CN111835360B - Sparse signal reconstruction method based on basis pursuit denoising - Google Patents

Abstract

A sparse signal reconstruction method based on basis pursuit denoising is very sensitive to noise through reconstruction performance of compressed sensing, namely the reconstruction performance of the compressed sensing can be greatly reduced through introduction of the noise, and filtering of electrocardiosignal noise and signal reconstruction are achieved through basis pursuit denoising. The original signal is accurately reconstructed through a numerical optimization algorithm, original wave crest and wave trough information is obtained, the original electrocardiosignal is reconstructed by utilizing the sparsity characteristic of the electrocardiosignal, noise is effectively removed, and the authenticity of the electrocardiosignal is ensured.

Description

Sparse signal reconstruction method based on basis pursuit denoising

Technical Field

The invention relates to the technical field of electrocardiogram signal denoising, in particular to a sparse signal reconstruction method based on basis pursuit denoising

Background

The compressed sensing theory breaks through the traditional data sampling theory, realizes data compression while sampling data, can accurately reconstruct an original signal through a few sampling data with the sampling frequency lower than Nyquist, and greatly reduces the sampling amount, sampling time and storage cost of the data. Based on the prior knowledge of the sparse characteristics of the signals, the original signals can be accurately reconstructed by calculating a certain optimization problem.

The compressed sensing theory comprises three main contents: sparse representation of signals, design of measurement matrices, and design of reconstruction algorithms. The sparse representation of the signal is a priori premise and theoretical basis for the establishment of the theory; the design of the measurement matrix and the reconstruction algorithm is a key step of the application of the compressive sensing theory. The quality of the reconstruction algorithm directly affects the reconstruction accuracy and quality of the signal, and even determines the number of required measurement values to some extent. Therefore, the design of the reconstruction algorithm is the research focus and core content of the compressed sensing theory, and determines whether the original signal can be accurately reconstructed from the measured value or not, and also determines the key factors of the compressed sensing from the theory to the practical application.

With the rapid development of computer-related technologies, in the processing of cardiac electrical signals, the noise reduction processing of cardiac electrical signals becomes a hot problem for research. How to reduce various interferences in the process of detecting the electrocardio signals and improve the noise reduction performance, and how to recover the original signals from the complex noise signals as far as possible is a key problem of electrocardio signal processing.

Disclosure of Invention

In order to overcome the defects of the technology, the invention provides a method for effectively filtering noise of a electrocardiosignal and realizing the reconstruction of a clean signal based on a basis tracking method.

The technical scheme adopted by the invention for overcoming the technical problems is as follows:

a sparse signal reconstruction method based on basis pursuit denoising comprises the following steps:

a) establishing an L multiplied by N dimensional random Gaussian matrix gamma from L measured values in the N multiplied by 1 dimensional electrocardiosignals, orthogonalizing the random Gaussian matrix gamma to obtain an N multiplied by N dimensional matrix M, and calculating by using a computer according to a formula y which is Mx to obtain a sparse solution x, wherein y is an original electrocardiosignal;

b) adding noise to the original electrocardiosignal y to form the electrocardiosignal y with noise₁；

c) By the formula

Calculating to obtain the error minimum value of sparse solution x, wherein lambda regularization factor | · | | non-calculation₁ Represents 1 norm, | ·| non-conducting phosphor₂Represents a 2 norm;

d) by the formula H ═ I_N+δD′D]^-1Calculating to obtain sparse band matrix H in the formula

I_NIs a unit moment of length ND' is a transposed matrix of D, and delta is an optimization parameter;

e) by the formula

The parameter delta is optimized in an iterative manner, where u_i＝2-2cos(iπ/N)，i＝1,2,...,N；

f) Let x be u-v and b be HA' y by the formula

Calculating to obtain optimized noiseless electrocardiosignal x, wherein y₁' is y₁M 'is the transpose of M, N' is the transpose of N, I_2NIs an identity matrix of length N, I_2N＝(1，1,1......1)，I_2N' is I_2NB is HM' y₁，

By the formula

Calculating to obtain an intermediate matrix z, wherein z 'is the transpose of z, c' is the transpose of c,

further, step b) is performed by the formula y₁Calculating to obtain electrocardiosignal y with noise₁Where s is a noise signal.

The invention has the beneficial effects that: the reconstruction performance of the compressed sensing is very sensitive to noise, namely the reconstruction performance of the compressed sensing is greatly reduced by introducing the noise, and the filtering of electrocardiosignal noise and the reconstruction of signals are realized by basis tracking noise reduction. The original signal is accurately reconstructed through a numerical optimization algorithm, original wave crest and wave trough information is obtained, the original electrocardiosignal is reconstructed by utilizing the sparsity characteristic of the electrocardiosignal, noise is effectively removed, and the authenticity of the electrocardiosignal is ensured.

Drawings

FIG. 1 is a flow chart of the present invention;

FIG. 2 is a schematic diagram of the denoising effect of the electrocardiographic signal according to the present invention.

Detailed Description

The invention will be further explained with reference to fig. 1 and 2.

A sparse signal reconstruction method based on basis pursuit denoising comprises the following steps:

a) and extracting the length N of the electrocardiosignal by a formula N-length (y), wherein the sparsity is k, and the measurement frequency is L. The measurement matrix plays an important role in the compression feedback and signal reconstruction processes. The better the performance of the measurement matrix, the smaller the reconstructed mean square error. The stronger the nonlinear correlation of the measurement matrix column vector is, the higher the sparsity of the matrix elements is, and the better the signal reconstruction effect after compression feedback is. The measurement matrix M realizes that the N multiplied by 1 dimensional electrocardiosignals can be reconstructed from L measured values, the measurement matrix M is established according to the length N of the electrocardiosignals and the measurement times L, therefore, an L multiplied by N dimensional random Gaussian matrix gamma is established from the L measured values of the N multiplied by 1 dimensional electrocardiosignals, the random Gaussian matrix gamma is orthogonalized to obtain an N multiplied by N dimensional matrix M, and a sparse solution x is obtained by utilizing a computer through the calculation of a formula y Mx, wherein y is an original electrocardiosignal. Sparse representation of a signal means that when the signal is decomposed in a certain transform domain, it represents that the coefficients have only a few large coefficients, most of which are equal to zero or almost close to zero. The purpose of the sparse representation is to find the most sparse combinations of atoms from the redundant overcomplete dictionaries, thus obtaining the features that are the most essential of the original information.

b) The electrocardiosignal is expressed as a form with sparse waveform distribution on the whole, and the signal has certain periodicity. After the row differential calculation, the second-order differential calculation and the third-order differential calculation are carried out, so that the sparsity of the electrocardiosignals is more obvious. Therefore, the entire electrocardiographic signal can be regarded as a sparse signal. Because the electrocardiosignal can be interfered by various noise signals, the noise component is added on the basis, so that the original electrocardiosignal y is denoised to form the electrocardiosignal y with noise₁. The goal of the noise addition is to recover signal x from signal y.

c) The problem of recovering sparse signals can be translated into a problem of optimizing an objective function, as is well known, the L1 norm is representative of convex sparsity. Therefore, the sparse noise reduction problem is summarized as an optimization problem of the minimization problem at the L1 norm under data fidelity. Thus by the formula

Calculating to obtain the error minimum value of sparse solution x, wherein lambda regularization factor | · | | non-calculation₁ Represents 1 norm, | ·| non-conducting phosphor₂Representing a 2 norm. According to the signal model, expressing a non-convex optimization problem of sparse signal estimation as a basis tracking denoising problem model, and introducing regularization parameters and signal norm control errors and sparsity into the basis tracking denoising problem model.

d) By the formula H ═ I_N+δD′D]^-1Calculating to obtain sparse band matrix H in the formula

I_NIs an identity matrix with the length of N, D' is a transposed matrix of D, and delta is an optimization parameter. The function of delta is a compromise parameter between the filtering effect and the distortion of the peak value of the R wave, when the gain is too large, the noise removal effect is good, but the peak value of the R wave peak is affected, and when the gain is too small, the signal recovery effect is good and the noise removal effect is poor.

e) By the formula

The parameter delta is optimized in an iterative manner, where u_i＝2-2cos(iπ/N)，i＝1,2,...,N。

f) The basic goal of sparse reconstruction is to perform signal or image reconstruction through an optimization problem with fewer data samples. Regarding the sparse reconstruction process, an important aspect is how to efficiently and quickly reconstruct the original signal in the case of data interfered by noise. In the step, a quadratic programming method and the filter matrix in the previous step are adopted to realize the recovery of the sparse signal under the noise interference. Let x be u-v and b be HA' y by the formula

Calculating to obtain optimized noiseless electrocardiosignal x, wherein y₁' is y₁M 'is the transpose of M, N' is the transpose of N, I_2NIs an identity matrix of length N, I_2N＝(1，1,1......1)，I_2N' is I_2NB is HM' y₁，

By the formula

Calculating to obtain an intermediate matrix z, wherein z 'is the transpose of z, c' is the transpose of c,

the reconstruction performance of the compressed sensing is very sensitive to noise, namely the reconstruction performance of the compressed sensing is greatly reduced by introducing the noise, and the filtering of electrocardiosignal noise and the reconstruction of signals are realized by basis tracking noise reduction.

As can be seen from the graph 2, the original signal is accurately reconstructed through a numerical optimization algorithm, the original information of wave crests and wave troughs is obtained, the original electrocardiosignal is reconstructed by utilizing the sparsity characteristic of the electrocardiosignal, the noise is effectively removed, and the authenticity of the electrocardiosignal is ensured.

Further, step b) is performed by the formula y₁Calculating to obtain electrocardiosignal y with noise₁Where s is a noise signal.

Claims (2)

1. A sparse signal reconstruction method based on basis pursuit denoising is characterized by comprising the following steps:

a) establishing an L multiplied by N dimensional random Gaussian matrix gamma from L measured values in the N multiplied by 1 dimensional electrocardiosignals, orthogonalizing the random Gaussian matrix gamma to obtain an N multiplied by N dimensional matrix M, and calculating by using a computer according to a formula y which is Mx to obtain a sparse solution x, wherein y is an original electrocardiosignal;

b) for original electrocardio-letterAdding noise to the signal y to form an electrocardiosignal y with noise₁；

c) By the formula

Calculating to obtain the error minimum value of sparse solution x, wherein lambda regularization factor | · | | non-calculation₁Represents 1 norm, | ·| non-conducting phosphor₂Represents a 2 norm;

d) by the formula H ═ I_N+δD′D]^-1Calculating to obtain sparse band matrix H in the formula

I_NThe length is an identity matrix with the length of N, D' is a transposed matrix of D, and delta is an optimization parameter;

e) by the formula

The parameter delta is optimized in an iterative manner, where u_i＝2-2cos(iπ/N)，i＝1,2,...,N；

f) Let x be u-v and b be HA' y by the formula

Calculating to obtain optimized noiseless electrocardiosignal x, wherein y₁' is y₁M 'is the transpose of M, N' is the transpose of N, I_2NIs an identity matrix of length N, I_2N＝(1，1,1......1)，I_2N' is I_2NB is HM' y₁，

By the formula

Calculating to obtain an intermediate matrix z, wherein z 'is the transpose of z, c' is the transpose of c,

2. the sparse signal reconstruction method based on basis pursuit denoising of claim 1, wherein: in step b) by the formula y₁Calculating to obtain electrocardiosignal y with noise₁Where s is a noise signal.

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