CN113598785B - Electrocardiosignal denoising method based on wavelet domain sparse characteristic - Google Patents

Abstract

An electrocardiosignal denoising method based on wavelet domain sparse characteristics carries out optimization calculation on electrocardiosignal wavelet coefficients, realizes accurate and efficient denoising, and can better keep the waveform characteristics of original electrocardiosignals. The obtained threshold function is continuous, and noise spikes and pseudo Gibbs oscillations which occur at discontinuous points in the traditional wavelet threshold denoising method can be avoided. The sparse characteristic of the electrocardiosignals in the wavelet domain is fully utilized, and a sparse solution can be obtained by selecting a non-convex sparse penalty function. By selecting proper regularization parameters, strict convexity of the objective function is guaranteed, a unique solution of a denoising problem can be obtained by a convex optimization method, and the optimization algorithm is high in calculation efficiency and high in convergence speed.

Description

Electrocardiosignal denoising method based on wavelet domain sparse characteristic

Technical Field

The invention relates to the technical field of electrocardiosignal denoising, in particular to an electrocardiosignal denoising method based on wavelet domain sparse characteristics.

Background

Electrocardiogram (ECG) is the most commonly used tool for diagnosing human cardiac function, and it is the recording of the electrical activity of the heart that occurs during the cardiac cycle. The characteristic waveforms of a cardiac cycle of an electrocardiogram include the P-wave, QRS-wave and T-wave, these components contain important clinical information, and any changes in waveform pattern are characteristic of cardiac arrhythmias, e.g., the interval time of the QRS-wave provides information on heart rate variability, and other features extracted from the electrocardiographic signal can help identify various heart diseases. However, during the acquisition of the cardiac electrical signal, the signal is very susceptible to various noises, such as power frequency interference, baseline drift, electromyographic interference, motion artifacts, and the like, and most of the noises are in the frequency band of 0.05-100 Hz. These noise interferences make it difficult for a physician to extract valuable cardiac status information. Therefore, the effective analysis of the electrocardiosignals has important significance for preventing and diagnosing cardiovascular diseases, and the denoising is also an important component of an automatic electrocardiogram analysis system.

The electrocardiosignal not only has the sparse characteristic, but also the signal after wavelet transformation is still sparse, namely the electrocardiosignal also has the sparse characteristic in the wavelet domain. The denoising method of the electrocardiosignal comprises a filter bank, principal component analysis, Empirical Mode Decomposition (EMD), wavelet transformation, a denoising method based on sparse decomposition and the like. The traditional method for removing noise by filtering is based on time domain or frequency domain filtering, has a good effect of inhibiting noise outside a signal frequency range, but cannot well track the time-varying form of a signal. Denoising by using an empirical mode decomposition method usually discards some initial natural modes containing noise, thereby causing distortion of the reconstructed electrocardiosignal, especially distortion of a QRS waveform. The wavelet transformation method limits the electrocardiosignals to a specific frequency band through wavelet transformation, noise components with smaller amplitude are dispersed in different frequency bands, denoising of the electrocardiosignals is realized by setting a wavelet threshold, and a proper threshold is selected according to the signal-to-noise ratio of the signals, so that the noise can be effectively inhibited, and the effective components of the signals are not influenced or minimally influenced. However, the wavelet thresholding method is prone to introduce new noise, such as spurious noise peaks and pseudo gibbs' oscillations, while removing noise, due to the fact that the wavelet coefficients containing the noise exceed a set threshold or the non-zero wavelet coefficients are set to zero.

Disclosure of Invention

In order to overcome the defects of the technology, the invention provides a method for optimizing and estimating all wavelet coefficients by selecting a non-convex sparse penalty function, so as to achieve an accurate and efficient electrocardiosignal denoising effect.

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

an electrocardiosignal denoising method based on wavelet domain sparse characteristics comprises the following steps:

a) establishing a mathematical model of the electrocardiosignal such as y (l) ═ x (l) + N (l) · l ═ 1,2^N，R^NIs a real number space of N dimensions, x is a clean electrocardiosignal, and x belongs to R^NN is white Gaussian noise, n belongs to R^NPerforming wavelet transformation on the electrocardiosignal x, wherein the wavelet coefficient of the electrocardiosignal x is s, s is Wx, W is a wavelet transformation matrix, and W satisfies W^TW ═ γ I, I is the identity matrix, T is the matrix transpose, γ is a constant, γ > 0, s_j,kIs the element of s, j is the wavelet scale, k is time;

b) establishing a convex optimization problem for the variable s, by formula

Calculating to obtain the optimal solution s of the convex optimization problem^*In the formula | · | non-conducting phosphor₂Is L2 norm, | · | | non-woven₁L1 norm, F(s) is a non-convex cost function, arg represents a variable corresponding to the minimum value of F(s), D is a first order differential matrix, and D is equal to R^(N-1)×N，

λ_j、a_jA is a constant, λ_j＞0、a_j＞0、α＞0，

For a scaled very small maximum concave penalty function,

c) based on an alternating direction multiplier method, making s ═ v, and v as an auxiliary variable, and obtaining a convex optimization problem about the variables s and v;

d) by the formula

Iteratively solving the minimization problem to obtain the optimal solution s of the ith step⁽ⁱ⁾Where w is (Wy + ρ (v-d))/(1+ ρ), ρ is a constant, ρ > 0, d is an optimization variable, and w is an integer_j,kIs an element of w by the formula

Calculating to obtain the optimal solution of the (i + 1) th iteration

Sign (. cndot.) in the formulaThe function of the number of the signals,

is w⁽ⁱ⁾Element of (a), w⁽ⁱ⁾Is the value of the ith iteration of w, w⁽ⁱ⁾＝(Wy+ρ(v⁽ⁱ⁾-d⁽ⁱ⁾))/(1+ρ)，v⁽ⁱ⁾Is the value of v step i iteration, d⁽ⁱ⁾D is the value of the ith iteration;

e) by the formula

Iteratively solving the minimization problem to obtain the optimal solution v of the ith step⁽ⁱ⁾Let u be d + s to obtain

Selecting a non-linear function

According to the semi-orthogonal linear transformation property of the adjacent operator, obtaining

In the formula

z is an auxiliary variable, h (z, u) is a function of the variable z, u, the maximum function of h (z, u) is selected as G (z, u),

z_ifor the value of z step i iteration, Λ_iIs a matrix of the order of N,

is Λ_iThe inverse of the matrix of (a) is,

g (z, u) satisfies the condition that G (z, u) ≧ h (z, u), G (z)_i,u)＝h(z_iU) then

Obtained according to the inverse theorem of the matrix

By the formula

Calculating to obtain the optimal solution v of the (i + 1) th iteration^{(i +1)}，u⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾+s⁽ⁱ⁺¹⁾，d⁽ⁱ⁾Is the value of the i-th iteration, s⁽ⁱ⁺¹⁾The value of the iteration of the (i + 1) th step is obtained;

f) by the formula d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(v⁽ⁱ⁺¹⁾-s⁽ⁱ⁺¹⁾) Calculating to obtain the optimal solution d of the (i + 1) th iteration⁽ⁱ⁺¹⁾；

g) Given constant c₀> 0, if a convergence condition is satisfied

Executing step h), if the convergence condition is not met, repeatedly executing steps d) to f) until the convergence condition is met;

h) by the formula x ═ γ^-1W^TAnd s, calculating to obtain a clean electrocardiosignal x, and denoising the electrocardiosignal x. Further, in step c), the formula is used

Establishing an optimization problem, s^*Is the optimal solution of s, v^*Is an optimal solution of v, wherein

The invention has the beneficial effects that: by optimizing and calculating wavelet coefficients of the electrocardiosignal, accurate and efficient denoising is realized, and meanwhile, the waveform characteristics of the original electrocardiosignal can be better reserved. The obtained threshold function is continuous, and noise spikes and pseudo Gibbs oscillations which occur at discontinuous points in the traditional wavelet threshold denoising method can be avoided. The sparse characteristic of the electrocardiosignals in the wavelet domain is fully utilized, and a sparse solution can be obtained by selecting a non-convex sparse penalty function. By selecting proper regularization parameters, strict convexity of the objective function is guaranteed, a unique solution of a denoising problem can be obtained by a convex optimization method, and the optimization algorithm is high in calculation efficiency and high in convergence speed.

Detailed Description

The present invention is further explained below.

An electrocardiosignal denoising method based on wavelet domain sparse characteristics comprises the following steps:

a) establishing a mathematical model of the electrocardiosignal such as y (l) ═ x (l) + N (l) · l ═ 1,2^N，R^NIs a real number space of N dimensions, x is a clean electrocardiosignal, and x belongs to R^NN is white Gaussian noise, n belongs to R^NPerforming wavelet transformation on the electrocardiosignal x, wherein the wavelet coefficient of the electrocardiosignal x is s, s is Wx, W is a wavelet transformation matrix, and W satisfies W^TW ═ γ I, I is the identity matrix, T is the matrix transpose, γ is a constant, γ > 0, s_j,kIs the element of s, j is the wavelet scale, k is time;

b) establishing a convex optimization problem for the variable s, by formula

Calculating to obtain the optimal solution s of the convex optimization problem^*In the formula | · | non-conducting phosphor₂Is L2 norm, | · | | non-woven₁L1 norm, F(s) is a non-convex cost function, arg represents a variable corresponding to the minimum value of F(s), D is a first order differential matrix, and D is equal to R^(N-1)×N，

λ_j、a_jA is a constant, λ_j＞0、a_j＞0、α＞0，

Is a scaled minimax-concave penalty function,

c) based on an alternating direction multiplier method, making s ═ v, and v as an auxiliary variable, and obtaining a convex optimization problem about the variables s and v;

d) by the formula

Iteratively solving the minimization problem to obtain the optimal solution s of the ith step⁽ⁱ⁾Where w is (Wy + ρ (v-d))/(1+ ρ), ρ is a constant, ρ > 0, d is an optimization variable, and w is an integer_j,kIs an element of w by the formula

Calculating to obtain the optimal solution of the (i + 1) th iteration

Wherein sign (. cndot.) is a sign function,

is w⁽ⁱ⁾Element of (a), w⁽ⁱ⁾Is the value of the ith iteration of w, w⁽ⁱ⁾＝(Wy+ρ(v⁽ⁱ⁾-d⁽ⁱ⁾))/(1+ρ)，v⁽ⁱ⁾Is the value of v step i iteration, d⁽ⁱ⁾D is the value of the ith iteration;

e) by the formula

Iteratively solving the minimization problem to obtain the optimal solution v of the ith step⁽ⁱ⁾Let u be d + s to obtain

Selecting a non-linear function

According to the semi-orthogonal linear transformation property of a Proximity Operator (Proximity Operator), obtaining

In the form ofIn

z is an auxiliary variable, h (z, u) is a function of the variable z, u, the maximum function of h (z, u) is selected as G (z, u),

z_ifor the value of z step i iteration, Λ_iIs a matrix of the order of N,

is Λ_iThe inverse of the matrix of (a) is,

g (z, u) satisfies the condition that G (z, u) ≧ h (z, u), G (z)_i,u)＝h(z_iU) then

Obtained according to the inverse theorem of the matrix

By the formula

Calculating to obtain the optimal solution v of the (i + 1) th iteration^{(i +1)}，u⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾+s⁽ⁱ⁺¹⁾，d⁽ⁱ⁾Is the value of the i-th iteration, s⁽ⁱ⁺¹⁾The value of the iteration of the (i + 1) th step is obtained;

f) by the formula d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(v⁽ⁱ⁺¹⁾-s⁽ⁱ⁺¹⁾) Calculating to obtain the optimal solution d of the (i + 1) th iteration⁽ⁱ⁺¹⁾；

g) Given constant c₀> 0, if a convergence condition is satisfied

Executing step h), if the convergence condition is not met, repeatedly executing steps d) to f) until the convergence condition is met;

h) by the formula x ═ γ^-1W^TAnd s, calculating to obtain a clean electrocardiosignal x, and denoising the electrocardiosignal x. By optimizing and calculating wavelet coefficients of the electrocardiosignal, accurate and efficient denoising is realized, and meanwhile, the waveform characteristics of the original electrocardiosignal can be better reserved. The obtained threshold function is continuous, and noise spikes and pseudo Gibbs oscillations which occur at discontinuous points in the traditional wavelet threshold denoising method can be avoided. The sparse characteristic of the electrocardiosignals in the wavelet domain is fully utilized, and a sparse solution can be obtained by selecting a non-convex sparse penalty function. By selecting proper regularization parameters, strict convexity of the objective function is guaranteed, a unique solution of a denoising problem can be obtained by a convex optimization method, and the optimization algorithm is high in calculation efficiency and high in convergence speed.

Further, in step c), the formula is used

Establishing an optimization problem with s as s^*Of the optimal solution, v^*Is an optimal solution of v, wherein

Finally, it should be noted that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that changes may be made in the embodiments and/or equivalents thereof without departing from the spirit and scope of the invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (2)

1. An electrocardiosignal denoising method based on wavelet domain sparse characteristics is characterized by comprising the following steps:

a) establishing a mathematical model of the electrocardiosignal such as y (l) ═ x (l) + N (l) · l ═ 1,2^N，R^NIs a real number space of N dimensions, x is a clean electrocardiosignal, and x belongs to R^NN is white Gaussian noise, n belongs to R^NPerforming wavelet transformation on the electrocardiosignal x, wherein the wavelet coefficient of the electrocardiosignal x is s, s is Wx, W is a wavelet transformation matrix, and W satisfies W^TW ═ γ I, I is the identity matrix, T is the matrix transpose, γ is a constant, γ > 0, s_j,kIs the element of s, j is the wavelet scale, k is time;

b) establishing a convex optimization problem for the variable s, by formula

Calculating to obtain the optimal solution s of the convex optimization problem^*In the formula | · | non-conducting phosphor₂Is L2 norm, | · | | non-woven₁L1 norm, F(s) is a non-convex cost function, arg represents a variable corresponding to the minimum value of F(s), D is a first order differential matrix, and D is equal to R^(N-1)×N，

λ_j、a_jA is a constant, λ_j＞0、a_j＞0、α＞0，

For a scaled very small maximum concave penalty function,

c) based on an alternating direction multiplier method, making s ═ v, and v as an auxiliary variable, and obtaining a convex optimization problem about the variables s and v;

d) by the formula

Iteratively solving the minimization problem to obtain the optimal solution s of the ith step⁽ⁱ⁾Where w is (Wy + ρ (v-d))/(1+ ρ), ρ is a constant, ρ > 0, d is an optimization variable, and w is an integer_j,kIs an element of w by the formula

Calculating to obtain the optimal solution of the (i + 1) th iteration

Wherein sign (. cndot.) is a sign function,

is w⁽ⁱ⁾Element of (a), w⁽ⁱ⁾Is the value of the ith iteration of w, w⁽ⁱ⁾＝(Wy+ρ(v⁽ⁱ⁾-d⁽ⁱ⁾))/(1+ρ)，v⁽ⁱ⁾Is the value of v step i iteration, d⁽ⁱ⁾D is the value of the ith iteration;

e) by the formula

Iteratively solving the minimization problem to obtain the optimal solution v of the ith step⁽ⁱ⁾Let u be d + s to obtain

Selecting a non-linear function

According to the semi-orthogonal linear transformation property of the adjacent operator, obtaining

In the formula

H (z, u) is a function of the variables z, u, the maximum function of h (z, u) is selected as G (z, u),

z_ifor the value of z step i iteration, Λ_iIs a matrix of the order of N,

is Λ_iThe inverse of the matrix of (a) is,

g (z, u) satisfies the condition that G (z, u) ≧ h (z, u), G (z)_i,u)＝h(z_iU) then

Obtained according to the inverse theorem of the matrix

By the formula

Calculating to obtain the optimal solution v of the (i + 1) th iteration^{(i +1)}，u⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾+s⁽ⁱ⁺¹⁾，d⁽ⁱ⁾Is the value of the i-th iteration, s⁽ⁱ⁺¹⁾The value of the iteration of the (i + 1) th step is obtained;

f) by the formula d⁽ⁱ⁺¹⁾＝d⁽ⁱ⁾-(v⁽ⁱ⁺¹⁾-s⁽ⁱ⁺¹⁾) Calculating to obtain the optimal solution d of the (i + 1) th iteration⁽ⁱ⁺¹⁾；

g) Given constant c₀> 0, if a convergence condition is satisfied

Executing step h), if the convergence condition is not met, repeatedly executing steps d) to f) until the convergence condition is met;

h) by the formula x ═ γ^-1W^TAnd s, calculating to obtain a clean electrocardiosignal x, and denoising the electrocardiosignal x.

2. The electrocardiosignal denoising method based on the wavelet domain sparse characteristic as claimed in claim 1, wherein: in step c) by the formula s^*,

Establishing an optimization problem, s^*Is the optimal solution of s, v^*Is an optimal solution of v, wherein