CN112515637A - Electrocardiosignal noise reduction method based on group sparsity characteristic - Google Patents

Abstract

A group sparsity characteristic-based electrocardiosignal noise reduction method fully utilizes the group sparsity characteristic of electrocardiosignals and promotes the group sparsity of understanding by reasonably selecting a signal group sparsity measurement function. Based on the characteristics of a banded system, only one tri-diagonal equation set needs to be solved for each iteration, the effectiveness and the calculation efficiency of the algorithm are improved, the method is suitable for non-overlapped group sparse signals, and when the group sparse signals are overlapped, the method is still effective. By selecting the strictly convex cost function, the convergence of the algorithm is ensured, the only optimal solution is finally converged, and the accurate and efficient noise reduction is achieved while the waveform characteristics of the original electrocardiosignal are maintained.

Description

Electrocardiosignal noise reduction method based on group sparsity characteristic

Technical Field

The invention relates to the technical field of electrocardiosignal noise reduction, in particular to an electrocardiosignal noise reduction method based on group sparsity.

Background

The aging problem of population and unhealthy life style of people lead to the rapid increase of the number of residents suffering from cardiovascular diseases in China in recent years. Nowadays, cardiovascular diseases are the first killers threatening human life and health, and have the characteristics of high morbidity, disability rate and fatality rate, and low treatment rate and control rate. Electrocardiogram (ECG), which is a bioelectrical signal for recording the electrical activity of the heart, has been widely used for diagnosis of cardiovascular diseases because of its simplicity and non-invasiveness. However, the electrocardiographic signal is an extremely weak electrophysiological signal, and is very susceptible to various external noises and interferences during the acquisition process. Such as power line interference, baseline drift, muscle noise, electrode motion artifacts, etc., which will cause waveform distortion of the original electrocardiographic signal, and have an important impact on the subsequent analysis of the electrocardiographic signal, even cause a doctor to make an erroneous evaluation on the patient, therefore, the noise reduction processing of the electrocardiographic signal is very important for the clinical diagnosis and classification of cardiovascular diseases.

The electrocardiosignal and the differential signal thereof have the group sparsity characteristic, namely, the electrocardiosignal not only has the sparsity characteristic, but also has the clustering or grouping characteristic. In particular, the large values of the cardiac signal itself and its derivative are not isolated, but generally occur near or adjacent to other large values. At present, the denoising method for electrocardiosignals comprises adaptive filtering, principal component analysis, empirical mode decomposition, neural network, wavelet transformation, sparse denoising and the like. The traditional filtering method has the phenomenon that the detail information of the signal and noise are discarded together, and the self sparse characteristic of the electrocardiosignal cannot be well utilized. The sparse noise reduction method based on the L1 norm can guarantee the sparsity of the solution, but the peak value of the electrocardiosignal is easily under-estimated, and the original signal information is lost. The full-variation noise reduction method improves the problem of signal peak value underestimation, retains more signal detail characteristics, and is easy to generate sawtooth waveforms, so that the signal waveforms are not smooth enough. Therefore, how to better utilize the group sparsity of the electrocardiosignal itself is a challenging problem in the academic and medical fields while maintaining the waveform characteristics of the original electrocardiosignal and achieving accurate and efficient noise reduction effects.

Disclosure of Invention

In order to overcome the defects of the technology, the invention provides the electrocardiosignal noise reduction method based on the group sparsity characteristic, and the electrocardiosignal is more accurately and efficiently estimated by selecting the strictly convex cost function and utilizing the characteristic of a banded system.

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

an electrocardiosignal noise reduction method based on group sparsity characteristics comprises the following steps:

a) establishing an electrocardiosignal mathematical model such as y ═ x + w, wherein y ∈ R^NIs a noisy electrocardiosignal, x is a clean electrocardiosignal, x ═ x (0), x (1), x (n)]^T∈R^N，w∈R^NFor added noise signals, R^NReal number space of N dimensions, x_n,K＝[x(n),...,x(n+K-1)]^T∈R^K，0≤n≤N-K，x_n,KRepresenting a vector with a starting index n and consisting of K points in the vector x;

b) building a convex optimization problem for a variable x by solving an optimal solution x of the convex optimization problem^*Obtaining a clean electrocardiosignal x, namely x^*＝x；

c) Based on an alternating direction multiplier method, enabling x to be equal to z, and enabling z to be an auxiliary variable, and obtaining a convex optimization problem about the variables x and z;

d) calculating an optimal solution through an iterative algorithm based on an optimization minimum method;

e) judging whether the iteration result meets the set convergence condition, if not, returning to the step d) to continue the iteration until the convergence condition is met, and finally obtaining the optimal solution x of the convex optimization problem^*Optimal solution x of the convex optimization problem^*Is a clean electrocardiosignal.

Further, in step b), the formula is used

Computing an optimal solution x^*In the formula

|x_iI denotes the ith element x of x_iF (x) is a convex optimization function, arg represents a variable corresponding to the minimum value of f (x), and D isFirst order differential matrix, D ∈ R^(N-1)×N，

λ₁And λ₂Are all constant, phi₁(x) Phi and phi₂(Dx) is the selected electrocardiosignal group sparsity measuring function,

0≤k≤K-1，s＝Dx。

further, the formula is used in step c)

Establishing an optimization problem, wherein z^*Is the optimal solution for z.

Further, step d) comprises the following steps:

d-1) by the formula

Iteratively solving the minimization problem to obtain the optimal solution x of the ith step⁽ⁱ⁾Where eta is a constant, eta > 0, d is an optimization variable, F₁(x) Has a maximum function of G₁(x,r)，

r is an auxiliary variable, C₁Is a constant that is independent of x,

0≤j≤K-1，[Λ(r)]_n,nis the element of the N-th row and the N-th column of Λ (r), and Λ (r) is an NxNth order matrix, when x ≠ r, G₁(x,r)≥F₁(x) When x is r, G₁(r,r)＝F₁(x) Wherein, in the step (A),

by the formula

Calculating to obtain the (i + 1) th stepIterative optimal solution x⁽ⁱ⁺¹⁾，

I is an identity matrix [ ·]^-1Is the inverse of the matrix, z⁽ⁱ⁾Is the optimal solution of step Z, d⁽ⁱ⁾Is the optimal solution of the ith step, wherein

[Λ(x⁽ⁱ⁾)]_n,nIs Λ (x)⁽ⁱ⁾) The nth row and the nth column of elements,

d-2) by the formula

Iteratively solving the minimization problem to obtain the optimal solution z of the step i⁽ⁱ⁾Wherein v is Dz, v_n,KIs a vector of vector v with a starting index n and consisting of K successive points, F₂(z) has a maximum function of G₂(z,u)，

0≤j≤K-1，[Λ(Du)]_n,nRow n and column n of Λ (Du), where ξ Du, T is the matrix transpose, C₂Is a constant independent of z, u is an auxiliary variable, and G is a constant independent of z when z ≠ u₂(z,u)≥F₂(z) when z is u, G₂(u,u)＝F₂(u) wherein,

by the formula

Calculating to obtain the i +1 st step iterative optimal solution z⁽ⁱ⁺¹⁾，

In the formula

0≤j≤K-1，[Λ(Dz⁽ⁱ⁾)]_n,nIs Λ (Dz)⁽ⁱ⁾) Of the nth row and the nth column of (1), wherein v⁽ⁱ⁾＝Dz⁽ⁱ⁾；

d-3) by the formula d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(z⁽ⁱ⁺¹⁾-x⁽ⁱ⁺¹⁾) Iterating to obtain the optimal solution d of the (i + 1) th step⁽ⁱ⁺¹⁾，d⁽ⁱ⁾And (4) obtaining the optimal solution of the step (i).

Further, step e) is based on the formula

Calculating the convergence criterion F (x)⁽ⁱ⁾) In the formula c₀For given constant, when the iteration result is not satisfied, returning to the step d) by the formula

And d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(z⁽ⁱ⁺¹⁾-x^{(i +1)}) The iteration continues.

The invention has the beneficial effects that: the staircase artifact caused by the noise reduction of the electrocardiosignals based on the total variation method is effectively solved, so that the waveform characteristics of the original electrocardiosignals are kept, the group sparsity characteristic of the electrocardiosignals is fully utilized by reasonably selecting a signal group sparsity measurement function, and the group sparsity of the electrocardiosignals is promoted to be known. Based on the characteristics of a banded system, only one tri-diagonal equation set needs to be solved for each iteration, the effectiveness and the calculation efficiency of the algorithm are improved, the method is suitable for non-overlapped group sparse signals, and when the group sparse signals are overlapped, the method is still effective. By selecting the strictly convex cost function, the convergence of the algorithm is ensured, the only optimal solution is finally converged, and the accurate and efficient noise reduction is achieved while the waveform characteristics of the original electrocardiosignal are maintained.

Detailed Description

The present invention is further explained below.

An electrocardiosignal noise reduction method based on group sparsity characteristics comprises the following steps:

a) establishing an electrocardiosignal mathematical model such as y ═ x + w, wherein y ∈ R^NIs a noisy electrocardiosignal, x is a clean electrocardiosignal, x ═ x (0), x (1), x (n)]^T∈R^N，w∈R^NFor added noise signals, R^NReal number space of N dimensions, x_n,K＝[x(n),...,x(n+K-1)]^T∈R^K，0≤n≤N-K，x_n,KRepresents that the initial subscript in the vector x is N and the vector consisting of K points is not less than 0 and not more than N and not more than N-K;

b) due to the electrocardiosignal x and the differentiation thereof

All are group sparse, in order to promote the group sparsity of the solution, an electrocardiosignal group sparsity measurement function is selected, a convex optimization problem about a variable x is established, and the optimal solution x of the convex optimization problem is solved^*Obtaining a clean electrocardiosignal x, namely x^*＝x；

c) Based on an alternating direction multiplier method, enabling x to be equal to z, and enabling z to be an auxiliary variable, and obtaining a convex optimization problem about the variables x and z;

d) calculating an optimal solution through an iterative algorithm based on an optimization minimum method;

e) judging whether the iteration result meets the set convergence condition, if not, returning to the step d) to continue the iteration until the convergence condition is met, and finally obtaining the optimal solution x of the convex optimization problem^*Optimal solution x of the convex optimization problem^*The electrocardiosignal is clean, thereby realizing the aim of noise reduction of the electrocardiosignal.

By the method, stair artifacts caused by noise reduction of the electrocardiosignals based on a total variation method are effectively solved, so that the waveform characteristics of the original electrocardiosignals are kept, the group sparsity characteristic of the electrocardiosignals is fully utilized by reasonably selecting a signal group sparsity measurement function, and the group sparsity of understanding is promoted. Based on the characteristics of a banded system, only one tri-diagonal equation set needs to be solved for each iteration, the effectiveness and the calculation efficiency of the algorithm are improved, the method is suitable for non-overlapped group sparse signals, and when the group sparse signals are overlapped, the method is still effective. By selecting the strictly convex cost function, the convergence of the algorithm is ensured, the only optimal solution is finally converged, and the accurate and efficient noise reduction is achieved while the waveform characteristics of the original electrocardiosignal are maintained.

The purpose of electrocardiosignal noise reduction is to recover clean electrocardiosignal x from electrocardiosignal y containing noise, and the electrocardiosignal noise reduction is realized by an optimization method because of the electrocardiosignal x and the differential thereof

Are all group sparse, and further, in step b), the group sparsity of the solution is promoted by a formula

Computing an optimal solution x^*In the formula

|x_iI denotes the ith element x of x_iF (x) is a convex optimization function, arg denotes the variable corresponding to the minimum of f (x), D is a first order differential matrix,

λ₁and λ₂Are all constant, phi₁(x) Phi and phi₂(Dx) is the selected electrocardiosignal group sparsity measuring function,

0≤k≤K-1，s＝Dx。

further, the formula is used in step c)

Establishing an optimization problem, wherein z^*Is the optimal solution for z.

Further, step d) comprises the following steps:

d-1) by the formula

Iteratively solving the minimization problem to obtain the optimal solution x of the ith step⁽ⁱ⁾Where eta is a constant, eta > 0, d is an optimization variable, F₁(x) Has a maximum function of G₁(x,r)，

r is an auxiliary variable, C₁Is a constant that is independent of x,

0≤j≤K-1，[Λ(r)]_n,nis the element of the N-th row and the N-th column of Λ (r), and Λ (r) is an NxNth order matrix, when x ≠ r, G₁(x,r)≥F₁(x) When x is r, G₁(r,r)＝F₁(x) Wherein, in the step (A),

by the formula

The (i + 1) th step iteration optimal solution x is obtained through calculation⁽ⁱ⁺¹⁾，

I is an identity matrix [ ·]^-1Is the inverse of the matrix, z⁽ⁱ⁾Is the optimal solution of step Z, d⁽ⁱ⁾Is the optimal solution of the ith step, wherein

[Λ(x⁽ⁱ⁾)]_n,nIs Λ (x)⁽ⁱ⁾) The nth row and the nth column of elements,

d-2) by the formula

Iteratively solving the minimization problem to obtain the optimal solution z of the step i⁽ⁱ⁾Wherein v is Dz, v_n,KIs a vector of vector v with a starting index n and consisting of K successive points, F₂(z) has a maximum function of G₂(z,u)，

0≤j≤K-1，[Λ(Du)]_n,nRow n and column n of Λ (Du), where ξ Du, T is the matrix transpose, C₂Is a constant independent of z, u is an auxiliary variable, and G is a constant independent of z when z ≠ u₂(z,u)≥F₂(z) when z is u, G₂(u,u)＝F₂(u) wherein,

by the formula

Calculating to obtain the i +1 st step iterative optimal solution z⁽ⁱ⁺¹⁾，

In the formula

0≤j≤K-1，[Λ(Dz⁽ⁱ⁾)]_n,nIs Λ (Dz)⁽ⁱ⁾) Of the nth row and the nth column of (1), wherein v⁽ⁱ⁾＝Dz⁽ⁱ⁾；

d-3) by the formula d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(z⁽ⁱ⁺¹⁾-x⁽ⁱ⁺¹⁾) Iterating to obtain the optimal solution d of the (i + 1) th step⁽ⁱ⁺¹⁾，d⁽ⁱ⁾And (4) obtaining the optimal solution of the step (i).

Further, step e) is based on the formula

Calculating the convergence criterion F (x)⁽ⁱ⁾) In the formula c₀For given constant, when the iteration result is not satisfied, returning to the step d) by the formula

And d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(z⁽ⁱ⁺¹⁾-x^{(i +1)}) The iteration continues.

Claims (5)

1. An electrocardiosignal noise reduction method based on group sparsity characteristics is characterized by comprising the following steps:

a) establishing an electrocardiosignal mathematical model such as y ═ x + w, wherein y ∈ R^NIs a noisy electrocardiosignal, x is a clean electrocardiosignal, x ═ x (0), x (1), x (n)]^T∈R^N，w∈R^NFor added noise signals, R^NReal number space of N dimensions, x_n,K＝[x(n),...,x(n+K-1)]^T∈R^K，0≤n≤N-K，x_n,KRepresenting a vector with a starting index n and consisting of K points in the vector x;

b) building a convex optimization problem for a variable x by solving an optimal solution x of the convex optimization problem^*Obtaining a clean electrocardiosignal x, namely x^*＝x；

c) Based on an alternating direction multiplier method, enabling x to be equal to z, and enabling z to be an auxiliary variable, and obtaining a convex optimization problem about the variables x and z;

d) calculating an optimal solution through an iterative algorithm based on an optimization minimum method;

e) judging whether the iteration result meets the set convergence condition, if not, returning to the step d) to continue the iteration until the convergence condition is met, and finally obtaining the optimal solution x of the convex optimization problem^*Optimal solution x of the convex optimization problem^*Is a clean electrocardiosignal.

2. The electrocardiosignal noise reduction method based on the group sparsity characteristic as claimed in claim 1, wherein:

in step b) by the formula

Computing an optimal solution x^*In the formula

|x_iI denotes the ith element x of x_iIs given by f (x), arg denotes the variable corresponding to the minimum of f (x), D is a first order differential matrix, D ∈ R^(N-1)×N，

λ₁And λ₂Are all constant, phi₁(x) Phi and phi₂(Dx) is the selected electrocardiosignal group sparsity measuring function,

0≤k≤K-1，s＝Dx。

3. the electrocardiosignal noise reduction method based on the group sparsity characteristic as claimed in claim 3, wherein: in step c) using the formula x^*,

Establishing an optimization problem, wherein z^*Is the optimal solution for z.

4. The electrocardiosignal noise reduction method based on the group sparsity characteristic as claimed in claim 4, wherein the step d) comprises the following steps:

d-1) by the formula

Iteratively solving the minimization problem to obtain the optimal solution x of the ith step⁽ⁱ⁾Where eta is a constant, eta > 0, and d is an optimization variable，F₁(x) Has a maximum function of G₁(x,r)，

r is an auxiliary variable, C₁Is a constant that is independent of x,

0≤j≤K-1，[Λ(r)]_n,nis the element of the N-th row and the N-th column of Λ (r), and Λ (r) is an NxNth order matrix, when x ≠ r, G₁(x,r)≥F₁(x) When x is r, G₁(r,r)＝F₁(x) Wherein, in the step (A),

by the formula

The (i + 1) th step iteration optimal solution x is obtained through calculation⁽ⁱ⁺¹⁾，

I is an identity matrix [ ·]^-1Is the inverse of the matrix, z⁽ⁱ⁾Is the optimal solution of step Z, d⁽ⁱ⁾Is the optimal solution of the ith step, wherein

[Λ(x⁽ⁱ⁾)]_n,nIs Λ (x)⁽ⁱ⁾) The nth row and the nth column of elements,

d-2) by the formula

Iteratively solving the minimization problem to obtain the optimal solution z of the step i⁽ⁱ⁾Wherein v is Dz, v_n,KIs a vector of vector v with a starting index n and consisting of K successive points, F₂(z) has a maximum function of G₂(z,u)，

0≤j≤K-1，[Λ(Du)]_n,nRow n and column n of Λ (Du), where ξ Du, T is the matrix transpose, C₂Is a constant independent of z, u is an auxiliary variable, and G is a constant independent of z when z ≠ u₂(z,u)≥F₂(z) when z is u, G₂(u,u)＝F₂(u) wherein,

by the formula

Calculating to obtain the i +1 st step iterative optimal solution z⁽ⁱ⁺¹⁾，

In the formula

0≤j≤K-1，[Λ(Dz⁽ⁱ⁾)]_n,nIs Λ (Dz)⁽ⁱ⁾) Of the nth row and the nth column of (1), wherein v⁽ⁱ⁾＝Dz⁽ⁱ⁾；

d-3) by the formula d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(z⁽ⁱ⁺¹⁾-x⁽ⁱ⁺¹⁾) Iterating to obtain the optimal solution d of the (i + 1) th step⁽ⁱ⁺¹⁾，d⁽ⁱ⁾And (4) obtaining the optimal solution of the step (i).

5. The electrocardiosignal noise reduction method based on the group sparsity characteristic as claimed in claim 5, wherein: in step e) by formula

Calculating the convergence criterion F (x)⁽ⁱ⁾) In the formula c₀For given constant, when the iteration result is not satisfied, returning to the step d) by the formula

And d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(z⁽ⁱ⁺¹⁾-x^{(i +1)}) The iteration continues.