CN108992062A - A kind of quadrature analysis and reconstructing method of electrocardiosignal - Google Patents
A kind of quadrature analysis and reconstructing method of electrocardiosignal Download PDFInfo
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Abstract
Description
技术领域technical field
本发明涉及心电信号处理技术领域，具体涉及一种心电信号的正交分析与重构方法。The invention relates to the technical field of electrocardiographic signal processing, in particular to an orthogonal analysis and reconstruction method of electrocardiographic signals.
背景技术Background technique
心电信号是将测量电极放置在人体表面的某一部位，从宏观上记录了心脏不断进行的有节奏的收缩和舒张运动的电信号变化曲线，心电信号在一定程度上客观反映了心脏各部位的生理状况，因此在临床医学中有重要意义；但是，目前已有的心电信号正交分析与重构的方法主要有以下缺点：(1)傅里叶变换：该方法通过相互正交的正弦与余弦函数对心电信号进行分解，得到心电信号中所包含的频率成分，经过傅里叶反变换，心电信号就能精确重构，但是该方法是在信号具有平稳特性的假设条件下进行的，而心电信号是一种非平稳的信号；(2)小波变换：该方法通过相互正交的小波基对心电信号进行分解，该方法具有良好的时频局部化特性，适用于非平稳信号，经过小波反变换，心电信号能精确重构，但是，该方法也存在着不足，例如如何选取合适的小波基，如何设置合适的参数等等，此外，由于小波基一旦选择，在心电信号的分析中就不能改变，所以小波变换也缺乏自适应性；(3)经验模式分解：该方法是一种新型自适应信号时频处理方法，依据数据自身的时间尺度特征来进行信号分解，非常适合对非线性、非平稳的心电信号进行分析处理，通过将所有固有模态函数相加，使心电信号能精确重构，但是该方法分解所得到的固有模态函数只是近似正交，而并非完全正交，因此会导致严重的能量泄露问题。The electrocardiographic signal is to place the measuring electrode on a certain part of the human body surface, and record the electrical signal change curve of the heart's continuous rhythmic contraction and diastolic movement from a macroscopic perspective. The electrocardiographic signal objectively reflects the various aspects of the heart. Therefore, it is of great significance in clinical medicine; however, the existing methods for orthogonal analysis and reconstruction of ECG signals mainly have the following shortcomings: (1) Fourier transform: this method uses mutual orthogonal The sine and cosine functions of the ECG signal are decomposed to obtain the frequency components contained in the ECG signal. After the inverse Fourier transform, the ECG signal can be accurately reconstructed, but this method is based on the assumption that the signal has a stationary characteristic. conditions, and the ECG signal is a non-stationary signal; (2) wavelet transform: This method decomposes the ECG signal through mutually orthogonal wavelet bases, and this method has good time-frequency localization characteristics, It is suitable for non-stationary signals. After wavelet inverse transformation, ECG signals can be accurately reconstructed. However, this method also has shortcomings, such as how to select a suitable wavelet base, how to set appropriate parameters, etc. In addition, because the wavelet base once The choice cannot be changed in the analysis of ECG signals, so the wavelet transform also lacks adaptability; (3) Empirical mode decomposition: This method is a new type of adaptive signal time-frequency processing method, which is based on the time scale characteristics of the data itself. Signal decomposition is very suitable for the analysis and processing of nonlinear and non-stationary ECG signals. By adding all the intrinsic mode functions, the ECG signals can be accurately reconstructed, but the inherent mode functions obtained by this method are decomposed It is only approximately orthogonal, but not completely orthogonal, so it will cause serious energy leakage problems.
发明内容Contents of the invention
本发明的目的在于克服现有技术的缺点与不足，提供一种心电信号的正交分析与重构方法，该方法通过奇异值分解来获取完全正交固有模态函数，防止能量泄露同时能够精确重构心电信号。The purpose of the present invention is to overcome the shortcomings and deficiencies of the prior art, and provide a method for orthogonal analysis and reconstruction of ECG signals. The method obtains completely orthogonal intrinsic mode functions through singular value decomposition, prevents energy leakage and can Precise reconstruction of ECG signals.
本发明的目的通过下述技术方案实现：The object of the present invention is achieved through the following technical solutions:
一种心电信号的正交分析与重构方法，包括下述步骤：A method for orthogonal analysis and reconstruction of electrocardiographic signals, comprising the steps of:
S1，对心电信号进行经验模式分解，从而得到对应的固有模态函数，具体步骤如下：S1. Carry out empirical mode decomposition on the ECG signal to obtain the corresponding intrinsic mode function. The specific steps are as follows:
S1.1，初始化残余信号为心电信号；S1.1, initializing the residual signal as an ECG signal;
S1.2，初始化细节信号为残余信号；S1.2, initializing the detail signal as a residual signal;
S1.3，查找细节信号的极大值点和极小值点；S1.3, find the maximum and minimum points of the detail signal;
S1.4，分别对S1.3中的极大值点和极小值点进行三次样条插值从而得到上下包络线；S1.4, respectively perform cubic spline interpolation on the maximum and minimum points in S1.3 to obtain the upper and lower envelopes;
S1.5，求S1.4中得到的上下包络线的均值；S1.5, find the mean value of the upper and lower envelope obtained in S1.4;
S1.6，用S1.3中的细节信号减去S1.5中的上下包络线均值后得到的信号作为新的细节信号，该新的细节信号用于判断以下情况：若该新的细节信号满足预设的固有模态函数的条件，则将该新的细节信号作为一个固有模态函数，并进行S1.7；否则将该新的细节信号作为S1.3中的细节信号并重复进行S1.3-S1.6；S1.6, the signal obtained by subtracting the mean value of the upper and lower envelopes in S1.5 from the detail signal in S1.3 is used as a new detail signal, and the new detail signal is used to judge the following situations: if the new detail If the signal satisfies the preset intrinsic mode function conditions, then use the new detail signal as an intrinsic mode function, and perform S1.7; otherwise, use the new detail signal as the detail signal in S1.3 and repeat S1.3-S1.6;
S1.7，将S1.2中的残余信号减去S1.6中得到的固有模态函数作为新的残差信号，若该残差信号满足固有模态函数条件，则结束；否则将该新的残差信号作为S1.2中的残余信号重复进行S1.2-S1.6；S1.7, subtract the intrinsic mode function obtained in S1.6 from the residual signal in S1.2 as a new residual signal, if the residual signal satisfies the intrinsic mode function condition, then end; otherwise, the new Repeat S1.2-S1.6 as the residual signal in S1.2;
S2，对S1中得到的固有模态函数进行标准化操作，即将每一个固有模态函数除以其对应的向量二范数，并以列向量的形式排列成一个矩阵；S2, standardize the intrinsic mode functions obtained in S1, that is, divide each intrinsic mode function by its corresponding vector two-norm, and arrange them into a matrix in the form of column vectors;
S3，建立心电信号完全正交固有模态函数的优化模型，通过奇异值分解的方法得到心电信号的完全正交固有模态函数，并进行正交误差分析，具体步骤如下：S3, establish the optimization model of the fully orthogonal intrinsic mode function of the ECG signal, obtain the fully orthogonal intrinsic mode function of the ECG signal through the method of singular value decomposition, and perform an orthogonal error analysis, the specific steps are as follows:
S3.1，心电信号完全正交固有模态函数的优化模型采用下式表示：S3.1, the optimization model of the fully orthogonal intrinsic mode function of the ECG signal is expressed by the following formula:
subject to UTU＝I，subject to U T U = I,
其中，U为需要求解的心电信号完全正交固有模态函数，为S2中的标准化固有模态函数排列成的矩阵，||·||F为矩阵的Frobenius范数，UTU为矩阵U的转置乘以矩阵U，I为单位矩阵；Among them, U is the fully orthogonal intrinsic mode function of the ECG signal to be solved, is the matrix arranged by the standardized intrinsic mode functions in S2 , ||||
S3.2，对进行奇异值分解，得到其正交矩阵和对角矩阵DA，则所求完全正交固有模态函数U为：S3.2, yes Perform singular value decomposition to obtain its orthogonal matrix and diagonal matrix D A , then the completely orthogonal intrinsic mode function U is:
S3.3，采用下述公式分析完全正交固有模态函数的正交误差ε1：S3.3, use the following formula to analyze the orthogonal error ε 1 of the fully orthogonal intrinsic mode function:
ε1＝||UTU-I||F；ε 1 ＝||U T UI|| F ;
S4，通过最小二乘法找到心电信号在完全正交固有模态函数上的最佳投影，具体步骤如下：S4, find the optimal projection of the ECG signal on the fully orthogonal intrinsic mode function by the least square method, the specific steps are as follows:
心电信号在完全正交固有模态函数上的最佳投影w用下述公式表示：The optimal projection w of the ECG signal on the fully orthogonal intrinsic mode function is expressed by the following formula:
w＝(UTU)UTx，w = (U T U) U T x,
其中x为原心电信号；Where x is the original ECG signal;
S5，重构心电信号，并进行重构误差的分析，具体步骤如下：S5, reconstructing the ECG signal, and analyzing the reconstruction error, the specific steps are as follows:
S5.1，根据S4中求得的最佳投影w，并通过下述公式求得重构信号 S5.1, according to the optimal projection w obtained in S4, and obtain the reconstructed signal by the following formula
S5.2，采用下述公式分析完全正交固有模态函数的重构误差ε2：S5.2, use the following formula to analyze the reconstruction error ε 2 of the fully orthogonal intrinsic mode function:
本发明与现有技术相比具有以下的有益效果：Compared with the prior art, the present invention has the following beneficial effects:
与现有的傅里叶和小波技术相比，本发明是基于经验模式分解来对心电信号进行正交分析与精确重构，从而继承了经验模式分解的自适应性，适合于分析非平稳信号；与传统的经验模式分解技术相比，本发明所得到的固有模态函数和三角函数基和小波基一样，是完全正交的，因此可以防止能量泄露，并且本发明同样可以精确重构心电信号；实验数据表明，本发明的正交误差达到数量级10-15，重构误差达到数量级10-14，因此误差可视为0，所以由本发明得到的固有模态函数完全正交并且能精确重构心电信号。通过分析心电信号的完全正交的固有模态函数，防止固有模态函数的能量泄露问题，更彻底分离信号内在模式，凸显信号特征。让本发明可以得到心电信号的信息例如心电信号的QRS波群的位置，心电信号T波的信息，心电信号的基线漂移状况等。Compared with the existing Fourier and wavelet techniques, the present invention is based on empirical mode decomposition to perform orthogonal analysis and accurate reconstruction of ECG signals, thereby inheriting the adaptability of empirical mode decomposition and suitable for analyzing non-stationary signal; compared with the traditional empirical mode decomposition technology, the inherent mode function obtained by the present invention is completely orthogonal to the trigonometric function base and the wavelet base, so energy leakage can be prevented, and the present invention can also be accurately reconstructed Electrocardiographic signal; experimental data shows that the orthogonality error of the present invention reaches an order of magnitude of 10 -15 , and the reconstruction error reaches an order of magnitude of 10 -14 , so the error can be regarded as 0, so the intrinsic mode function obtained by the present invention is completely orthogonal and can Precise reconstruction of ECG signals. By analyzing the fully orthogonal intrinsic mode functions of the ECG signal, the energy leakage of the intrinsic mode functions can be prevented, the inner mode of the signal can be more thoroughly separated, and the signal characteristics can be highlighted. The present invention can obtain the information of the electrocardiographic signal, such as the position of the QRS complex of the electrocardiographic signal, the information of the T wave of the electrocardiographic signal, the baseline drift status of the electrocardiographic signal, and the like.
附图说明Description of drawings
图1为本发明的流程图；Fig. 1 is a flowchart of the present invention;
图2为本发明的心电信号和所得的前三个完全正交的固有模态函数的波形图，其中第二个正交固有模态函数有效地反映了心电信号的QRS波群的位置；Fig. 2 is the oscillogram of the first three fully orthogonal intrinsic mode functions of the electrocardiogram of the present invention and the result, wherein the second orthogonal intrinsic mode function reflects the position of the QRS wave group of the electrocardiogram effectively ;
图3为本发明的心电信号和所得的后三个完全正交的固有模态函数的波形图，其中第四个正交固有模态函数包含了心电信号T波的信息，第六个正交固有模态函数反映了心电信号的基线漂移状况。Fig. 3 is the oscillogram of electrocardiographic signal of the present invention and the back three fully orthogonal intrinsic mode functions of gain, wherein the fourth orthogonal intrinsic mode function has included the information of electrocardiographic signal T wave, the sixth Orthogonal intrinsic mode functions reflect the baseline drift of ECG signals.
具体实施方式Detailed ways
下面结合实施例及附图对本发明作进一步详细的描述，但本发明的实施方式不限于此。The present invention will be further described in detail below in conjunction with the embodiments and the accompanying drawings, but the embodiments of the present invention are not limited thereto.
如图1～3所示，一种心电信号的正交分析与重构方法，包括下述步骤：As shown in Figures 1 to 3, a method for orthogonal analysis and reconstruction of ECG signals includes the following steps:
S1，对心电信号进行经验模式分解，从而得到对应的固有模态函数，具体步骤如下：S1. Carry out empirical mode decomposition on the ECG signal to obtain the corresponding intrinsic mode function. The specific steps are as follows:
S1.1，初始化残余信号为心电信号；S1.1, initializing the residual signal as an ECG signal;
S1.2，初始化细节信号为残余信号；S1.2, initializing the detail signal as a residual signal;
S1.3，查找细节信号的极大值点和极小值点；S1.3, find the maximum and minimum points of the detail signal;
S1.4，分别对S1.3中的极大值点和极小值点进行三次样条插值从而得到上下包络线；S1.4, respectively perform cubic spline interpolation on the maximum and minimum points in S1.3 to obtain the upper and lower envelopes;
S1.5，求S1.4中得到的上下包络线的均值；S1.5, find the mean value of the upper and lower envelope obtained in S1.4;
S1.6，用S1.3中的细节信号减去S1.5中的上下包络线均值后得到的信号作为新的细节信号，该新的细节信号用于判断以下情况：若该新的细节信号满足预设的固有模态函数的条件，则将该新的细节信号作为一个固有模态函数，并进行S1.7；否则将该新的细节信号作为S1.3中的细节信号并重复进行S1.3-S1.6；S1.6, the signal obtained by subtracting the mean value of the upper and lower envelopes in S1.5 from the detail signal in S1.3 is used as a new detail signal, and the new detail signal is used to judge the following situations: if the new detail If the signal satisfies the preset intrinsic mode function conditions, then use the new detail signal as an intrinsic mode function, and perform S1.7; otherwise, use the new detail signal as the detail signal in S1.3 and repeat S1.3-S1.6;
S1.7，将S1.2中的残余信号减去S1.6中得到的固有模态函数作为新的残差信号，若该残差信号满足固有模态函数条件，则结束；否则将该新的残差信号作为S1.2中的残余信号重复进行S1.2-S1.6；S1.7, subtract the intrinsic mode function obtained in S1.6 from the residual signal in S1.2 as a new residual signal, if the residual signal satisfies the intrinsic mode function condition, then end; otherwise, the new Repeat S1.2-S1.6 as the residual signal in S1.2;
S2，对S1中得到的固有模态函数进行标准化操作，即将每一个固有模态函数除以其对应的向量二范数，并以列向量的形式排列成一个矩阵；S2, standardize the intrinsic mode functions obtained in S1, that is, divide each intrinsic mode function by its corresponding vector two-norm, and arrange them into a matrix in the form of column vectors;
S3，建立心电信号完全正交固有模态函数的优化模型，通过奇异值分解的方法得到心电信号的完全正交固有模态函数，并进行正交误差分析，具体步骤如下：S3, establish the optimization model of the fully orthogonal intrinsic mode function of the ECG signal, obtain the fully orthogonal intrinsic mode function of the ECG signal through the method of singular value decomposition, and perform an orthogonal error analysis, the specific steps are as follows:
S3.1，心电信号完全正交固有模态函数的优化模型采用下式表示：S3.1, the optimization model of the fully orthogonal intrinsic mode function of the ECG signal is expressed by the following formula:
subject to UTU＝I，subject to U T U = I,
其中，U为需要求解的心电信号完全正交固有模态函数，为S2中的标准化固有模态函数排列成的矩阵，||·||F为矩阵的Frobenius范数，UTU为矩阵U的转置乘以矩阵U，I为单位矩阵；Among them, U is the fully orthogonal intrinsic mode function of the ECG signal to be solved, is the matrix arranged by the standardized intrinsic mode functions in S2 , ||||
S3.2，对(为矩阵的转置乘以矩阵)进行奇异值分解，得到其正交矩阵和对角矩阵DA，则所求完全正交固有模态函数U为：S3.2, yes ( for the matrix The transpose of is multiplied by the matrix ) for singular value decomposition to obtain its orthogonal matrix and diagonal matrix D A , then the completely orthogonal intrinsic mode function U is:
S3.3，采用下述公式分析完全正交固有模态函数的正交误差ε1：S3.3, use the following formula to analyze the orthogonal error ε 1 of the fully orthogonal intrinsic mode function:
ε1＝||UTU-I||F；ε 1 ＝||U T UI|| F ;
S4，通过最小二乘法找到心电信号在完全正交固有模态函数上的最佳投影，具体步骤如下：S4, find the optimal projection of the ECG signal on the fully orthogonal intrinsic mode function by the least square method, the specific steps are as follows:
心电信号在完全正交固有模态函数上的最佳投影w用下述公式表示：The optimal projection w of the ECG signal on the fully orthogonal intrinsic mode function is expressed by the following formula:
w＝(UTU)UTx，w = (U T U) U T x,
其中x为原心电信号；Where x is the original ECG signal;
S5，重构心电信号，并进行重构误差的分析，具体步骤如下：S5, reconstructing the ECG signal, and analyzing the reconstruction error, the specific steps are as follows:
S5.1，根据S4中求得的最佳投影w，并通过下述公式求得重构信号 S5.1, according to the optimal projection w obtained in S4, and obtain the reconstructed signal by the following formula
S5.2，采用下述公式分析完全正交固有模态函数的重构误差ε2：S5.2, use the following formula to analyze the reconstruction error ε 2 of the fully orthogonal intrinsic mode function:
与现有的傅里叶和小波技术相比，本发明是基于经验模式分解来对心电信号进行正交分析与精确重构，从而继承了经验模式分解的自适应性，适合于分析非平稳信号；与传统的经验模式分解技术相比，本发明所得到的固有模态函数和三角函数基和小波基一样，是完全正交的，因此可以防止能量泄露，并且本发明同样可以精确重构心电信号；实验数据表明，本发明的正交误差达到数量级10-15，重构误差达到数量级10-14，因此误差可视为0，所以由本发明得到的固有模态函数完全正交并且能精确重构心电信号。通过分析心电信号的完全正交的固有模态函数，防止固有模态函数的能量泄露问题，更彻底分离信号内在模式，凸显信号特征。让本发明可以得到心电信号的信息例如心电信号的QRS波群的位置，心电信号T波的信息，心电信号的基线漂移状况等。Compared with the existing Fourier and wavelet techniques, the present invention is based on empirical mode decomposition to perform orthogonal analysis and accurate reconstruction of ECG signals, thereby inheriting the adaptability of empirical mode decomposition and suitable for analyzing non-stationary signal; compared with the traditional empirical mode decomposition technology, the inherent mode function obtained by the present invention is completely orthogonal to the trigonometric function base and the wavelet base, so energy leakage can be prevented, and the present invention can also be accurately reconstructed Electrocardiographic signal; experimental data shows that the orthogonality error of the present invention reaches an order of magnitude of 10 -15 , and the reconstruction error reaches an order of magnitude of 10 -14 , so the error can be regarded as 0, so the intrinsic mode function obtained by the present invention is completely orthogonal and can Precise reconstruction of ECG signals. By analyzing the fully orthogonal intrinsic mode functions of the ECG signal, the energy leakage of the intrinsic mode functions can be prevented, the inner mode of the signal can be more thoroughly separated, and the signal characteristics can be highlighted. The present invention can obtain the information of the electrocardiographic signal, such as the position of the QRS complex of the electrocardiographic signal, the information of the T wave of the electrocardiographic signal, the baseline drift status of the electrocardiographic signal, and the like.
上述为本发明较佳的实施方式，但本发明的实施方式并不受上述内容的限制，其他的任何未背离本发明的精神实质与原理下所作的改变、修饰、替代、组合、简化，均应为等效的置换方式，都包含在本发明的保护范围之内。The above is a preferred embodiment of the present invention, but the embodiment of the present invention is not limited by the above content, and any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention are all Replacement methods that should be equivalent are all included within the protection scope of the present invention.
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