Magnetic resonance sounding signal de-noising method based on backtracking generalized OMP

Abstract

The invention belongs to the field of magnetic resonance sounding signal noise filtering, and particularly relates to a magnetic resonance sounding signal de-noising method based on backtracking generalized OMP, and the method comprises the steps: employing a magnetic resonance sounding water detector to carry out the sampling of a sparse signal at a frequency lower than the Nyquist frequency, and obtaining a noisy MRS signal; converting the noisy MRS signal into a two-dimensional matrix signal; dictionary learning is carried out on the two-dimensional matrix signals through a KSVD algorithm, and an updated dictionary is obtained after iteration; calculating a recovery matrix and measurement data by using an independently distributed Gaussian random matrix; performing iterative operation on the measurement data by using a backtracking generalized orthogonal matching pursuit algorithm, selecting an atomic sequence of maximum projection of the measurement data and a recovery matrix module, and obtaining a sparse coefficient matrix of the measurement data under the recovery matrix; and calculating a recovery signal according to the dictionary and the sparse coefficient matrix of the measurement data under the recovery matrix.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of magnetic resonance sounding signal noise filtering, and particularly relates to a magnetic resonance sounding signal denoising method based on backtracking generalized OMP. BACKGROUND

[0002] Magnetic resonance sounding (MRS) is a promising geophysical method for direct and quantitative detection of groundwater. The technology collects the induced magnetic field of the excited hydrogen protons in groundwater, analyzes the characteristics and occurrence state of groundwater quantitatively according to the relaxation characteristics. MRS is an efficient geophysical detection technology with the advantages of non-invasiveness and high sensitivity, and is widely used in water resource investigation, environmental monitoring, soil research, geological survey and earthquake research.

[0003] The MRS signal obtained in the process of ground magnetic resonance detection is extremely weak, and the MRS signal is very weak, usually in nanovolt level, and is often submerged in environmental noise. During the detection process, external environmental interference factors cannot be completely shielded, resulting in that the MRS signal is affected by random noise, power harmonic and spike noise, the quality of the signal is reduced, the extraction of MRS signal parameters and inversion are hindered, and finally the accuracy of regional water resource content evaluation is affected. In order to solve the problems of noise suppression and effective signal extraction, domestic and foreign scholars have carried out related research. However, the noise suppression method at the present stage has great limitations, and has good denoising performance under certain conditions, but for various uncertain noises in the MRS signal, noise suppression and effective extraction of the MRS signal are still technical problems in the field. SUMMARY

[0004] The embodiment of the application provides a magnetic resonance sounding signal denoising method based on backtracking generalized OMP, which solves the problem that the MRS signal is easily interfered by random noise and spike noise in the measurement process.

[0005] According to the magnetic resonance sounding signal denoising method based on backtracking generalized OMP, the method comprises the following steps: A magnetic resonance sounding water detector is used to sample a sparse signal at a frequency lower than the Nyquist frequency, and a noisy MRS signal is obtained; After removing the power harmonic noise in the noisy MRS signal by using a harmonic modeling method, the noisy MRS signal is converted into a two-dimensional matrix signal; A KSVD algorithm is used for dictionary learning of the two-dimensional matrix signal, a random matrix is used as an initial dictionary, the two-dimensional matrix signal is used as original data, and an updated dictionary is obtained after iteration; An independently distributed Gaussian random matrix is used to calculate a recovery matrix and measurement data; The backtracking generalized orthogonal matching pursuit algorithm is used to perform iterative calculations on the measurement data. The atomic sequence with the maximum projection of the measurement data onto the recovery matrix is ​​selected to obtain the sparse coefficient matrix of the measurement data under the recovery matrix. The recovered signal is calculated by multiplying the dictionary and the sparse coefficient matrix of the measurement data under the recovery matrix, and the first row of the recovered signal is taken to restore it to a one-dimensional signal.

[0006] Furthermore, the recovery matrix is The measurement data is ,in To recover the matrix, It is a Gaussian random matrix. It is a two-dimensional matrix signal. For a dictionary, For measurement data.

[0007] Furthermore, the formula for calculating the recovered signal is: , This is the sparse coefficient matrix of the measurement data under the recovery matrix. To restore the signal, For dictionaries.

[0008] Furthermore, the two-dimensional matrix signal is subjected to dictionary learning through the KSVD algorithm. The random matrix is ​​used as the initial dictionary and the two-dimensional matrix signal is used as the original data. After iteration, the updated dictionary is obtained, including: using the backtracking generalized orthogonal matching pursuit algorithm to calculate the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary. Learn the dictionary and update the dictionary. The formula used for this column is: , For the first in the dictionary Column vectors The sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary is the first... row vectors For signal residual, Denotes the square of the F-norm. For the first in the dictionary Column vectors The sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary is the first... row vectors It is a two-dimensional matrix signal. For a dictionary, This is the sparse coefficient matrix of the current dictionary; Extract the sparse coefficient matrix of the corresponding two-dimensional matrix signal in the current dictionary from the signal residual. The column vector of the row vector which is not 0, constitutes a new signal residual error, and singular value decomposition is performed on the new signal residual error to obtain the first column left singular vector to replace the first column of the dictionary Column vector; Repeat the same number of dictionary atoms to obtain the updated dictionary.

[0009] Further, the singular value decomposition of the new signal residual error comprises: The product of three matrices, is an orthogonal matrix with the same number of rows as the new signal residual error, and the column vector is called the left singular vector, is a rectangular diagonal matrix with the same size as the new signal residual error, and the non-negative elements on the diagonal in descending order are singular values, is an orthogonal matrix with the same number of columns as the new signal residual error, and the row vector is the right singular vector.

[0010] Further, the calculation of the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary using the backtracking generalized orthogonal matching pursuit algorithm comprises: Input the number of dictionary atoms to generate an initialized dictionary; Use the backtracking generalized orthogonal matching pursuit algorithm to calculate the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary, input the initial dictionary, the two-dimensional matrix signal and the first atomic constraint degree Sparse coding, loop calculation of the column vector of the two-dimensional matrix signal, initialization of the residual error: , is the column vector of the two-dimensional matrix signal, is the first residual error; Calculate the projection of the transpose of the current dictionary and the current residual error: , is the projection of the current residual error, is the transpose of the current dictionary, and the index of the maximum projection is added to the index set; Use the least squares method to solve the sparse coefficient atom corresponding to the index of the column vector of the two-dimensional matrix signal under the current loop; Update the first residual error according to the sparse coefficient atom; Repeat times, fill the sparse coefficient atom into the first sparse coefficient matrix A( ), represents the position of the th signal in the signal length, process all column vectors of the two-dimensional matrix signal, and finally obtain the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary.

[0011] Further, the update of the first residual error according to the sparse coefficient atom is represented as: , For the first residual, For sparse coefficient atoms, , This is the pseudo-inverse operation of the sub-dictionary set corresponding to the index set of the dictionary. This represents the set of sub-dictionaries corresponding to the index set of the dictionary.

[0012] Furthermore, the backtracking generalized orthogonal matching pursuit algorithm is used to iteratively calculate the measurement data, selecting the atomic sequence that projects the measurement data onto the maximum magnitude of the recovery matrix, thus obtaining the sparse coefficient matrix of the measurement data under the recovery matrix, including: Input recovery matrix, measurement data, second atomic constraint degree And select the number of atoms S, perform iterative calculations on the column vector of the measurement data, and initialize the second residual to be equal to the column vector of the measurement data; Calculate the modulus of the projection of the current second residual onto the transpose of the recovery matrix, and select the indices of the S largest moduli to add to the index set; Select a sub-dictionary set of the index set corresponding to the recovery matrix, and use pseudo-inverse operation to solve for the first coefficient; Update the second residual based on the first coefficient; repeat Next, fill the first coefficient into the second sparse coefficient matrix. ( ), Indicates the length of the signal. The position of each signal is used to process all atoms of the measurement data and obtain the sparse coefficient matrix of the measurement data under the recovery matrix.

[0013] Furthermore, the second residual is updated based on the first coefficient, expressed as: , A column vector representing the measurement data. This represents a set of sub-dictionaries corresponding to the index set of the recovery matrix. As the first coefficient, This is the second residual.

[0014] Compared with the prior art, the advantages of this application are as follows: The application is significantly superior to traditional OMP (orthogonal matching pursuit) and other sparse signal reconstruction in performance. The block atom selection mechanism selects multiple candidate atoms in each iteration, greatly reduces the number of iterations, thereby accelerating the convergence speed and improving the efficiency in high-dimensional or high-sparsity scenarios. The atom set is optimized through a dynamic backtracking mechanism to eliminate redundant or noise interference atoms, significantly improving the reconstruction accuracy and noise immunity, and avoiding the problem of cumulative error caused by early misselection in the traditional OMP method. In addition, in terms of computational complexity, although the pseudo-inverse calculation of a single iteration increases the computational complexity, the overall efficiency is still superior to the traditional OMP method, especially when processing large-scale data or high-sparsity signals. While maintaining low computational complexity, the reconstruction accuracy, convergence speed and robustness are significantly improved, and it is suitable for sparse representation tasks in complex environments, such as non-stationary signal processing, image restoration and low sampling rate compressed sensing scenarios. BRIEF DESCRIPTION OF DRAWINGS

[0015] Figure 1 A flowchart of a magnetic resonance sounding signal denoising method based on backtracking generalized OMP provided by an embodiment of the application is shown in the figure. Figure 2 An ideal MRS signal and spectrum provided by an embodiment of the application, (A) is a time-amplitude graph, and (B) is a frequency-amplitude graph. Figure 3 The original MRS signal, the denoised signal, and the spectrum of the ideal MRS signal provided by an embodiment of the application, (A) is a time-amplitude graph, and (B) is a frequency-amplitude graph. Figure 4 The measured noisy MRS signal and the spectrum of the denoised signal provided by an embodiment of the application, (A) is a time-amplitude graph, and (B) is a frequency-amplitude graph. DETAILED DESCRIPTION

[0016] In order to make the purpose, technical scheme and advantages of the application more clear and explicit, the following will further describe the application in combination with embodiments. It should be understood that the specific embodiments described herein are only used to explain the application and not to limit the application.

[0017] In order to make the purpose, technical scheme and advantages of the application more clear and explicit, the following will further describe the application in combination with embodiments. It should be understood that the specific embodiments described herein are only used to explain the application and not to limit the application.

[0018] Referring to Figure 1 A magnetic resonance sounding signal denoising method based on backtracking generalized OMP, as shown in the figure, comprises: S1 uses a magnetic resonance depth sounding instrument to sample sparse signals at a frequency lower than the Nyquist frequency to obtain a noisy MRS signal; the number of sampling points is , wherein the Nyquist frequency refers to the highest frequency that can effectively restore the signal in the sampled signal, and should be greater than or equal to twice the highest frequency of the effective signal.

[0019] S2 uses a harmonic modeling method to remove power harmonic noise in the noisy MRS signal, and then converts the noisy MRS signal into a two-dimensional matrix signal; wherein the sampling order is set to The harmonic modeling method considers electromagnetic field distribution and electromagnetic induction phenomena, establishes an electromagnetic model through electromagnetic theory such as Maxwell's equations, analyzes the generation mechanism of harmonics, and removes power harmonic noise according to the generation mechanism of harmonics.

[0020] The two-dimensional matrix signal is represented as: , wherein is the two-dimensional matrix signal, related to the number of sampling points and the sampling order, represents the element of the first row and the first column, is the element of the first row and the first column, is the element of the first row and the first column, and other elements have similar meanings, the value of P is the sampling order, and the value of P is the sampling point number.

[0021] S3 performs dictionary learning on the two-dimensional matrix signal through the KSVD algorithm (K times iteration singular value decomposition), uses a random matrix as the initial dictionary, and uses the two-dimensional matrix signal as the original data to obtain an updated dictionary after iteration; S4 uses an independently distributed Gaussian random matrix to calculate the recovery matrix and the measurement data, wherein the recovery matrix is calculated as and the measurement data is calculated as , wherein is the recovery matrix, is the Gaussian random matrix, is the two-dimensional matrix signal, is the dictionary, is the measurement data; , wherein the Gaussian random matrix is a matrix related to the sampling order and the parameter , is a number much lower than the Nyquist frequency, and the Gaussian random matrix is also represented as

[0022] ​​​​​S5 uses the backtracking generalized orthogonal matching pursuit algorithm to perform iterative calculations on the measurement data, selects the atomic sequence of the maximum projection of the measurement data onto the recovery matrix, and obtains the sparse coefficient matrix of the measurement data under the recovery matrix. S6 calculates the recovered signal based on the sparse coefficient matrix of the dictionary and measurement data under the recovery matrix, and takes the first row of the recovered signal to restore it to a one-dimensional signal, where: the recovered signal , This is the sparse coefficient matrix of the measurement data under the recovery matrix. To restore the signal.

[0023] In one embodiment, a dictionary learning process is performed on a two-dimensional matrix signal using the KSVD algorithm. A random matrix is ​​used as the initial dictionary, and the two-dimensional matrix signal is used as the original data. An updated dictionary is obtained after iteration, including: using the backtracking generalized orthogonal matching pursuit algorithm to calculate the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary. Learn the dictionary and update the dictionary. The formula used in this column is: , For the first in the dictionary Column vectors The sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary is the first... row vectors For signal residual, Denotes the square of the F-norm. For the first in the dictionary Column vectors The sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary is the first... row vectors This is the sparse coefficient matrix of the current dictionary; Here, the signal residual refers to the difference between the noisy signal Y and the signal reconstructed from the current dictionary and the sparse coefficient matrix under the current dictionary. Extract the sparse coefficient matrix of the corresponding two-dimensional matrix signal in the current dictionary from the signal residual. The column vectors with non-zero row vectors form the new signal residual, and singular value decomposition is performed on the new signal residual. This singular value decomposition includes transforming the new signal residual into... The product of three matrices, It is an orthogonal matrix with the same number of rows as the residual of the new signal, and its column vectors are called left singular vectors. It is a rectangular diagonal matrix of the same size as the residual of the new signal, with the non-negative elements on the diagonal arranged in descending order being the singular values. It is an orthogonal matrix with the same column number as the new signal residual, and its row vectors are right singular vectors. The first column of the left singular vector is used to replace the dictionary. column vector; Repeat the same number of dictionary atoms to obtain the updated dictionary.

[0024] When obtaining the updated dictionary, it is necessary to calculate the sparse coefficient matrix under the current dictionary. In an embodiment, the backtracking generalized orthogonal matching pursuit algorithm is used to calculate the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary, including: Input the number of dictionary atoms, adopt to represent, generate the initialized dictionary, and the dictionary is related to the number of dictionary atoms and the sampling order, which can be represented as , in order to facilitate the following ; The backtracking generalized orthogonal matching pursuit algorithm is used to calculate the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary, and the input includes the initial dictionary, the two-dimensional matrix signal and the atom constraint degree Sparse coding, loop calculation of column vectors of the two-dimensional matrix signal, initialization of the residual: , is the column vector of the two-dimensional matrix signal, is the first residual; Calculate the projection of the transpose of the current dictionary and the current residual: , is the projection of the current residual, is the transpose of the current dictionary, select the index of the maximum projection and add it to the index set, and the index set is represented by , is a number from 1 to , which is the current loop number; Use the least square method to solve the sparse coefficient atom corresponding to the index of the column vector of the two-dimensional matrix signal under the current loop; Update the first residual according to the sparse coefficient atom: , wherein is the first residual, is the sparse coefficient atom, , is the pseudo-inverse operation of the sub-dictionary set corresponding to the index set of the dictionary, represents the sub-dictionary set corresponding to the index set of the dictionary; Repeat times, that is, the number of times is related to the atom constraint degree, fill the sparse coefficient atom into the first sparse coefficient matrix A( ), represents the position of the signal in the signal length , process all column vectors of the two-dimensional matrix signal, and finally obtain the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary. The sparse coefficient matrix under the current dictionary is the last obtained first sparse coefficient matrix A( ).

[0025] In an embodiment, the calculation of the recovery signal also requires an important parameter, i.e., the sparse coefficient matrix of the measurement data under the recovery matrix. The sparse coefficient matrix of the measurement data under the recovery matrix is similar to the sparse coefficient matrix under the current dictionary, both of which are iteratively operated using the backtracking generalized orthogonal matching pursuit algorithm, the difference being that: The backtracking generalized orthogonal matching pursuit algorithm is used to iteratively operate the measurement data, the atom sequence with the maximum projection of the measurement data and the recovery matrix is selected, and the sparse coefficient matrix of the measurement data under the recovery matrix is obtained , comprising: Inputting the recovery matrix, the measurement data, the second atom constraint degree , and the number of selected atoms S, performing loop calculation on the column vectors of the measurement data, and initializing the second residual to be equal to the column vectors of the measurement data; Calculating the projection of the current second residual and the transpose of the recovery matrix, which is expressed by the formula: , is the second residual, is the projection of the transpose of the recovery matrix, indicates taking the modulus, indicates the transpose of the recovery matrix, and the indices of the maximum S modulus are added to the index set; Selecting the sub-dictionary set of the recovery matrix corresponding to the index set, using pseudo-inverse operation to solve the first coefficient, which is expressed as: , is the column vector of the measurement data, indicates pseudo-inverse operation, indicates the sub-dictionary set of the recovery matrix corresponding to the index set; Updating the second residual according to the first coefficient: ; Repeat times, fill the first coefficient into the second sparse coefficient matrix ( ), indicates the position of the th signal in the signal length, process all atoms of the measurement data, and obtain the sparse coefficient matrix of the measurement data under the recovery matrix , i.e., the final second sparse coefficient matrix ( ).

[0026] Simulation experiments are performed on the method of the present application.

[0027] This embodiment is a simulation experiment of the method of the present application carried out in the MATLAB R2023b programming environment.

[0028] According to the mathematical formula Construct a pure MRS signal, amplitude 200nV, Larmor frequency The frequency is 2325Hz, and the relaxation time is... It takes 0.25 seconds. For the length of time, ,like Figure 2 As shown. Among them, Figure 2 (A) in the graph is a time-amplitude plot. Figure 2 (B) in the diagram is a frequency-amplitude graph. The signal sampling frequency... Length of time The number of sampling points is P.

[0029] Set the standard deviation to Random noise is added to the MRS signal. Power frequency harmonic noise is removed from the signal using harmonic modeling methods. The sampling order is set. =2000, converting the signal into a two-dimensional matrix signal, setting the number of measurement points M=6000, generating independently distributed Gaussian random matrices for calculating the reconstruction matrix and measurement data. Comparative analysis of the time and frequency domain information of the original MRS signal, the ideal MRS signal, and the denoised signal (reconstructed signal), such as... Figure 3 As shown, where Figure 3 (A) in the graph is a time-amplitude plot. Figure 3 (B) in the diagram is a frequency-amplitude diagram. Calculations show that the initial amplitude error is -0.4043%, the relaxation time error is -2.2084%, and the SNR is 23.4293dB. The obtained parameter errors all meet the application requirements.

[0030] In another embodiment of this application, measured data collected on-site at Changchun Cultural Square is used as the processing object of the method of this application. To verify the effectiveness of the method of this application, as follows... Figure 4 As shown, the measured noisy MRS signal and the denoised signal (reconstructed signal) are compared and analyzed in the time and frequency domains. Figure 4 (A) is a time-amplitude plot. Figure 4 (B) shows the frequency-amplitude graph. After denoising, the measured data were calculated and found to show good parameter fitting under different signal-to-noise ratios (SNRs), with a high SNR improvement exceeding 23 dB. The obtained key parameter errors all meet application requirements.

[0031] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A method for magnetic resonance sounding signal denoising based on backtracking generalized OMP, characterized in that, The method comprises the following steps: A magnetic resonance depth sounder is used to sample sparse signals at a frequency lower than the Nyquist frequency to obtain a noisy MRS signal; A harmonic modeling method is used to remove power harmonic noise in the noisy MRS signal, and the noisy MRS signal is converted into a two-dimensional matrix signal; A KSVD algorithm is used to learn a dictionary from the two-dimensional matrix signal, a random matrix is used as an initial dictionary, and the two-dimensional matrix signal is used as original data, and an updated dictionary is obtained after iteration; An independent distributed Gaussian random matrix is used to calculate a recovery matrix and measurement data; A backtracking generalized orthogonal matching pursuit algorithm is used to iteratively operate on the measurement data, an atom sequence with the maximum projection of the measurement data and the recovery matrix is selected, and a sparse coefficient matrix of the measurement data under the recovery matrix is obtained; A recovery signal is calculated according to the product of the dictionary and the sparse coefficient matrix of the measurement data under the recovery matrix, and the first row of the recovery signal is restored to a one-dimensional signal.

2. The method according to claim 1, wherein, The recovery matrix is , the measurement data is , wherein is the recovery matrix, is a Gaussian random matrix, is a two-dimensional matrix signal, is a dictionary, is the measurement data.

3. The method of claim 1, wherein, The calculation formula of the recovery signal is: , is the sparse coefficient matrix of the measurement data under the recovery matrix, is the recovery signal, is the dictionary.

4. The method of claim 3, wherein, The KSVD algorithm is used to learn a dictionary from the two-dimensional matrix signal, a random matrix is used as an initial dictionary, and the two-dimensional matrix signal is used as original data, and an updated dictionary is obtained after iteration, comprising: The dictionary learning is performed, and the first column of the dictionary is updated The formula used is: , is the i-th column vector of the dictionary D, is the i-th column vector of the dictionary D, is the i-th column vector of the dictionary D, is the i-th column vector of the dictionary D, is the signal residual, denotes the square of the F-norm, is the i-th column vector of the dictionary D, is the i-th column vector of the dictionary D, is the i-th column vector of the dictionary D, is the i-th column vector of the dictionary D, is the two-dimensional matrix signal, is the dictionary, is the sparse coefficient matrix under the current dictionary; extracting the first column of the sparse coefficient matrix of the corresponding two-dimensional matrix signal in the signal residual under the current dictionary The column vector with a row vector of 0 forms a new signal residual, and singular value decomposition is performed on the new signal residual to obtain the first column left singular vector to replace the first column vector of the dictionary The column vector with a row vector of 0 forms a new signal residual, and singular value decomposition is performed on the new signal residual to obtain the first column left singular vector to replace the first column vector of the dictionary A backtracking generalized orthogonal matching pursuit algorithm is used to calculate a sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary; 5. The method of claim 4, wherein, The singular value decomposition of the new signal residual includes: a product of three matrices, is an orthogonal matrix with the same number of rows as the new signal residual, and its column vectors are called left singular vectors, is a rectangular diagonal matrix with the same size as the new signal residual, and the non-negative elements arranged in descending order on the diagonal are singular values, is an orthogonal matrix with the same number of columns as the new signal residual, and its row vectors are right singular vectors.

6. The method of claim 4, wherein, The above steps are repeated the same number of times as the number of dictionary atoms to obtain the updated dictionary. The backtracking generalized orthogonal matching pursuit algorithm is used to calculate the two-dimensional matrix signal under the current dictionary, comprising: calculating a sparse coefficient matrix of a two-dimensional matrix signal under a current dictionary using a backtracking generalized orthogonal matching pursuit algorithm, inputting an initial dictionary, the two-dimensional matrix signal and a first atom constraint degree performing sparse coding, cyclically calculating column vectors of the two-dimensional matrix signal, initializing a residual error: , column vectors of the two-dimensional matrix signal, is a first residual error; Compute the projection of the current dictionary transpose with the current residual: , is the projection of the current residual, is the transpose of the current dictionary, select the index of the maximum projection to add to the index set; The number of dictionary atoms is inputted to generate an initialized dictionary; A least square method is used to solve sparse coefficient atoms corresponding to the index of the column vector of the two-dimensional matrix signal in the current cycle; repeat Next, fill the sparse coefficient atoms into the first sparse coefficient matrix A( ), Indicates the length of the signal. The position of each signal is used to process all column vectors of the two-dimensional matrix signal, and finally the sparse coefficient matrix of the two-dimensional matrix signal under the current dictionary is obtained.

7. The method according to claim 6, wherein, The first residual is updated according to the sparse coefficient atom, denoted as: wherein, is the first residual, is the sparse coefficient atom, , is a pseudo-inverse operation of the dictionary corresponding index set sub-dictionary set, denotes the dictionary corresponding index set sub-dictionary set.

8. The method of claim 1, wherein, The first residual is updated according to the sparse coefficient atoms; inputting a recovery matrix, measurement data, and a second atomic constraint degree and selecting an atomic number S, performing loop calculation on a column vector of the measurement data, and initializing a second residual error to be equal to the column vector of the measurement data; The backtracking generalized orthogonal matching pursuit algorithm is used to iteratively operate on the measurement data, an atom sequence with the maximum projection of the measurement data and the recovery matrix is selected, and a sparse coefficient matrix of the measurement data under the recovery matrix is obtained, comprising: The modulus of the projection of the current second residual and the transpose of the recovery matrix is calculated, and the indexes of the maximum S modulus are added to the index set; A sub-dictionary set corresponding to the index set of the recovery matrix is selected, and a pseudo-inverse operation is used to solve the first coefficient; The second residual is updated according to the first coefficient; repetition second sparse coefficient matrix with the first coefficient ( ), denotes the position of the atom, all atoms of the measurement data are processed to obtain a sparse coefficient matrix of the measurement data under the recovery matrix.

9. The method of claim 8, wherein, updating the second residual according to the first coefficient, denoted as: , denotes a column vector of the measurement data, denotes a sub-dictionary set of the recovery matrix corresponding to the index set, is the first coefficient, is the second residual.

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