents

Optimal design method of M-channel oversampling image filter bank based on convex optimization Download PDF

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CN107239623B
CN107239623B CN201710426377.7A CN201710426377A CN107239623B CN 107239623 B CN107239623 B CN 107239623B CN 201710426377 A CN201710426377 A CN 201710426377A CN 107239623 B CN107239623 B CN 107239623B
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蒋俊正
刘松辽
欧阳缮
刘庆华
谢跃雷
晋良念
丁勇
曹想
杨玉琳
江庆
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Guilin University of Electronic Technology
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Abstract

The invention discloses an optimization design method of an M-channel oversampling image filter bank based on convex optimization, which utilizes a two-step method to carry out target optimization and simultaneously comprehensively considers the spectrum selectivity and the complete reconstruction condition of the image filter bank, thereby ensuring that the overall performance of the image filter bank is improved under the condition of smaller reconstruction error. The invention provides a simple and effective solution for reducing the reconstruction error of the image filter bank and realizing the recovery reconstruction of the signal.

Description

Optimal design method of M-channel oversampling image filter bank based on convex optimization
Technical Field
The invention relates to the technical field of multi-rate signal processing, in particular to an optimization design method of an M-channel oversampling image filter bank based on convex optimization.
Background
The graph signal processing has important application significance in the aspects of social interaction, biology, traffic, sensor network processing, image processing and the like. In practical application, the data volume of the graph is huge, and certain difficulty is brought to graph signal processing. The multiresolution analysis is a localized time-frequency analysis, can carry out multiscale detailed analysis on signals, effectively extracts the characteristic information of the signals, but has certain limitation on big data processing. However, for big data, the wavelet and filter bank are more suitable for processing large-scale image signals, and the wavelet and filter bank can perform sparse representation on the image signals, so that the big data can be processed more advantageously. The graph filter bank can be divided into two categories of orthogonality and biorthogonal from the aspect of frequency spectrum design, critical sampling and oversampling from the aspect of signal sampling, and can be divided into two categories of channels and M channels according to the number of the channels. The design of the two-channel image filter bank is the design method of the wavelet filter bank of the complete reconstruction orthogonal critical sampling proposed by Narang and Ortega at first, and the bipartite graph is processed by the method of orthogonal mirror image filtering, and the filter designed by the method has the characteristics of aliasing elimination, complete reconstruction and orthogonality, but the design method is directed at the bipartite graph or graph signals capable of being decomposed into the bipartite graph, and has certain limitation. Therefore, they have proposed a method for a tightly-supported biorthogonal wavelet filter bank of any undirected graph, where the relaxed orthogonality condition becomes biorthogonal, the biorthogonal graph wavelet is tightly-supported in frequency domain and node K order is localized, and still satisfies the complete reconstruction condition, and can process any graph signal, but this method does not consider the spectral selectivity of the filter, and thus has certain limitations. The design method of the image filter bank generates the transition from quadrature to biorthogonal, critical sampling to oversampling and two-channel to M-channel.
The image signal processing breaks through the original limitation from critical sampling to oversampling, and the signal after oversampling contains all the information of the original image signal, thereby being more convenient for further processing of the image signal. The over-sampled graph Laplace matrix is crucial to the development and application of graph signal processing, and the over-sampling has greater design freedom for the processing of regular signals. The oversampling has great advantages for the design of an M-channel image filter bank, and the M-channel image filter bank has more sub-band division and has great significance for large-scale signal processing. Currently, relatively few studies are conducted on M-channel map filter banks, and further studies are to be proposed.
Disclosure of Invention
The invention aims to solve the problem of poor overall performance of the existing M-channel oversampling map filter bank, and provides an optimization design method of the M-channel oversampling map filter bank based on convex optimization.
In order to solve the problems, the invention is realized by the following scheme:
the optimal design method of the M-channel oversampling map filter bank based on convex optimization specifically comprises the following steps:
step 1, designing an analysis filter of an M-channel diagram filter bank in consideration of spectral characteristics, and solving the analysis filter which enables the pass-band distortion and the stop-band energy of the analysis filter bank to be minimum by using the pass-band ripple and the stop-band energy of the analysis filter bank as objective functions through a semi-positive definite programming solver under the constraint condition of 3 dB; the optimization problem is that
Figure BDA0001316280000000021
Figure BDA0001316280000000022
i＝0,1,…,M/2-1；j＝1,2,…,M/2-1
In the formula, Ep(h) Analyzing the passband ripple energy of the filter; es(h) To analyze the stop band energy of the filter, α is the analysis weight, h is the analysis filter bank, h is h [ h ]0；h1；…；hM/2-1]；
Figure BDA0001316280000000027
(. is) th iAn analysis filter hiLhiFor the ith analysis filter hiLength of (h)iIs the ith analysis filter; when i is 1,2, …, M/2-2, hi＝[0i1,Ii,0i2]h; when i is 0, hi＝[Ii,0i2]h; when i is M/2-1, hi＝[0i1,Ii]h；0i1Is L in sizehi×(Lh0+Lh1+…+Lh(i-1)) All-zero matrix of, 0i2Is L in sizehi×(Lh(i+1)+Lh(i+2)…+Lh(M/2-1)) All-zero matrix of (I)iIs L in sizehi×LhiThe identity matrix of (1);
Figure BDA0001316280000000023
(. h) is the jth analysis filter hjLhjFor the jth analysis filter hjLength of (h)jFor the jth analysis filter, h is given when j is 1,2, …, M/2-2j＝[0j1,Ij，0j2]h; when j is M/2-1, hj＝[0j1,Ij]h；0j1Is L in sizehj×(Lh0+Lh1+…+Lh(j-1)) All-zero matrix of, 0j2Is L in sizehj×(Lh(j+1)+Lh(j+2)…+Lh(M/2-1)) All-zero matrix of (I)jIs L in sizehj×LhjThe identity matrix of (1); t represents transposition; m is the number of channels;
step 2, taking the analysis filter solved in the step 1 as a known condition, designing a comprehensive filter of the M-channel diagram filter bank from the aspect of complete reconstruction characteristics, and under the constraint condition of complete reconstruction, solving the comprehensive filter which enables the stop band attenuation of the comprehensive filter to be maximum by taking the stop band energy of the comprehensive filter bank as an objective function through a semi-positive definite programming solver; the optimization problem is that
Figure BDA0001316280000000024
s.t.|aT(λk)g-bk|≤r；g1(0)＝0
Figure BDA0001316280000000025
k＝0,1,…,N；l＝0,1,…,M/2-1
In the formula, Es(g) The stop band energy of the synthesis filter, β the synthesis weight, g the synthesis filter set, g ═ g0；g1；…；gM/2-1]；
Figure BDA0001316280000000026
(. h) is the first synthesis filter glLglFor the first synthesis filter glLength of (g)lIs the first synthesis filter, when l is 1,2, …, M/2-2, gl＝[0gl1,Igl,0gl2]g; when l is 0, gl＝[Igl,0gl2]g; when l is M/2-1, gl＝[0gl1,Igl]g；0gl1Is L in sizegl×(Lg0+Lg1+…+Lg(l-1)) All-zero matrix of, 0gl2Is L in sizegl×(Lg(l+1)+Lg(l+2)+…+Lg(M/2-1)) All-zero matrix of (I)glIs L in sizegl×LglThe identity matrix of (1);
Figure BDA0001316280000000031
(. h) is the ith analysis filter hlLhlFor the first analysis filter hlLength of (h)lIs the l analysis filter; when l is 1,2, …, M/2-2, hl＝[0hl1,Ihl,0hl2]h; when l is 0, hl＝[Ihl,0hl2]h; when l is M/2-1, hl＝[0hl1,Ihl]h; h is an analysis filterbank, Ohl1Is L in sizehl×(Lh0+Lh1+…+Lh(l-1)) All-zero matrix of, Ohl2Is L in sizehl×(Lh(l+1)+Lh(l+2)…+Lh(M/2-1)) All-zero matrix of (I)hlIs L in sizehl×LhlThe identity matrix of (1); t represents transposition; lambda [ alpha ]kFor the k-th frequency discrete point,
Figure BDA0001316280000000032
bkis a full reconstruction value;rto reconstruct the error tolerance; a isT(λk) For the k frequency discrete point lambdakThe response vector of (a); g1(0) Is the 1 st synthesize filter g1The value at frequency zero; n +1 is the number of frequency discrete points; m is the number of channels;
and 3, forming an M-channel diagram filter bank by the analysis filter h obtained in the step 1 and the synthesis filter g obtained in the step 2.
In the above steps 1 and 2, the analysis filter and the synthesis filter are effectively solved by using a semi-positive definite programming solver SDPT 3.
In the above scheme, the analysis weight α is equal to the comprehensive weight β.
In the above scheme, the number M of channels is an even number greater than 2.
In the above scheme, λkHas a value range of [0,2]]。
In the above scheme, the value b is completely reconstructedk＝2。
Compared with the prior art, the method utilizes a two-step method to carry out target optimization, and simultaneously comprehensively considers the spectrum selectivity and the complete reconstruction condition of the graph filter bank, thereby ensuring that the overall performance of the graph filter bank is improved under the condition of smaller reconstruction error. The invention provides a simple and effective solution for reducing the reconstruction error of the image filter bank and realizing the recovery reconstruction of the signal.
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Fig. 1 is a basic structure of an M-channel oversampled picture filter bank.
Fig. 2 shows the magnitude response of the resulting filter of the graph after optimization in example 1 of the present invention.
Fig. 3 is the magnitude response of the resulting graph filter after optimization in example 2 of the present invention.
FIG. 4 is a simulation result of a denoising experiment performed by using the graph filter system obtained in example 2 and the graph filter system obtained by the conventional method according to the present invention. Wherein (a) is an input signal; (b) is a noise signal (σ 1/2); (c) de-noised output signals for a graph filter bank designed based on prior method 3 (critical sampling biorthogonal); (d) the output signal of the graph filter bank designed by the invention after denoising.
Detailed Description
For ease of understanding, the following describes the present invention in further detail by taking a four-channel filter bank as an example.
Analysis subband filter H for filter bank of M-channel mapiAnd a synthesis subband filter GiExpressed as a function with respect to the analysis spectral kernel and the synthesis spectral kernel, respectively, representing the full reconstruction condition of the graph filter bank; namely, it is
Figure BDA0001316280000000041
In the formula, HiFor the ith analysis subband Filter, GiFor the ith synthesis subband filter, hi(lambda) is the value of the ith analysis subband filter spectral kernel at the characteristic root lambda, gi(λ) is the value of the ith synthesis subband filter spectral kernel at the eigenroot λ, λ is the eigenroot of the laplacian matrix of the graph G, σ (G) is the eigenspace formed by all the eigenroots 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.
If g isk(λ)＝hk+M/2(2-λ),gk+M/2(λ)＝hk(2- λ), then the full reconstruction condition can be expressed as:
Figure BDA0001316280000000042
in the formula, λ ∈ [0,2], λ is a continuous variable.
The analysis spectral kernel and the synthesis spectral kernel of the four-channel map filter bank with respect to the subband filter are represented as:
Figure BDA0001316280000000043
in the formula, Lh0,Lh1,Lg0,Lg1Respectively representing analysis filters h0,h1And a synthesis filter g0,g1Length of (d).
Figure BDA0001316280000000044
Notation and representation of 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 a low pass filter can be measured by integrating:
Figure BDA0001316280000000045
when i is 0, a is 0, b is λpd0When the above formula is EP(h0) I.e. h0The energy of the passband ripple of (a); when i is 1, a is lambdapd1,b＝λpd2When the above formula is EP(h1) I.e. h1The energy of the passband ripple. The stopband attenuation is determined by the stopband energy:
Figure BDA0001316280000000051
Figure BDA0001316280000000052
in a four-channel filter bank, if g0(λ)＝h2(2-λ)，g1(λ)＝h3(2-λ)，g2(λ)＝h0(2-λ)，g3(λ)＝h1(2- λ), then the graph filter bank is biorthogonal. The full reconstruction condition can be expressed as:
g0(λ)h0(λ)+g0(2-λ)h0(2-λ)+g1(λ)h1(λ)+g1(2-λ)h1(2-λ)＝2 (8)
for convenience of presentation, note
Figure BDA0001316280000000053
The full reconstruction condition can be expressed as:
Figure BDA0001316280000000054
the reconstruction error can be expressed as E (λ) ═ l (λ) + l (2- λ) -2, and it is desirable in the design that the reconstruction error be as small as possible.
The design problem of the graph filter bank can be summarized into a band-constrained optimization problem by fully considering the spectral characteristics and the complete reconstruction conditions of the graph filter bank, and the graph filter bank is designed by a two-step method. The two constraint optimization problems are both semi-definite planning problems and can be solved effectively. Design in-figure filter g0,g1With respect to the filter g0,g1For simplicity and ease of calculation, let h ═ h0；h1]；g＝[g0；g1](ii) a The above-mentioned indices can be converted into functions with respect to the filters h, g, and the optimal solution is solved by a semi-positive programming.
Fig. 1 shows an oversampling map filter bank system with M channels, and a design method based on the above structure, that is, an optimal design method of an M-channel oversampling map filter bank based on convex optimization, includes the following steps:
the first step is as follows: designing an analysis filter, taking the passband ripple and the stopband energy of an analysis filter bank as objective functions, solving the analysis filter which enables the passband distortion and the stopband energy to be minimum under the constraint condition of 3dB, wherein the optimization is a semi-positive definite programming problem and can be effectively solved.
Figure BDA0001316280000000061
3dB constraint, h0(0.5)＝1,h1(0.5)＝1,h1(1) The weights are 1 and α, α is usually β, and for easy solution, it is remembered
h＝[h0；h1]；h0＝[I0,0]h＝B0h；h1＝[0,I1]h＝B1h (11)
Wherein, B0Is L in sizeh0×(Lh0+Lh1) Matrix of (A), B1Is L in sizeh1×(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 as follows:
Figure BDA0001316280000000062
the second step is that: and taking the solved analysis filter as a known condition, and under the condition of complete reconstruction constraint, taking the maximum stop band attenuation of the synthesis filter into consideration, and taking the stop band energy of the synthesis filter bank as an objective function to solve the synthesis filter.
Figure BDA0001316280000000063
Wherein
Figure BDA0001316280000000064
rIs 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 convenience of calculation, the following substitutions are made:
g＝[g0；g1]；g0＝[I0,0]g＝C0g；g1＝[0,I1]g＝C1g； (14)
wherein, C0Is L in sizeg0×(Lg0+Lg1) Matrix of C1Is L in sizeg1×(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 BDA0001316280000000065
wherein b isk2, k 0, …, N
Figure BDA0001316280000000066
Vector aT(λk) Is represented as follows:
aT(λk)＝d(λk)+d(2-λk),k＝0,…,N (16)
the optimization problems are all semi-definite programming problems, can be effectively solved through a semi-definite programming solver SDPT3, 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 map filter bank based on convex optimization comprises the following steps:
step 1, designing an analysis filter of an M-channel diagram filter bank in consideration of spectral characteristics, and solving the analysis filter which enables the pass-band distortion and the stop-band energy of the analysis filter bank to be minimum by using the pass-band ripple and the stop-band energy of the analysis filter bank as objective functions through a semi-positive definite programming solver under the constraint condition of 3 dB; the optimization problem is that
Figure BDA0001316280000000071
Figure BDA0001316280000000072
i＝0,1,…,M/2-1；j＝1,2,…,M/2-1
In the formula, Ep(h) Analyzing the passband ripple energy of the filter; es(h) To analyze the stop band energy of the filter, α is the analysis weight, h is the analysis filter bank, h is h [ h ]0；h1；…；hM/2-1]；
Figure BDA0001316280000000073
(. h) is the ith analysis filter hiLhiFor the ith analysis filter hiLength of (h)iIs the ith analysis filter; when i is 1,2, …, M/2-2, hi＝[0i1,Ii,0i2]h; when i is 0, hi＝[Ii,0i2]h; when i is M/2-1, hi＝[0i1,Ii]h；0i1Is L in sizehi×(Lh0+Lh1+…+Lh(i-1)) All-zero matrix of, 0i2Is L in sizehi×(Lh(i+1)+Lh(i+2)…+Lh(M/2-1)) All-zero matrix of (I)iIs L in sizehi×LhiThe identity matrix of (1);
Figure BDA0001316280000000074
(. h) is the jth analysis filter hjLhjFor the jth analysis filter hjLength of (h)jFor the jth analysis filter, h is given when j is 1,2, …, M/2-2j＝[0j1,Ij，0j2]h; when j is M/2-1, hj＝[0j1,Ij]h；0j1Is L in sizehj×(Lh0+Lh1+…+Lh(j-1)) All-zero matrix of, 0j2Is L in sizehj×(Lh(j+1)+Lh(j+2)…+Lh(M/2-1)) All-zero matrix of (I)jIs L in sizehj×LhjThe identity matrix of (1); t represents transposition; m is the number of channels;
step 2, taking the analysis filter solved in the step 1 as a known condition, designing a comprehensive filter of the M-channel diagram filter bank from the aspect of complete reconstruction characteristics, and under the constraint condition of complete reconstruction, solving the comprehensive filter which enables the stop band attenuation of the comprehensive filter to be maximum by taking the stop band energy of the comprehensive filter bank as an objective function through a semi-positive definite programming solver; the optimization problem is that
Figure BDA0001316280000000075
s.t.|aT(λk)g-bk|≤r；g1(0)＝0
Figure BDA0001316280000000076
k＝0,1,…,N；l＝0,1,…,M/2-1
In the formula, Es(g) The stop band energy of the synthesis filter, β the synthesis weight, g the synthesis filter set, g ═ g0；g1；…；gM/2-1]；
Figure BDA0001316280000000077
(. h) is the first synthesis filter glLglFor the first synthesis filter glLength of (g)lIs the first synthesis filter, g when l is 1,2l＝[0gl1,Igl,0gl2]g; when l is 0, gl＝[Igl,Ogl2]g; when l is M/2-1, gl＝[Ogl1,Igl]g；Ogl1Is L in sizegl×(Lg0+Lg1+…+Lg(l-1)) All-zero matrix of, Ogl2Is L in sizegl×(Lg(l+1)+Lg(l+2)+…+Lg(M/2-1)) All-zero matrix of (I)glIs L in sizegl×LglThe identity matrix of (1);
Figure BDA0001316280000000078
(. h) is the ith analysis filter hlLhlFor the first analysis filter hlIs longDegree of, hlIs the l analysis filter; when l is 1,2, …, M/2-2, hl＝[0hl1,Ihl,0hl2]h; when l is 0, hl＝[Ihl,0hl2]h; when l is M/2-1, hl＝[0hl1,Ihl]h; h is the analysis filterbank, 0hl1Is L in sizehl×(Lh0+Lh1+…+Lh(l-1)) All-zero matrix of, 0hl2Is L in sizehl×(Lh(l+1)+Lh(l+2)…+Lh(M/2-1)) All-zero matrix of (I)hlIs L in sizehl×LhlThe identity matrix of (1); t represents transposition; lambda [ alpha ]kFor the k-th frequency discrete point,
Figure BDA0001316280000000081
bkis a full reconstruction value;rto reconstruct the error tolerance; a isT(λk) For the k frequency discrete point lambdakThe response vector of (a); g1(0) Is the 1 st synthesize filter g1The value at frequency zero; n +1 is the number of frequency discrete points; m is the number of channels;
and 3, forming an M-channel diagram filter bank by the analysis filter h obtained in the step 1 and the synthesis filter g obtained in the step 2.
The optimization problems are all semi-definite programming problems, can be effectively solved through a semi-definite programming solver SDPT3, and the solved graph filter is a real-valued function of lambda.
The performance of the present invention is illustrated by the following specific simulation examples.
Simulation example 1:
design a graph filter bank with the parameters of Lh0＝12，Lh1＝12，Lg0＝11，Lg1＝11，λpd0＝0.4，λpd1＝0.6，λpd2＝0.8，λsd0＝0.75，λsd1＝0.25，λsd2＝1.3，α＝0.1，r＝10-9The resulting magnitude response of the filter bank of the graph is shown in FIG. 2. PR in the graph denotes l (λ) + l (2- λ), λ ∈ [0,2]]The value of (a) is selected,reconstruction errors for the metric map filter bank. The maximum reconstruction error and the orthogonality value obtained by simulation calculation are respectively Emax＝3.9346×10-9Table 1 shows the reconstruction performance and boundary ratio of the graph filter bank of the present invention compared to the graph filter bank of the prior art method 1 (near orthogonal M-channel oversampling) design, under the same graph filter bank length and operating environment, Θ being 0.9353.
TABLE 1
Figure BDA0001316280000000082
RBTo be the boundary ratio,
Figure BDA0001316280000000083
taking R in simulationBAre compared with the maximum value of RBA value equal to 1 indicates that the filter bank is fully reconstructed, when the reconstruction characteristic of the filter bank is better. The maximum boundary ratio of the graph filter bank designed by the invention is 1, which is obviously superior to the graph filter bank designed by the existing method. In general, the reconstruction error of the image filter bank designed by the orthogonal method is larger than that of the biorthogonal filter bank, so the method has obvious advantages in reconstruction characteristics. It can be seen from table 1 that the bi-orthogonal filter bank designed by the present invention significantly reduces the reconstruction error. From the graph filter bank magnitude response, it can be seen that the overall spectral characteristics of the graph filter banks of the two algorithm designs are comparable. In the design of the graph filter bank, in order to recover the original signal without distortion as much as possible, the expected reconstruction error of the graph filter bank is smaller, and the design method has better reconstruction performance, so the overall performance of the graph filter bank designed by the invention is better.
Simulation example 2:
design a graph filter bank with the parameters of Lh0＝8，Lh1＝8，Lg0＝7，Lg1＝7，λpd0＝0.3，λpd1＝0.6，λpd2＝0.8，λsd0＝0.75，λsd1＝0.25，λsd2＝1.3，α＝0.1，r＝10-13To obtainThe resulting graph filter bank magnitude response is shown in fig. 3. Table 2 shows the reconstruction performance and orthogonality comparison of the graph filter bank of the present invention with the graph filter bank of the prior art method 2 (M-channel oversampling) design under the same graph filter bank length and operating environment.
TABLE 2
Figure BDA0001316280000000091
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 invention is obviously smaller than that of the existing method, and the orthogonality and the overall spectrum characteristic are slightly better than those of the existing method. Compared with the example 1, the parameter setting of the graph filter bank is changed, the spectrum characteristic of the synthesis filter is changed, but smaller reconstruction errors can be obtained, and the reconstruction errors are main performance indexes for measuring the overall performance of the graph filter bank, so that the overall performance of the graph filter bank designed by the invention is better. Proper relaxation of the reconstruction error margin results in a more spectrally selective graph filter, and the graph filter bank can be flexibly designed as required.
The simulation results of de-noising of the graph filter bank designed by the method of the present invention and the prior art 3 (critical sampling biorthogonal) are shown in fig. 4, wherein (a) is an input signal, when the noise standard deviation σ of the graph filter bank designed by the parameters of the above example 2 is 1/2; (b) is a noise signal (σ 1/2); (c) de-noised output signals for a graph filter bank designed based on prior method 3 (critical sampling biorthogonal); (d) the output signal of the graph filter bank designed by the invention after denoising. Table 3 gives the signal-to-noise ratio comparisons when the noise sigma takes different values.
TABLE 3
Figure BDA0001316280000000092
Compared with the prior method 3 (critical sampling biorthogonal), the method has higher signal-to-noise ratio under the same parameters and operation environment, and the signal-to-noise ratio is slightly worse by about 1.5dB compared with the prior method 2(M channel oversampling).

Claims (5)

1. The optimal design method of the M-channel oversampling map filter bank based on convex optimization is characterized by comprising the following steps:
step 1, designing an analysis filter of an M-channel diagram filter bank in consideration of spectral characteristics, and solving the analysis filter which enables the pass-band distortion and the stop-band energy of the analysis filter bank to be minimum by using the pass-band ripple and the stop-band energy of the analysis filter bank as objective functions through a semi-positive definite programming solver under the constraint condition of 3 dB; the optimization problem is that
Figure FDA0002371648700000011
Figure FDA0002371648700000012
In the formula, Ep(h) Analyzing the passband ripple energy of the filter; es(h) To analyze the stop band energy of the filter, α is the analysis weight, h is the analysis filter bank, h is h [ h ]0；h1；…；hM/2-1]；
Figure FDA0002371648700000013
For the ith analysis filter hiLhiFor the ith analysis filter hiLength of (h)iIs the ith analysis filter; when i is 1,2, …, M/2-2, hi＝[0i1,Ii,0i2]h; when i is 0, hi＝[Ii,0i2]h; when i is M/2-1, hi＝[0i1,Ii]h；0i1Is L in sizehi×(Lh0+Lh1+…+Lh(i-1)) All-zero matrix of, 0i2Is L in sizehi×(Lh(i+1)+Lh(i+2)…+Lh(M/2-1)) All-zero matrix of (I)iIs L in sizehi×LhiThe identity matrix of (1);
Figure FDA0002371648700000014
for the jth analysis filter hjLhjFor the jth analysis filter hjLength of (h)jFor the jth analysis filter, h is given when j is 1,2, …, M/2-2j＝[0j1,Ij，0j2]h; when j is M/2-1, hj＝[0j1,Ij]h；0j1Is L in sizehj×(Lh0+Lh1+…+Lh(j-1)) All-zero matrix of, 0j2Is L in sizehj×(Lh(j+1)+Lh(j+2)…+Lh(M/2-1)) All-zero matrix of (I)jIs L in sizehj×LhjThe identity matrix of (1); t represents transposition; m is the number of channels;
step 2, taking the analysis filter solved in the step 1 as a known condition, designing a comprehensive filter of the M-channel diagram filter bank from the aspect of complete reconstruction characteristics, and under the constraint condition of complete reconstruction, solving the comprehensive filter which enables the stop band attenuation of the comprehensive filter to be maximum by taking the stop band energy of the comprehensive filter bank as an objective function through a semi-positive definite programming solver; the optimization problem is that
Figure FDA0002371648700000015
s.t.|aT(λk)g-bk|≤r；g1(0)＝0
Figure FDA0002371648700000016
In the formula, Es(g) The stop band energy of the synthesis filter, β the synthesis weight, g the synthesis filter set, g ═ g0；g1；…；gM/2-1]；
Figure FDA0002371648700000021
For the first synthesis filter glLglFor the first synthesis filter glLength of (g)lIs the first synthesis filter, when l is 1,2, …, M/2-2, gl＝[0gl1,Igl,0gl2]g; when l is 0, gl＝[Igl,0gl2]g; when l is M/2-1, gl＝[0gl1,Igl]g；0gl1Is L in sizegl×(Lg0+Lg1+…+Lg(l-1)) All-zero matrix of, 0gl2Is L in sizegl×(Lg(l+1)+Lg(l+2)+…+Lg(M/2-1)) All-zero matrix of (I)glIs L in sizegl×LglThe identity matrix of (1);
Figure FDA0002371648700000023
for the first analysis filter hlLhlFor the first analysis filter hlLength of (h)lIs the l analysis filter; when l is 1,2, …, M/2-2, hl＝[0hl1,Ihl,0hl2]h; when l is 0, hl＝[Ihl,0hl2]h; when l is M/2-1, hl＝[0hl1,Ihl]h; h is the analysis filterbank, 0hl1Is L in sizehl×(Lh0+Lh1+…+Lh(l-1)) All-zero matrix of, 0hl2Is L in sizehl×(Lh(l+1)+Lh(l+2)…+Lh(M/2-1)) All-zero matrix of (I)hlIs L in sizehl×LhlThe identity matrix of (1); t represents transposition; lambda [ alpha ]kFor the k-th frequency discrete point,
Figure FDA0002371648700000022
bkis a full reconstruction value;rto reconstruct the error tolerance; a isT(λk) For the k frequency discrete point lambdakThe response vector of (a); g1(0) Is the 1 st synthesize filter g1The value at frequency zero; n +1 is the number of frequency discrete points; m is the number of channels;
step 3, an M-channel diagram filter bank can be formed by the analysis filter h obtained in the step 1 and the synthesis filter g obtained in the step 2;
wherein the number of channels M is an even number greater than 2.
2. The method as claimed in claim 1, wherein in steps 1 and 2, a semi-positive definite programming solver SDPT3 is used to solve the analysis filter and the synthesis filter effectively.
3. The optimal design method for the M-channel oversampled filter bank based on convex optimization as claimed in claim 1, wherein in step 1, the analysis weights α are equal to the synthesis weights β.
4. The method of claim 1, wherein λ is λkHas a value range of [0,2]]。
5. The optimal design method for M-channel oversampled filter banks based on convex optimization as claimed in claim 1, wherein the fully reconstructed value b isk＝2。
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