Method for denoising magnetic resonance signal by combining shaping prony algorithm with spatial prediction filter

Abstract

The application discloses a magnetic resonance signal denoising method combining a shaping Prony algorithm and a spatial prediction filter. Three-dimensional coordinate axis parameters are determined, and a spatial adaptive prediction filter structure is constructed. After data preparation is performed on received noisy signal data of multiple measuring points, a three-dimensional data body is established, Prony decomposition is performed on signal data in the three-dimensional data body by using a Prony algorithm, a shaping regularization method is introduced to solve a least square value, Prony transformation values of the noisy signal of the measuring points and spatial prediction filter coefficient values corresponding to the measuring points are obtained. A pure signal of a target measuring point is predicted and approximated by using Prony components of adjacent measuring points and the spatial prediction filter coefficient values, and suppression of random noise of the noisy signal of the target measuring point is realized. By using the method, coil laying work is reduced, detection efficiency is improved, random noise can be further effectively suppressed, complex effective signals are protected, and the signal-to-noise ratio of the magnetic resonance signal is improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of nuclear magnetic resonance data processing, and particularly relates to a magnetic resonance signal denoising method combining a shaping Prony algorithm and a spatial prediction filter. BACKGROUND

[0002] The magnetic resonance groundwater detection technology is the only non-invasive geophysical method for directly detecting groundwater at present, and has been widely applied to groundwater resource exploration, disaster water detection and interpretation in recent years. However, due to the extremely weak magnetic resonance signal, only nV level (1nV=10 -9 V) can be achieved, and the field geomagnetic field cannot be shielded, so the detection process is inevitably disturbed by the surrounding environmental noise, including spike noise, power harmonic noise and random noise. Due to the wide frequency band of random noise, the statistical characteristics are random and have no uniform rule, and the effective signal is often mixed with the random noise, which is difficult to suppress.

[0003] For the noise of the nuclear magnetic resonance groundwater detection data, experts in the field have researched a variety of data processing methods to eliminate, such as the notch method, the wavelet transform method, the far-end reference denoising method and the modeling method, etc. However, these methods are mostly for spike noise and power harmonic noise with certain characteristics and statistical rules. For random noise, the stacking method is generally used in the international groundwater magnetic resonance detection field. In the use of the stacking method to suppress random noise, a higher stacking number is needed to be provided for the signal at the cost of detection efficiency, but only part of the random noise can be suppressed.

[0004] The patent CN107783200B proposes a full-wave magnetic resonance signal random noise reduction method combining EMD and TFPF algorithms. The decomposition characteristics of the EMD algorithm are used to decompose the full-wave magnetic resonance signal into different intrinsic mode components, and then the TFPF algorithm is used to encode the signal dominant mode component into the instantaneous frequency of the unit amplitude analytical signal. The time-frequency distribution along the instantaneous frequency concentration characteristics of the analytical signal are used to suppress random noise. For the multi-dimensional data obtained by the field experiment, the single-point signal data is used for denoising processing by the algorithm, the processing process is relatively cumbersome, the calculation time is long, and the anti-interference ability is weak.

[0005] Li Bang published in Journal of Jilin University (Earth Sciences Edition), 2022, 52 (3); 775-784, the paper "Groundwater magnetic resonance data random noise suppression method based on convolutional neural network", the random noise suppression problem of MRS signal in strong noise is studied, based on the convolutional neural network framework, the training method of supervised learning is adopted, the nonlinear mapping relationship between the time-frequency spectrum of noisy signal and the time-frequency spectrum of original noise-free signal is obtained, and then the noise suppression of magnetic resonance signal is realized. This method processes random noise for single MRS signal data, and the training model is time-consuming, iterative, and cumbersome in the early stage. SUMMARY

[0006] The technical problem to be solved by the present application is to solve the existing problems in the field of magnetic resonance signal random noise suppression. A magnetic resonance signal denoising method combining shaping Prony algorithm and spatial prediction filter is provided. Under complex background noise, random noise is effectively suppressed, the signal-to-noise ratio of MRS signal is improved, the measurement efficiency is improved, the number of emission pulses is reduced, and more accurate signal data is provided for subsequent magnetic resonance signal inversion work.

[0007] The present application is implemented as follows,

[0008] A magnetic resonance signal denoising method combining shaping Prony algorithm and spatial prediction filter, the method comprises:

[0009] Step A: using a ground magnetic resonance groundwater detection instrument to collect multiple pulse moment noisy data at multiple measurement points on a measurement line at a sampling rate fs, preparing data for noisy data S(t) of all measurement points, arranging the noisy signal S(t) as S(t,x,y) according to t as time, x as pulse moment, and y as three-dimensional coordinate system of measurement point;

[0010] Step B: constructing a spatial prediction filter, dividing the spatial data body, determining the selection time L, the measurement point number M and the pulse moment number N according to the number of emission pulse moments and the number of measurement points, and forming an MxNxL three-dimensional data body;

[0011] Step C: for the noisy signal S(t,x,y) in the MxNxL three-dimensional data body, the Prony component S pi,j,k (t,x,y) of the adjacent detection point is calculated to obtain the least square error prediction value between the noisy signal S(t,x,y) of the target detection point and the spatial prediction filter generated value , that is, the random noise value:

[0012]

[0013] Wherein, the spatial prediction filter coefficient B i,j,k (t,x,y) is the noisy signal Si,j,k The Prony component S of (t,x,y) pi,j,k The filter coefficients corresponding to (t,x,y) Let S(t,x,y) be the sum of the Prony components of the signals of neighboring detection points around the noisy signal S(t,x,y) of the target detection point, where i,j,k are the indices of the distance from the target detection point along time, pulse moment, and measurement point movement, respectively, and L,M,N control the size of the filter along the time-space direction.

[0014] Step D: Obtain the Prony component S of the signal from the neighboring detection point in C. pi,j,k (t,x,y) and its corresponding spatial prediction filter coefficient B i,j,k (t,x,y), for the noisy signal S of the neighboring probe points in the three-dimensional data volume. i,j,k (t,x,y) is subjected to a Prony transformation, and a set of P exponential functions with arbitrary amplitude, phase, frequency and attenuation factor are used to transform S. i,j,k The approximate representation of (t,x,y) is S. pi,j,k (t,x,y), using the noisy data S(t,x,y) at the target detection point in the original data as the fitting object, and using the Prony component S of the calculated neighboring detection point signal. pi,j,k (t,x,y) and the corresponding spatial prediction filter coefficients B i,j,k The problem is to construct a minimization problem (t, x, y), and then use the integer regularized least squares method to solve for each Prony component S. pi,j,k The spatial prediction filter coefficient B corresponding to (t,x,y) i,j,k (t,x,y);

[0015] Step E: Within the spatial prediction filter, determine the location of the target detection point, and use the Prony components S of the signals from each measurement point adjacent to the target detection point in the 3D data volume obtained in Step D. pi,j,k (t,x,y) and its corresponding filter coefficients B i,j,k (t,x,y) represents the pure signal at the target detection point. Perform predictive filtering.

[0016]

[0017] Using the same shaping regularization method, we introduce shaping regularization constraints to solve for the least squares solution, ensuring that the noisy signal S(t,x,y) at the target detection point is consistent with the pure signal generated by spatial prediction filtering. The predicted squared error between Minimum, meaning the random noise is minimized.

[0018] Furthermore, it also includes using the signal-to-noise ratio as an evaluation metric:

[0019]

[0020] where S(t,x,y) represents the measured noisy data of the target detection point, represents the predicted pure signal of the target detection point, and rms represents the root mean square, with the unit of dB.

[0021] Further, the shaping Prony transform in step D has the following specific steps:

[0022] D1, performing Hilbert transform on the MRS noisy signal S(t,x,y) of the three-dimensional data body to obtain complex data;

[0023] D2, approximating the noisy signal S(t,x,y) using a linear combination of complex exponential functions, where, n=1,2,3,...,N is the number of sampling points, Δt is the sampling period, p is the decomposition number of the Prony algorithm, the initial phase in radians rad, the attenuation factor in s -1 and the frequency in Hz;

[0024] D3, adding a shaping regularization method in the conjugate gradient to calculate the least square solution of the constant coefficient vector r j in the characteristic polynomial of the difference equation

[0025]

[0026] D4, calculating the root w j of the characteristic polynomial, and calculating the attenuation factor α j and the frequency f j of the Prony component according to

[0027] D5, using the same method as in step D3 to solve the least square solution of the coefficient q j in the matrix form of the linear combination W·q=S, and calculating the initial amplitude A j and the initial phase θ j of the Prony component according to A j =|q j |, θ j =arg(q j );

[0028] D6, allowing the filter coefficient B i,j,k (t,x,y) to vary with time, pulse number, and measurement point in the spatial prediction filter structure, i,j,k are the time and space indexes of the filter coefficient point S(t,x,y) to be solved, and the Prony component S pi,j,k ​​The spatial prediction filter coefficient B corresponding to (t, x, y) i,j,k (t, x, y) is solved by using a shaped regularization least square method.

[0029]

[0030] where S pi,j,k (t, x, y) represents the Prony component of the neighboring point signal of the measured point noise signal S(t, x, y) to be solved, S represents a shaped regularization operator, and δ represents a shaped regularization scale parameter.

[0031] Further, the shaped regularization least square solution includes:

[0032] The least square is a condition of minimizing the weighted combination of the residual term and the two norm of the boundary constraint, that is,

[0033]

[0034] λ is a scalar regularization parameter for controlling the weight of the objective function, and D is usually selected as a first or second order inverse operator;

[0035] The coefficient matrix W is regarded as a unit matrix I by the shaped regularization algorithm, and the form of the regularization solution is converted to The inverse matrix in the formula is defined as a shaped regularization operator S to control the shape of the parameter q, that is, S=(I+λ 2 D′D) -1 The shaped regularization solution is represented as:

[0036] The scale variable unit 1 / δ is introduced by the coefficient matrix W, and the shaped regularization solution is, The data fitting is performed in a specified direction by setting the shaped operator S, and the linear prediction coefficient r j and the least square solution of q j are calculated to obtain the root w j of the characteristic polynomial, and the amplitude, frequency, phase and attenuation coefficient of the Prony component are solved;

[0037] The Prony component of the noise signal is obtained A k , f k , θ k , α k are the amplitude, frequency, phase and attenuation coefficient of the kth component, respectively, and p is the model order.

[0038] Compared with the prior art, the present application has the beneficial effects that: the present application uses Prony components to approximately express MRS signal data, uses multi-data-point Prony components and target data-point filter coefficients to predictively estimate, suppresses random noise of a target region in a three-dimensional space prediction filter structure, and significantly improves detection efficiency by filtering and estimating long-short pulse moments and using adjacent measurement points to filter and estimate random noise of the target region, effectively suppresses random noise while retaining more signal effective components, has strong adaptability to particularly weak magnetic resonance signals, reduces the number of emission pulse moments while obtaining good noise reduction effect, significantly enhances signal-to-noise ratio, improves the accuracy of post-inversion interpretation, and has great practical application value. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 A magnetic resonance signal noise reduction method flowchart of the shaped Prony space prediction filter provided for the embodiment of the present application;

[0040] Figure 2 The three-dimensional shaped Prony prediction filter network structure constructed for the embodiment of the present application, (a) is a three-dimensional data body formed by the noisy data S(t) received by all measurement points of the water finder, (b) is to determine and select the time number L, the measurement point number M and the pulse moment number N according to the emission pulse moment number and the measurement point number near the target detection point, and form an MxNxL three-dimensional prediction filter structure for prediction filtering;

[0041] Figure 3 The target region magnetic resonance signal containing random noise provided for the embodiment of the present application;

[0042] Figure 4 The magnetic resonance signal after attenuation of random noise provided for the embodiment of the present application. DETAILED DESCRIPTION

[0043] In order to make the purpose, technical scheme and advantages of the present application clearer, the present application will be further described in detail below with embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application, and are not used to limit the present application.

[0044] Figure 1 The magnetic resonance signal noise reduction method flowchart based on the shaped Prony space prediction filter.

[0045] Referring to Figure 1 The specific steps of suppressing random noise of a magnetic resonance signal based on the shaped Prony space prediction filter are as follows:

[0046] A, the ground magnetic resonance groundwater detection instrument is used to collect multi-pulse moment noise signal of multiple measuring points on the measuring line with the sampling rate fs, data preparation is carried out on the noise signal S(t) of each measuring point, and the noise signal S(t) is arranged as S(t,x,y) according to t as time, x as pulse moment and y as three-dimensional coordinates of the measuring point;

[0047] Figure 2 The three-dimensional shaping Prony prediction filter network structure provided by the embodiment of the application is constructed. Referring to Figure 2 As shown in the figure, the following are specific steps for constructing the shaping Prony spatial prediction filter;

[0048] B, the spatial prediction filter is constructed, since the MRS full-wave signal data sampling frequency is high, the sampling quantity is more, the spatial data body needs to be divided, the selection time number L, the measuring point number M and the pulse moment number N are determined according to the number of transmitting pulse moments and the number of measuring points, and the M*N*L three-dimensional data body is formed;

[0049] Figure 3 The target area magnetic resonance signal containing random noise provided by the embodiment of the application;

[0050] Referring to Figure 3 As shown in the figure, the noise signal S(t,x,y) in the M*N*L three-dimensional data body used below is the target area magnetic resonance signal containing random noise provided in the figure;

[0051] C, for the noise signal S(t,x,y) in the M*N*L three-dimensional data body, the least square error prediction value between the noise signal S(t,x,y) of the target detection point and the spatial prediction filter generation value , that is, the random noise value, can be calculated through the Prony component Spi,j,k(t,x,y) of the adjacent detection point:

[0052]

[0053] Wherein, the spatial prediction filter coefficient B i,j,k (t,x,y) is the Prony component S i,j,k (t,x,y) of the adjacent detection point noise signal S pi,j,k (t,x,y) corresponding to the filter coefficient, The Prony component and of the adjacent detection point signal around the target detection point noise signal S(t,x,y), i,j,k are indexes moving along time, pulse moment and measuring point respectively. L, M, N control the size of the filter along the time space direction.

[0054] D, in order to obtain the Prony component Sp i,j,k(t, x, y) and the corresponding spatial prediction filter coefficient value B i,j,k (t, x, y) to the noisy signal S i,j,k (t, x, y) is approximated by a set of P exponential functions with arbitrary amplitude, phase, frequency and damping factor, that is, S i,j,k (t, x, y) is approximated by a set of P exponential functions with arbitrary amplitude, phase, frequency and damping factor, that is, S i,j,k (t, x, y) is approximated by a set of P exponential functions with arbitrary amplitude, phase, frequency and damping factor, that is, S i,j,k (t, x, y) and the corresponding spatial prediction filter coefficient value B i,j,k (t, x, y) is approximated by a set of P exponential functions with arbitrary amplitude, phase, frequency and damping factor, that is, S i,j,k (t, x, y) and the corresponding spatial prediction filter coefficient value B i,j,k (t, x, y);

[0055] E, in the spatial prediction filter, the position of the target detection point is determined, and the Prony component Sp of each measurement point signal of the target detection point in the three-dimensional data body is solved in step D i,j,k (t, x, y) and the corresponding filter coefficient B i,j,k (t, x, y) to the target detection point pure signal is predicted and filtered,

[0056]

[0057] The same as the above using the shaping regularization method, the shaping regularization constraint condition is introduced to solve the least square solution, so that the square error prediction value between the noisy signal S(t, x, y) at the target detection point and the pure signal generated by the spatial prediction filter is minimized, that is, the random noise is minimized, so as to achieve the suppression effect of the random noise in the noisy signal;

[0058] Figure 4 The magnetic resonance signal after the attenuation of the random noise provided by the embodiment of the application;

[0059] Referring to Figure 4 , the prediction filtering result of the pure signal of the target detection point is shown;

[0060] F, the evaluation index used is the signal-to-noise ratio:

[0061]

[0062] Wherein, S(t, x, y) represents the measured noisy data of the target detection point,​​ Prony represents the target detection point prediction pure signal, rms represents the root mean square, and the unit is dB.

[0063] Further, the specific steps of Prony transformation in step D are as follows:

[0064] D1, Hilbert transform is performed on the three-dimensional data body MRS noise signal S(t, x, y) to obtain complex data;

[0065] D2, a linear combination of a certain number of complex exponential functions is used to approximate the noise signal S(t, x, y), Wherein, n=1, 2, 3,..., N is the number of sampling points, Δt is the sampling period, p is the decomposition number of Prony algorithm, the initial phase is rad, the attenuation factor is s -1 unit, and the frequency is Hz;

[0066] D3, the Gaussian smoothing operator with a radius of p is specified as the shaping regularization operator S, and the median of the matrix T is the auxiliary variable δ;

[0067] D4, the shaping regularization method is added in the conjugate gradient, and the least square solution of the constant coefficient vector rj in the characteristic polynomial of the difference equation is calculated;

[0068]

[0069] D5, the root w j of the characteristic polynomial is calculated, and the attenuation factor α j and the frequency f j of the Prony component are calculated according to ;

[0070] D6, the median of the matrix W is re-specified as the shaping regularization auxiliary variable δ;

[0071] D7, the same method as in D4 is used to solve the least square solution of the coefficient qj in the matrix form of the linear combination W·q=S, and the initial amplitude A j and the initial phase θ j of the Prony component are calculated according to A j =|q j |, θ j =arg(q j );

[0072] D8, the filter coefficient B i,j,k(t, x, y) varies with time, pulse moment, measuring point, i, j, k are time and space indexes of the filter coefficient point S(t, x, y) to be solved, the Prony component Sp of the noisy signal S(t, x, y) obtained by Prony transformation is solved i,j,k (t, x, y) corresponding to the space prediction filter coefficient B i,j,k (t, x, y) is solved by using a shaped regularization least square method;

[0073]

[0074] wherein, Sp i,j,k (t, x, y) represents the Prony component of the adjacent measuring point signal of the noisy signal S(t, x, y) of the measuring point to be solved, S represents a shaped regularization operator, and δ represents a shaped regularization scale parameter;

[0075] Further, the shaped regularization least square solution in step D is characterized in that,

[0076] d1, the least square purpose is to make the weighted combination of the residual term and the boundary constraint two norms reach the minimum value, that is,

[0077]

[0078] λ is a scalar regularization parameter for controlling the weight of the objective function, and D is usually selected as a first or second order inverse operator;

[0079] d2, the shaped regularization algorithm is to regard the coefficient matrix W as a unit matrix I, and the form of the regularization solution is converted to The inverse matrix in the formula is defined as a shaped regularization operator S to control the shape of the parameter q, that is, S=(I+λ 2 D′D) -1 That is, the shaped regularization solution is represented as

[0080] d3, a scale variable unit 1 / δ is introduced for the coefficient matrix W, and the shaped regularization solution is, By setting the shaping operator S, the data fitting is performed in a specified direction, and the least square solution of the linear prediction coefficient r j and q j is calculated, the root w j of the characteristic polynomial is obtained, and the amplitude, frequency, phase and attenuation coefficient of the Prony component are solved.

[0081] d4, the Prony component A , f k , θ k , α k of the noisy signal is obtained krespectively the amplitude, frequency, phase and attenuation coefficient of the kth component, and p is the model order.

[0082] The above description is only the preferred embodiment of the present application, and is not intended to limit the present application. Any modification, equivalent replacement and improvement made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A method for magnetic resonance signal denoising by combining a shaped Prony algorithm with a spatial prediction filter, characterized in that, The method comprises: Step A: using a ground magnetic resonance groundwater detection instrument to collect a plurality of pulse moment noisy data under a plurality of measuring points on a measuring line at a sampling rate fs, performing data preparation on the noisy data S(t) of all measuring points, and arranging the noisy signal S(t) into S(t, x, y) according to t as time, x as a pulse moment, and y as a three-dimensional coordinate system of a measuring point; Step B: constructing a spatial prediction filter, dividing the spatial data body, determining the selection time number L, the measuring point number M and the pulse moment number N according to the number of transmitted pulse moments and the number of measuring points, and forming an MxNxL three-dimensional data body; Step C: For the noisy signal S(t,x,y) in the MxNxl three-dimensional data volume, the least square error prediction value, i.e. the random noise value, between the noisy signal S(t,x,y) of the target detection point and the spatial prediction filter generated value pi,j,k is calculated by the Prony component S(t,x,y) of the adjacent detection points . where B is the spatial prediction filter coefficient i,j,k S (t, x, y) is the noisy signal of the target probe point i,j,k S (t, x, y) is the Prony component of the noisy signal of the target probe point pi,j,k B is the filter coefficient corresponding to S (t, x, y) S (t, x, y) is the Prony component of the noisy signal of the target probe point, i, j, k are the indexes of the neighboring probe points along time, pulse width and measurement point, respectively, and L, M, N control the size of the filter along the time and space directions. Step D: Obtain Prony components S of the noisy signal of the neighboring probe point in C pi,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B i,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B i,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B i,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B pi,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B pi,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B i,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B pi,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B i,j,k (t,x,y) and its corresponding spatial prediction filter coefficient value B Step E: In the spatial prediction filter, the position of the target probe point is determined, and the Prony components S of each measuring point signal of the adjacent probe points in the three-dimensional data body solved in step D are adopted pi,j,k (t,x,y) and its corresponding filter coefficient B i,j,k (t,x,y) to the target probe point pure signal Prediction filtering is performed, Using the same shaping regularization method, we introduce shaping regularization constraints to solve for the least squares solution, ensuring that the noisy signal S(t,x,y) at the target detection point is consistent with the pure signal generated by spatial prediction filtering. The predicted squared error between Minimum, meaning minimum random noise; The shaping Prony transform in the step D is specifically as follows: D1, performing Hilbert transform on the three-dimensional data body MRS noisy signal S(t, x, y) to obtain complex data; D2, approximation of the noisy signal S(t, x, y) using a linear combination of complex exponential functions, wherein, n = 1, 2, 3,..., N is the number of sampling points, At is the sampling period, p is the decomposition number of the Prony algorithm, the initial phase in radians rad, the attenuation factor in s -1 -1, and the frequency in Hz; D3, adding a shaping regularization method to the conjugate gradient, computing the characteristic polynomial of the difference equation the constant coefficient vector r in j the least squares solution of D4, computing the roots w of the characteristic polynomial j , according to computing the damping factors a of the Prony components j and the frequencies f j ; D5. Solve the least-squares solution of the linear combination W - q = S in the matrix form of the coefficients q using the same method as in step D3 j j j j j j j ;​​​​​​ D6. Allow filter coefficient B in spatial prediction filter structure i,j,k (t,x,y) varies with time, pulse moment, measurement point, i,j,k are time and space indexes of the filter coefficient point S(t,x,y) to be solved, Prony component S pi,j,k (t,x,y) corresponding to the spatial prediction filter coefficient B i,j,k (t,x,y) is solved by using the regularized least square method. where S pi,j,k (t, x, y) denotes the Prony component of the neighboring probing point signal of the measured point noisy signal S(t, x, y) of the filter coefficient to be solved, S denotes a shaping regularization operator, and δ denotes a shaping regularization scale parameter.

2. The magnetic resonance signal denoising method of the shaping Prony algorithm combined with the spatial prediction filter according to claim 1, characterized in that, Further comprising using a signal-to-noise ratio as an evaluation index: Where S(t, x, y) represents the measured noisy data of the target detection point, represents the predicted pure signal of the target detection point, and rms represents the root mean square, with the unit of dB.

3. A method of magnetic resonance signal denoising according to the shaped Prony algorithm of claim 1 in combination with a spatial prediction filter, characterized in that, The shaping regularized solution least square solution comprises: The least square is a condition that makes a weighted combination of a residual term and a two-norm boundary constraint reach a minimum value, that is λ is a scalar regularization parameter for controlling the weight of the objective function, and D is selected as a first-order or second-order inverse operator; By considering the coefficient matrix W as the identity matrix I through the shape regularizing algorithm, the form of the regularized solution is converted to The inverse matrix in the formula is defined as the shape regularizing operator S to control the shape of the parameter q, that is, S=(I+λ 2 D′D) -1 The shape regularized solution is represented as: The coefficient matrix W introduces the scaling variable unit 1 / δ, and the shaping regularization solution is By setting the shaping operator S, the data fitting is performed in the specified direction, and the linear prediction coefficient r j And the least square solution of q j The root w j of the characteristic polynomial is obtained, and the amplitude, frequency, phase and attenuation coefficient of the Prony component are solved. Prony components of a noisy signal A k , f k , θ k , α k are the amplitude, frequency, phase and damping coefficient of the kth component, respectively, and p is the model order.

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