CN106374935A - Block sparse signal reconstruction method adopting self-adaptive coupling parameters - Google Patents

Abstract

The invention belongs to the field of compressed sensing, and provides a block sparse signal reconstruction method adopting self-adaptive coupling parameters, used for improving the block sparse reconstruction precision under a condition that a block structure is unknown. The block sparse signal reconstruction method disclosed by the invention comprises the following steps of: introducing self-adaptive coupling parameters beta1 and beta2 at first, establishing a new coupling hierarchical bayesian model, and estimating a block sparse signal according to a maximum posteriori probability criterion; then, estimating a hyper-parameter and a coupling parameter by using an expectation maximization criterion; re-estimating the block sparse signal according to the hyper-parameter and the coupling parameter; and performing iteration till a stop condition is satisfied, so that a reconstructed block sparse signal is obtained. According to the block sparse signal reconstruction method disclosed by the invention, the self-adaptive coupling parameters are introduced, so that the coupling degree between various hyper-parameters is controlled by different coupling coefficients; therefore, non-zero signal elements are encouraged to gather to form a block structure with relatively high flexibility; and thus, the block sparse reconstruction precision is obviously increased.

Description

Block sparse signal reconstruction method adopting self-adaptive coupling parameters

Technical Field

The invention belongs to the field of compressed sensing, and provides a block sparse signal reconstruction method adopting self-adaptive coupling parameters, aiming at the problem of block sparse signal reconstruction with an unknown block structure.

Technical Field

Sparse signal reconstruction is an important research topic in the compressed sensing field. The traditional sparse reconstruction theory usually assumes that the distribution of non-zero terms in the sparse signal is random and arbitrary, but in practical application, the sparse signal often has certain block structure characteristics, which is called as a block sparse signal. For the problem of sparse signal reconstruction with a Block structure, among the existing methods, a Block Orthogonal Matching Pursuit (OMP) method and a hybrid l₂/l₁Norm minimization (Mixed l)₂/l₁Norm Minimization) method, etc., requires information of the size or distribution of known blocks, which is often unknown in the actual signal reconstruction. For the condition of unknown Block structure, the Expanded Block Sparse Bayes (EBSBL) method assumes that the signal is formed by stacking blocks with the same structure, each Block obeys Gaussian distribution, and the posterior covariance matrix of the signal is Expanded into a Block diagonal matrix structure, so that the traditional Block Bayes method is applied to Sparse reconstruction; the Pattern-Coupled sparse bayesian (PCSBL) method uses Coupled superparameters to control signal elements, so that a block sparse structure is generated, that is, the sparsity of a signal element is not only related to its own superparameters, but also influenced by neighboring superparameters, but the method only introduces a fixed coupling parameter to control the influence degree of the signal element on neighboring elements, so that the performance of the signal element is limited.

Disclosure of Invention

The invention aims to provide a block sparse signal reconstruction method adopting self-adaptive coupling parameters, which is used for improving the accuracy of block sparse reconstruction under the condition of unknown block structure. According to the invention, a new coupling hierarchical Bayes model is established, a plurality of self-adaptive coupling parameters are introduced, and the coupling degree of each hyper-parameter is controlled by different coupling coefficients, so that the aggregation of non-zero signal elements is encouraged to form a block structure with higher flexibility, and the accuracy of block sparse reconstruction is improved.

In order to achieve the purpose, the invention adopts the following solution:

the method comprises the following specific steps:

a block sparse signal reconstruction method adopting adaptive coupling parameters comprises the following steps:

step 1, setting the received signal as: y is Ax + n, where a represents the sensing matrix, x represents the block sparse signal, and n represents gaussian noise;

initialization hyper-parameter α, coupling parameter β₁And β₂And the noise variance λ^-1Wherein α ═ α₁,α₁,…,α_n]，Each element in x obeys a gaussian distribution:

$p\left(x_i\right|{\mathrm{\α}}_i,{\mathrm{\α}}_{i-1},{\mathrm{\α}}_{i+1},{\mathrm{\β}}_{i-1}^i,{\mathrm{\β}}_{i+1}^i)=N\left(x_i\right|0,{\left({\mathrm{\α}}_i+{\mathrm{\β}}_{i-1}^i{\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i+1}^i{\mathrm{\α}}_{i+1}\right)}^{-1}),$

where i is 1,2, …, n, and the block sparse signal x has n elements;

gaussian noise n obeys a mean value of 0 and a covariance matrix of lambda^-1I gaussian distribution, then the posterior mean of x is deduced to be: mu-lambda sigma A^Ty, the covariance matrix is: sigma ═(λA^TA+S)^-1Where S is the diagonal matrix:

$S=diag({\mathrm{\α}}_1+{\mathrm{\β}}_2^1{\mathrm{\α}}_2,{\mathrm{\α}}_2+{\mathrm{\β}}_1^2{\mathrm{\α}}_1+{\mathrm{\β}}_3^2{\mathrm{\α}}_3,...,{\mathrm{\α}}_n+{\mathrm{\β}}_{n-1}^n{\mathrm{\α}}_{n-1});$

an estimate of the sparse signal x is obtained according to the maximum a posteriori criterion as

Step 2, estimating hyper-parameters and coupling parameters according to an expectation maximization criterion:

${\mathrm{\α}}_i=\frac{{\mathrm{\κ}}_{\mathrm{\α}}}{0.5({\mathrm{\ω}}_i+{\mathrm{\β}}_i^{i+1}{\mathrm{\ω}}_{i+1}+{\mathrm{\β}}_i^{i-1}{\mathrm{\ω}}_{i-1})+b}$

${\mathrm{\β}}_i^{i+1}=\frac{{\mathrm{\κ}}_{{\mathrm{\β}}_1}}{{\mathrm{\α}}_i{\mathrm{\ω}}_{i+1}}$

${\mathrm{\β}}_i^{i-1}=\frac{{\mathrm{\κ}}_{{\mathrm{\β}}_2}}{{\mathrm{\α}}_i{\mathrm{\ω}}_{i-1}}$

$\mathrm{\λ}=\frac{m+2(c-1)}{\left|\right|y-A\mathrm{\μ}|{|}_2^2+Tr\left({\mathrm{\ΣA}}^{T}A\right)+2d},$

wherein,κ_α∈(0.5,2)，a, b, c and d are parameters in gamma distribution obeyed by α and lambda, and m is the number of measured values;

step 3, according to the parameter set { α of step 2 estimation₁,β₂λ } and the formula obtained in step 1Re-estimating the block sparse signal to obtain

Step 4, judgmentWhether the number of iterations is less than a specified threshold or not, and if so, outputting a reconstructed signalOtherwise, go back to step 2 to continue the iteration.

The invention first introduces adaptive coupling parameters β₁And β₂Establishing a new coupling layered Bayes model, and estimating block sparse signals according to the maximum posterior probability criterion; then, applying an expectation maximization criterion to estimate the hyperparameter and the coupling parameter; re-estimating the block sparse signal according to the hyper-parameters and the coupling parameters; and iterating until the stop condition is met to obtain the reconstructed block sparse signal. The invention introduces the self-adaptive coupling parameters, so that the coupling degree between each hyper-parameter is controlled by different coupling coefficients, thereby encouraging the non-zero signal elements to gather to form a block structure with higher flexibility, and obviously improving the accuracy of block sparse reconstruction.

Drawings

Fig. 1 is a schematic flow chart of a block sparse signal reconstruction method using adaptive coupling parameters according to the present invention.

FIG. 2 is a comparison graph of Root Mean Square Error (R MSE) of the present invention with the PCSBL method and the EBSBL method as a function of signal sparsity K.

FIG. 3 is a graph comparing the Root Mean Square Error (R MSE) of the present invention with the PCSBL method and the EBSBL method as a function of the number m of measured values.

Detailed Description

The present invention will be described in further detail with reference to the following specific examples and drawings, but the present invention is not limited to the examples.

The basic idea of the invention is to establish a new coupling layered Bayes model, introduce a plurality of adaptive coupling parameters, couple the hyper-parameters of signal prior distribution with each other, and encourage the aggregation of non-zero signal elements to form a block structure with higher flexibility.

A block sparse signal reconstruction method adopting adaptive coupling parameters comprises the following steps:

step 1, estimating a block sparse signal according to observation data, a hypothesis model, initialized hyper-parameters and initialized coupling parameters;

let y be Ax + n, where A ∈ R^m×nRepresenting a sensing matrix, y ∈ R^m×1Representing the received signal, x ∈ R^n×1Representing block sparse signals, n ∈ R^m×1Representing gaussian noise;

assuming that each element in the signal x obeys a gaussian distribution:

$p\left(x_i\right|{\mathrm{\α}}_i,{\mathrm{\α}}_{i-1},{\mathrm{\α}}_{i+1},{\mathrm{\β}}_{i-1}^i,{\mathrm{\β}}_{i+1}^i)=N\left(x_i\right|0,{\left({\mathrm{\α}}_i+{\mathrm{\β}}_{i-1}^i{\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i+1}^i{\mathrm{\α}}_{i+1}\right)}^{-1})$

wherein i is 1,2, …, n, α_iFor controlling x in a conventional sparse reconstruction method_iA hyper-parameter of sparsity whose probability density obeys a gamma distribution:in the present invention, x is controlled_iSparsity variables other than just α_iThe sparsity of which is also subject to the hyperparameters α of neighboring coefficients_i-1And α_i+1The influence of (a) on the performance of the device,are respectively α_i-1And α_i+1Pair α_iSetting the hyper-parameter coupling coefficient to obey the uniform distribution from 0 to 1; such that the element x in the block sparse signal_iThe sparsity ofTo determine whenWhen it goes to infinity, x_iIs 0, and the setting vector α is [ α ]₁,α₁,…,α_n]，And

assuming that the noise n obeys a mean of 0 and the covariance matrix is λ^-1The Gaussian distribution of I can obtain p (y | x; lambda) with mean value of Ax and covariance matrix of lambda according to the mathematical model of the received signal^-1A Gaussian distribution of I, i.e. p (y | x; λ) ═ N (Ax, λ)^-1I) (ii) a These conditions can be used to obtain a posterior probability density of x, where p (x | y) ═ N (μ, Σ)

μ＝λΣA^Ty，Σ＝(λA^TA+S)^-1，

S is a diagonal matrix:

an estimate of the sparse signal x is obtained according to the maximum a posteriori criterion as

Step 2, estimating hyper-parameters and coupling parameters according to an expectation maximization criterion:

wherein Q ═ E_x|y[log p(α,β₁,β₂,λ|y)]Known as p (α)₁,β₂,λ|y)∝p(y,α,β₁,β₂λ), with x as the hidden variable, the Q function can be re-expressed as;

Q＝E_x|y[ln p(x,α,β₁,β₂)]+E_x|y[ln p(y,λ|x)]

ignoring the coefficients that are not related to the hyperparameter, the first term in Q can be rewritten as:

$\begin{array}{c}{Q}_1=E_{x|y}\left\{\ln \[p\left(\mathrm{\α}\right)p\left({\mathrm{\β}}_1\right)p\left({\mathrm{\β}}_2\right)p\left(x|\mathrm{\α},{\mathrm{\β}}_1,{\mathrm{\β}}_2\right)\]\right\}\\ =\underset{i=1}{\overset{n}{\mathrm{\Σ}}}(a{\mathrm{ln\α}}_i-{\mathrm{b\α}}_i+\frac{1}{2}\ln \left({\mathrm{\α}}_i+{\mathrm{\β}}_{i-1}^i{\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i+1}^i{\mathrm{\α}}_{i+1}\right)\\ -\frac{1}{2}\left({\mathrm{\α}}_i+{\mathrm{\β}}_{i-1}^i{\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i+1}^i{\mathrm{\α}}_{i+1}\right)\left({\mathrm{\μ}}_i^2+{\mathrm{\Σ}}_{i,i}\right))\end{array}$

wherein, mu_iI-th element of mean value mu, sigma_i,iFor the ith element on the covariance matrix sigma diagonal, use is made of the probability density p (λ) of λ^-1(c)d^cλ^c-1e^-dλThe second term in Q can be expressed as:

$\begin{array}{c}{Q}_2=E_{x|y}\{ln\[p\left(y\right|x,\mathrm{\λ})p\left(\mathrm{\λ}\right)\]\}\\ =(\frac{m}{2}+c-1)\ln \mathrm{\λ}-\frac{\mathrm{\λ}}{2}{E}_{x|y}\[\left|\right|y-Ax|{|}_2^2\]-d\mathrm{\λ}\end{array}$

will Q₁Are respectively paired with α_i，Andand (3) solving partial derivatives, wherein the obtained partial derivatives are as follows:

$\begin{array}{c}\frac{\∂Q_1}{\∂{\mathrm{\α}}_i}=\frac{a}{{\mathrm{\α}}_i}-b+\frac{1}{2\left({\mathrm{\α}}_i+{\mathrm{\β}}_{i-1}^i{\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i+1}^i{\mathrm{\α}}_{i+1}\right)}-\frac{{\mathrm{\ω}}_i}{2}\\ +\frac{{\mathrm{\β}}_i^{i-1}}{2\left({\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i-2}^{i-1}{\mathrm{\α}}_{i-2}+{\mathrm{\β}}_i^{i-1}{\mathrm{\α}}_i\right)}-\frac{{\mathrm{\ω}}_{i-1}}{2}{\mathrm{\β}}_i^{i-1}\\ +\frac{{\mathrm{\β}}_i^{i+1}}{2\left({\mathrm{\α}}_{i+1}+{\mathrm{\β}}_i^{i+1}{\mathrm{\α}}_i+{\mathrm{\β}}_{i+2}^{i+1}{\mathrm{\α}}_{i+2}\right)}-\frac{{\mathrm{\ω}}_{i+1}}{2}{\mathrm{\β}}_i^{i+1}\\ \frac{\∂Q_1}{\∂{\mathrm{\β}}_i^{i-1}}=\frac{{\mathrm{\α}}_i}{2\left({\mathrm{\α}}_{i-1}+{\mathrm{\β}}_{i-2}^{i-1}{\mathrm{\α}}_{i-2}+{\mathrm{\β}}_i^{i-1}{\mathrm{\α}}_i\right)}-\frac{{\mathrm{\ω}}_{i-1}}{2}{\mathrm{\α}}_i\\ \frac{\∂Q_1}{\∂{\mathrm{\β}}_i^{i+1}}=\frac{{\mathrm{\α}}_i}{2\left({\mathrm{\α}}_{i+1}+{\mathrm{\β}}_i^{i+1}{\mathrm{\α}}_i+{\mathrm{\β}}_{i+2}^{i+1}{\mathrm{\α}}_{i+2}\right)}-\frac{{\mathrm{\ω}}_{i+1}}{2}{\mathrm{\α}}_i\end{array}$

wherein,let the three partial derivatives be 0, apply inequality Andα is obtained_i，Andthe value range is as follows:

$\frac{a}{0.5\left({\mathrm{\&omega;}}_i+{\mathrm{\&beta;}}_i^{i+1}{\mathrm{\&omega;}}_{i+1}+{\mathrm{\&beta;}}_i^{i-1}{\mathrm{\&omega;}}_{i-1}\right)+b}<{\mathrm{\&alpha;}}_i<\frac{a+1.5}{0.5\left({\mathrm{\&omega;}}_i+{\mathrm{\&beta;}}_i^{i+1}{\mathrm{\&omega;}}_{i+1}+{\mathrm{\&beta;}}_i^{i-1}{\mathrm{\&omega;}}_{i-1}\right)+b}$

$0<{\mathrm{\&beta;}}_i^{i-1}<\frac{1}{{\mathrm{\&omega;}}_{i-1}{\mathrm{\&alpha;}}_i}$

$0<{\mathrm{\&beta;}}_i^{i+1}<\frac{1}{{\mathrm{\&omega;}}_{i+1}{\mathrm{\&alpha;}}_i}$

through experiments, in the embodiment, a is 0.5, and b is 10^-4And updated α using the following formula_i，And

${\mathrm{\&alpha;}}_i=\frac{{\mathrm{\&kappa;}}_{\mathrm{\&alpha;}}}{0.5({\mathrm{\&omega;}}_i+{\mathrm{\&beta;}}_i^{i+1}{\mathrm{\&omega;}}_{i+1}+{\mathrm{\&beta;}}_i^{i-1}{\mathrm{\&omega;}}_{i-1})+b}$

${\mathrm{\&beta;}}_i^{i-1}=\frac{{\mathrm{\&kappa;}}_{{\mathrm{\&beta;}}_1}}{{\mathrm{\&omega;}}_{i-1}{\mathrm{\&alpha;}}_i}$

${\mathrm{\&beta;}}_i^{i+1}=\frac{{\mathrm{\&kappa;}}_{{\mathrm{\&beta;}}_2}}{{\mathrm{\&omega;}}_{i+1}{\mathrm{\&alpha;}}_i}$

wherein, κ_α∈(0.5,2)，

Will Q₂Taking the derivative of λ and making the derivative 0, the value of λ is found to be

$\mathrm{\&lambda;}=\frac{m+2(c-1)}{\left|\right|y-A\mathrm{\&mu;}|{|}_2^2+Tr\left({\mathrm{\&Sigma;A}}^{T}A\right)+2d}$

Wherein, the parameters c and d are both 10^-4；

Step 3, according to the parameter set { α of step 2 estimation₁,β₂λ } and the formula obtained in step 1Re-estimating the block sparse signal to obtain

Step 4, judgmentWhether the number of iterations is less than a specified threshold or not, and if so, outputting a reconstructed signalOtherwise, go back to step 2 to continue the iteration.

The effect of the present invention is illustrated by the simulation results in fig. 2 and 3:

α was initialized to 1 and β was set in the simulation₁And β₂Initialization is 0 and variance of noise is initialized to lambda^-11 is ═ 1; setting the number n of signal elements as 100; the number L of the fixed nonzero blocks is 3, and the signal-to-noise ratio SNR is 2 dB; monte Carlo experiments were performed 500 times. The performance of each method was compared by the root mean square error RMSE, which is defined as:

whereinIs the ith element of the real signal in the kth Monte Carlo experiment;

fig. 2 and fig. 3 are comparison graphs of the root mean square error of the present invention with other methods (PCSBL method and EBSBL method) according to the variation of the signal sparsity K and the number m of measured values, respectively, and it can be seen from the graphs that the present invention has the smallest root mean square error compared with the block sparse reconstruction method proposed recently, which shows that the present invention has higher reconstruction accuracy compared with other methods.

Where specific embodiments of the invention are described above, any feature disclosed in this specification may, unless stated otherwise, be replaced by alternative features serving the same, equivalent or similar purpose; all of the disclosed features, or all of the method or process steps, may be combined in any combination, except mutually exclusive features and/or steps.

Claims (1)

1. A block sparse signal reconstruction method adopting adaptive coupling parameters comprises the following steps:

step 1, setting the received signal as: y is Ax + n, where a represents the sensing matrix, x represents the block sparse signal, and n represents gaussian noise;

initialization hyper-parameter α, coupling parameter β₁And β₂And the noise variance λ^-1Wherein α ═ α₁,α₁,…,α_n]，Each element in x obeys a gaussian distribution:

$p\left(x_i\right|{\mathrm{\&alpha;}}_i,{\mathrm{\&alpha;}}_{i-1},{\mathrm{\&alpha;}}_{i+1},{\mathrm{\&beta;}}_{i-1}^i,{\mathrm{\&beta;}}_{i+1}^i)=N\left(x_i\right|0,{\left({\mathrm{\&alpha;}}_i+{\mathrm{\&beta;}}_{i-1}^i{\mathrm{\&alpha;}}_{i-1}+{\mathrm{\&beta;}}_{i+1}^i{\mathrm{\&alpha;}}_{i+1}\right)}^{-1}),$

where i is 1,2, …, n, and the block sparse signal x has n elements;

gaussian noise n obeys a mean value of 0 and a covariance matrix of lambda^-1I gaussian distribution, then the posterior mean of x is deduced to be: mu-lambda sigma A^Ty, the covariance matrix is: Σ ═ λ a^TA+S)^-1Where S is the diagonal matrix:

$S=diag({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_2^1{\mathrm{\&alpha;}}_2,{\mathrm{\&alpha;}}_2+{\mathrm{\&beta;}}_1^2{\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_3^2{\mathrm{\&alpha;}}_3,...,{\mathrm{\&alpha;}}_n+{\mathrm{\&beta;}}_{n-1}^n{\mathrm{\&alpha;}}_{n-1});$

an estimate of the sparse signal x is obtained according to the maximum a posteriori criterion as

Step 2, estimating hyper-parameters and coupling parameters according to an expectation maximization criterion:

${\mathrm{\&alpha;}}_i=\frac{{\mathrm{\&kappa;}}_{\mathrm{\&alpha;}}}{0.5(s_i+{\mathrm{\&beta;}}_i^{i+1}{s}_{i+1}+{\mathrm{\&beta;}}_i^{i-1}{s}_{i-1})+b}$

${\mathrm{\&beta;}}_i^{i+1}=\frac{{\mathrm{\&kappa;}}_{{\mathrm{\&beta;}}_1}}{{\mathrm{\&alpha;}}_i{s}_{i+1}}$

${\mathrm{\&beta;}}_i^{i-1}=\frac{{\mathrm{\&kappa;}}_{{\mathrm{\&beta;}}_2}}{{\mathrm{\&alpha;}}_i{s}_{i-1}}$

$\mathrm{\&lambda;}=\frac{m+2(c-1)}{\left|\right|y-A\mathrm{\&mu;}|{|}_2^2+Tr\left({\mathrm{\&Sigma;A}}^{T}A\right)+2d},$

wherein,κ_α∈(0.5,2)，a, b, c and d are parameters in gamma distribution obeyed by α and lambda, and m is the number of measured values;

step 3, according to the parameter set { α of step 2 estimation₁,β₂λ } and the formula obtained in step 1Re-estimating the block sparse signal to obtain

Step 4, judgmentWhether the number of iterations is less than a specified threshold or not, and if so, outputting a reconstructed signalOtherwise, go back to step 2 to continue the iteration.