CN115508835A - Tomography SAR three-dimensional imaging method based on blind compressed sensing - Google Patents

Abstract

The invention discloses a tomography SAR three-dimensional imaging method based on blind compressed sensing, which relates to the technical field of tomography SAR imaging and comprises the following steps: s1: preprocessing an SAR sequence dataset acquired from an observation object region, and sequentially constructing high-latitude signals by pixels in each image and arranging the high-latitude signals into a matrix form; s2: reconstructing image information by using a blind compressed sensing frame by using the correlation of the elevation directions of adjacent azimuth-distance units, and modeling an image signal into a product of a sparse matrix and a dictionary matrix; establishing a tomography SAR imaging model based on blind compressed sensing; s3: continuously transforming and solving the imaging model; and solving the target optimization problem by using an alternative multiplier method, solving the target optimization problem through variable alternative circulation, and solving the sub-problem in a circulation mode to obtain a high-resolution tomography SAR imaging result. Aiming at the sparse target point scene, the invention can still ensure the imaging quality and reduce the complexity of the flow when the navigation number is less.

Description

A TomoSAR 3D Imaging Method Based on Blind Compressive Sensing

technical field

The invention relates to the technical field of tomographic SAR imaging, in particular to a tomographic SAR three-dimensional imaging method based on blind compressed sensing.

Background technique

Tomosynthetic aperture radar imaging (tomography synthetic aperture radar, TomoSAR) applies the principle of synthetic aperture to the elevation direction, and uses multiple 2D SAR images of the same scene to perform aperture synthesis in the elevation direction from different incident angles to obtain elevation resolution. Rate. It can reconstruct the three-dimensional information of the scatterer and invert the elevation profile, which can effectively solve the overlap effect that exists when the target scattering point in the same scattering unit and the radar are equal in slant distance in two-dimensional SAR imaging, so as to realize the three-dimensional imaging. Compared with Interferometric SAR (InSAR), tomographic SAR imaging technology can not only obtain the elevation information of target scatterers, but also obtain the distribution of scatterers in the elevation direction, which can completely restore the real 3D scene.

Blind Compressed Sensing (BCS) is a sparse linear combination of elevation basis functions that model signals as large dictionaries. This theory combines dictionary construction theory. Dictionary construction refers to the construction of optimal sparse basis under sparse representation. , it needs to satisfy the condition of the uniqueness of the coefficient, and optimize the solution to get more accurate results. The traditional compressive sensing technology uses a fixed analytical sparse transformation to reconstruct the image, and the signal is often unknown and complex, while the underlying sparse model in blind compressive sensing is unknown a priori, and its theory does not assume a fixed dictionary, nor The dictionary is learned separately based on prior information, but an adaptive compressed sensing model based on dictionary replacement learning, which avoids the need for a dictionary in the sampling and recovery process, and the sparse basis in this dictionary corresponds to a specific scene target. Therefore, the sparse basis in the dictionary is not constrained by orthogonality. Therefore, the blind CS algorithm enables the sparse model to adapt to the fixed data considered, and the final dictionary of its model will be suitable for all sparse images regardless of the image sparse basis.

The prior art discloses a full-polarization SAR super-resolution imaging method based on distributed compressed sensing, which includes the following steps: S1: Establishing a full-polarization synthetic aperture radar signal model according to the full-polarization channel set in the synthetic aperture radar; The full polarization channels include HH polarization channels, HV polarization channels and VV polarization channels; S2: using each polarization channel in the synthetic aperture radar full polarization channels to receive corresponding backscatter echo data; According to the backscatter echo data received by each polarization channel, the SAR backscatter coefficient matrix is obtained; S3: Aiming at the problem of full-polarization SAR super-resolution imaging, a distributed compressed sensing algorithm is used to establish a SAR The optimization problem model of the backscattering coefficient matrix; S4: solve the optimization problem of the backscattering coefficient matrix of the synthetic aperture radar, obtain the backscattering coefficient matrix of the synthetic aperture radar; S5: according to the backward scattering coefficient of the synthetic aperture radar Scattering coefficient matrix, perform super-resolution imaging processing for each polarization channel, and obtain the corresponding pseudo-color image; S6: For the pseudo-color image corresponding to each polarization channel, use the pseudo-color image fusion algorithm based on RGB space to perform pseudo-color fusion to obtain a pseudo-color fusion image. However, the targeted target scene is a distributed target scene, which cannot handle sparse point target scenes, and the imaging effect of SAR cannot be guaranteed when the number of voyages is small, and the complexity of the process is high.

Contents of the invention

In order to solve the problem that the current target scene is a distributed target scene, the sparse point target scene cannot be processed, the imaging effect of SAR cannot be guaranteed when the number of voyages is small, and the complexity of the process is high, the present invention proposes a method based on The tomographic SAR 3D imaging method based on blind compressive sensing constructs and optimizes the tomographic SAR model based on blind compressive sensing when the number of passes is small, so as to reduce the number of passes required for measurement and ensure imaging quality and reduce the complexity of the process.

In order to solve the problems of the technologies described above, the technical solution adopted in the present invention is:

A tomographic SAR three-dimensional imaging method based on blind compressed sensing, the steps are as follows:

S1: Preprocess several voyage SAR sequence data sets of the same imaging area in the acquired observation object, and the pixels in each SAR image construct high-latitude signals one by one in order and arrange them in a matrix form; the voyage It refers to the repeated flight of the airborne SAR at different altitudes, the flight trajectory is a straight line, and repeated observations of the same scene;

S2: Using the correlation of the elevation direction of the adjacent azimuth-distance units of the observed object, the image information is reconstructed using the blind compressed sensing framework, and the image signal is modeled as a product of a sparse matrix and a dictionary matrix; the establishment of blind compression based Perceived tomographic SAR imaging model; the image signal is the image signal in the preprocessed data set;

S3: Continue to transform and solve the tomographic SAR imaging model based on blind compressed sensing; use the alternating multiplier method to solve the target optimization problem, solve the problem through alternating variables, and solve the sub-problems in a loop, Obtain high resolution tomographic SAR imaging results.

The working principle of the present invention is as follows:

The present invention combines the blind compressed sensing algorithm to establish a tomographic SAR imaging model combined with the blind compressed sensing, utilizes the sparse structural characteristics of the target, introduces the blind compressed sensing algorithm to process the azimuth-distance unit elevation of the target, and converts the target's Considering the sparse characteristics and its own structural characteristics, the blind compressed sensing scheme models the image as a product of a sparse matrix and a dictionary matrix, and uses the alternate multiplier method to optimize the algorithm to decompose multiple complex problems into simple sub-problems , through variable alternating loop to solve, and loop to optimize the solution to the sub-problem, so as to obtain high-resolution tomographic SAR imaging results.

Preferably, the preprocessing includes: single-view multiple image sequence registration, de-corrosion, phase compensation, and baseline estimation.

Preferably, the expression of the tomographic SAR imaging model based on blind compressed sensing is:

In formula (1), U is the data fidelity item and sparse coefficient, V is the dictionary matrix, A represents the tomographic operator of tomographic SAR, y is the echo data, λ is the regularization parameter, s.t. is the constraint condition; The matrix V adds a unit F-norm;

用于确保数据的一致性；

通过在U上使用非凸项l_P(p<1)的半范数来促进空间系数的稀疏性；通过引入约束S＝UV，将U和V进行解耦，其中S是UV的辅助变量。In formula (1)

Used to ensure data consistency;

The sparsity of spatial coefficients is promoted by using the semi-norm of the non-convex term l _P (p<1) on U; U and V are decoupled by introducing the constraint S=UV, where S is an auxiliary variable for UV.

Further, the steps for solving the tomographic SAR imaging model based on blind compressed sensing are as follows:

S301: Carry out the convergence of the decoupling acceleration equation through variable decomposition, introduce the constraint Z=S, where Z is an auxiliary variable, and the constrained optimization problem is written as:

其中L是辅助变量，β是正则化参数；The p penalty of U in Eq. (2) is approximated as

where L is an auxiliary variable and β is a regularization parameter;

S302: Use the enhanced Lagrangian framework to enforce the constraints in S301, the expression of the AL function is:

In the formula, Q is an auxiliary variable of V, Λ _S , Λ _V , Λ _Z are augmented Lagrangian multipliers, Λ′ _S , Λ′ _V , Λ′ _Z are inverse operations of Lagrangian multipliers, β _S , β _V , β _U , and β _z are penalty parameters, and use the alternate learning minimization strategy to solve the variables U, V, Q, L, S, and Z;

S303: Decompose into six sub-problems, all sub-problems are analyzed and solved by minimizing formula (3), one variable is analyzed at a time, and other variables are kept constant; the six sub-problems include: L sub-problem, U Sub-problems, Q sub-problems, V sub-problems, S sub-problems, Z sub-problems.

Further, the method of parsing the L variable is as follows:

L sub-problem: Ignoring all items irrelevant to the L variable, formula (3) can be written as:

Solve using the contraction rule:

"+" in the formula is defined as the convergence operator of (τ) ₊ = max{0,τ}.

Further, the method of parsing the U variable is as follows:

U sub-problem: Minimize formula (3) to get the quadratic sub-problem about U variable:

The analytical solution to the quadratic subproblem is as follows:

是惩罚参数。In the formula, S _n , V _n , V′ _n are the process variables for solving U, I is the identity matrix,

is the penalty parameter.

The way to parse the Q variable is as follows:

Q sub-problem: through the minimization formula (3) about the Q variable:

Formula (8) is solved using the projection method specified in the following formula;

The projection equation is as follows:

If the F-norm of the Q variable is less than 1, then Q=V; otherwise, the variable V is a unit F-norm; since Q _n is obtained by scaling V _n , the F-norm is uniform.

Further, the method of parsing the V variable is as follows:

V sub-problem: get the quadratic sub-problem about the V variable according to formula (3):

Minimizing Equation (10) with respect to V yields the following closed-form solution:

V _n+1 ＝(β _S U′ _n+1 U _n+1 +β _V I) ^-1 (β _S U′ _n+1 S _n +U′ _n+1 Λ _x +β _V Q _n+1 - Λ _V ) (11)

Further, the method of parsing the S variable is as follows:

S sub-problem: According to the formula (3), remove the items irrelevant to the S variable, and get:

Minimize equation (12) with respect to S, and obtain the following closed-form solution:

S _n+1 ＝(β _S I+β _Z ) ^-1 (β _S U _n+1 V _n+1 -Λ _s +β _Z Z _n +Λ _Z ) (13) Further, the method of analyzing the Z variable as follows:

Z sub-problem: According to formula (3) ignoring the constants irrelevant to the Z variable:

Equation (14) is the Fourier domain replacement problem, and the analysis is as follows:

In formula (15), A' is the inverse operation of A, and y is the echo data;

The calculation is carried out through the loop iteration of six sub-problems, and the solution of formula (1) is obtained.

Compared with prior art, the beneficial effect of the present invention is:

1. Introduce the alternating multiplier method to decompose multiple complex problems into simple sub-problems for the optimization algorithm, and optimize and solve the sub-problems in a loop to obtain high-resolution tomographic SAR imaging results;

2. For the scene of sparse target points, the reconstruction results of the target area can still be guaranteed when the number of passes is small, and the imaging of the ground and canopy areas can be realized more comprehensively;

3. Ensure the quality of imaging and reduce the complexity of the process.

Description of drawings

FIG. 1 is a flow chart of a tomographic SAR three-dimensional imaging method based on blind compressed sensing.

Fig. 2 is a parameter diagram of the L-band radar system in the embodiment.

detailed description

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

Example 1

In this embodiment, as shown in Figure 1, a tomographic SAR three-dimensional imaging method based on blind compressed sensing, the steps are as follows:

S1: Preprocess several voyage SAR sequence data sets of the same imaging area in the acquired observation object, and the pixels in each SAR image construct high-latitude signals one by one in order and arrange them in a matrix form; the voyage It refers to the repeated flight of the airborne SAR at different altitudes, the flight trajectory is a straight line, and repeated observations of the same scene;

S2: Using the correlation of the elevation direction of the adjacent azimuth-distance units of the observed object, the image information is reconstructed using the blind compressed sensing framework, and the image signal is modeled as a product of a sparse matrix and a dictionary matrix; the establishment of blind compression based Perceived tomographic SAR imaging model; the image signal is the image signal in the preprocessed data set;

S3: Continue to transform and solve the tomographic SAR imaging model based on blind compressed sensing; use the alternating multiplier method to solve the target optimization problem, solve the problem through alternating variables, and solve the sub-problems in a loop, Obtain high resolution tomographic SAR imaging results.

The working principle of the present invention is as follows:

In the process of tomographic SAR imaging, too many passes will increase the amount of calculation and complexity, and too few passes will affect the quality of imaging;

The present invention combines the blind compressed sensing algorithm to establish a tomographic SAR imaging model combined with the blind compressed sensing, utilizes the sparse structural characteristics of the target, introduces the blind compressed sensing algorithm to process the azimuth-distance unit elevation of the target, and converts the target's Considering the sparse characteristics and its own structural characteristics, the blind compressed sensing scheme models the image as a product of a sparse matrix and a dictionary matrix, and uses the alternate multiplier method to optimize the algorithm to decompose multiple complex problems into simple sub-problems , through variable alternating loop to solve, and loop to optimize the solution to the sub-problem, so as to obtain high-resolution tomographic SAR imaging results.

In this embodiment, the preprocessing includes: single-view complex image sequence registration, evil removal, phase compensation, and baseline estimation.

Specifically, the expression of the tomographic SAR imaging model based on blind compressed sensing is:

In formula (1), U is the data fidelity item and sparse coefficient, V is the dictionary matrix, A represents the tomographic operator of tomographic SAR, y is the echo data, λ is the regularization parameter, s.t. is the constraint condition; The matrix V adds a unit F-norm;

用于确保数据的一致性；

通过在U上使用非凸项l_P(p<1)的半范数来促进空间系数的稀疏性；通过引入约束S＝UV，将U和V进行解耦，其中S是UV的辅助变量。In formula (1)

Used to ensure data consistency;

The sparsity of spatial coefficients is promoted by using the semi-norm of the non-convex term l _P (p<1) on U; U and V are decoupled by introducing the constraint S=UV, where S is an auxiliary variable for UV.

More specifically, the steps to solve the tomographic SAR imaging model based on blind compressed sensing described in S3 are as follows:

S301: Carry out the convergence of the decoupling acceleration equation through variable decomposition, introduce the constraint Z=S, where Z is an auxiliary variable, and the constrained optimization problem is written as:

其中L是辅助变量，β是正则化参数；The p penalty of U in Eq. (2) is approximated as

where L is an auxiliary variable and β is a regularization parameter;

S302: Use the enhanced Lagrangian framework to enforce the constraints in S301, the expression of the AL function is:

In the formula, Q is an auxiliary variable of V, Λ _S , Λ _V , A _z are augmented Lagrangian multipliers, Λ′ _S , Λ′ _V , Λ′ _z are inverse operations of Lagrangian multipliers, β _S , β _V , β _U , and β _Z are penalty parameters, and use the alternate learning minimization strategy to solve the variables U, V, Q, L, S, and Z;

S303: Decompose into six sub-problems, all sub-problems are analyzed and solved by minimizing formula (3), one variable is analyzed at a time, and other variables are kept constant; the six sub-problems include: L sub-problem, U Sub-problems, Q sub-problems, V sub-problems, S sub-problems, Z sub-problems.

Example 2

Based on a tomographic SAR three-dimensional imaging method based on blind compressed sensing described in Embodiment 1, in this embodiment, the six sub-problems and their resolution methods are as follows:

The method of parsing the L variable is as follows:

L sub-problem: Ignoring all items irrelevant to the L variable, formula (3) can be written as:

Solve using the contraction rule:

"+" in the formula is defined as the convergence operator of (τ) ₊ = max{0,τ}.

The method of parsing the U variable is as follows:

U sub-problem: Minimize formula (3) to get the quadratic sub-problem about U variable:

The analytical solution to the quadratic subproblem is as follows:

是惩罚参数。In the formula, S _n , V _n , V′ _n are the process variables for solving U, I is the identity matrix,

is the penalty parameter.

The way to parse the Q variable is as follows:

Q sub-problem: through the minimization formula (3) about the Q variable:

Formula (8) is solved using the projection method specified in the following formula;

The projection equation is as follows:

If the F-norm of the Q variable is less than 1, then Q=V; otherwise, the variable V is a unit F-norm; since Q _n is obtained by scaling V _n , the F-norm is uniform.

The method of parsing the V variable is as follows:

V sub-problem: get the quadratic sub-problem about the V variable according to formula (3):

Minimizing Equation (10) with respect to V yields the following closed-form solution:

V _n+1 ＝(β _S U′ _n+1 U _n+1 +β _V I) ^-1 (β _S U′ _n+1 S _n +U′ _n+1 Λ _x +β _V Q _n+1 - Λ _V ) (11)

The method of parsing the S variable is as follows:

S sub-problem: According to the formula (3), remove the items irrelevant to the S variable, and get:

Minimize equation (12) with respect to S, and obtain the following closed-form solution:

S _n+1 ＝(β _S I+β _Z ) ^-1 (β _S U _n+1 V _n+1 -Λ _s +β _Z Z _n +Λ _Z ) (13)

Here's how to parse the Z variable:

Z sub-problem: According to formula (3) ignoring the constants irrelevant to the Z variable:

Equation (14) is the Fourier domain replacement problem, and the analysis is as follows:

In formula (15), A' is the inverse operation of A, and y is the echo data;

The calculation is carried out through the loop iteration of six sub-problems, and the solution of formula (1) is obtained. After solving the tomographic SAR imaging model based on blind compressed sensing, a high-resolution tomographic SAR imaging result is obtained; while ensuring the imaging quality, the complexity of the process is reduced, and the complexity of calculation is reduced.

Example 3

Based on a kind of tomographic SAR three-dimensional imaging method based on blind compressed sensing described in embodiment 1 and embodiment 2, construct relevant experiments;

As shown in Figure 2, the experimental data is the L-band data of the HV polarization channel provided by the airborne F-SAR system of the German Aerospace Agency (Deutsches zentrum für Luftund Raumfahrt, DLR), and the collection area is the part of the range position of Trockenbornn, Germany area; the area is mainly composed of forests and flat land.

In this embodiment, experiment is carried out using the parameters therein:

The radar center frequency of the experimental parameters is 1.325GHz, the azimuth resolution is 0.4m, and the range resolution is 1.5m. Different types of methods are used to reconstruct the data. The methods are the classic spectrum estimation algorithm CAPON compressed sensing method and Tomographic SAR imaging method based on blind compressed sensing method;

This embodiment illustrates the effectiveness of the imaging method by mean square error (Mean Squared Error, MSE). When the signal-to-noise ratio is 5dB, the mean square error value of this imaging method is 0.0610, and the mean square error value of CAPON is 0.0932. CS The mean square error value of the imaging method is 0.0854; when the signal-to-noise ratio is 25dB, the mean square error value of this imaging method is 0.0132, the mean square error value of CAPON is 0.0297, and the mean square error value of CS is 0.0215; CAPON and other methods are comparable The ratio error is large and not stable enough. CS and this imaging method have a tendency to gradually decrease with the change of signal-to-noise ratio, but this imaging method has a lower mean square error and the imaging result is more stable. Therefore, compared with the traditional method, this imaging method can more accurately detect the ground and canopy areas in the data, and the reconstruction accuracy of the target area is higher.

Apparently, the above-mentioned embodiments of the present invention are only examples for clearly illustrating the present invention, rather than limiting the implementation of the present invention. All modifications, equivalent replacements and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the claims of the present invention.

Claims (10)

1. A chromatography SAR three-dimensional imaging method based on blind compressed sensing is characterized by comprising the following steps:

s1: preprocessing a plurality of acquired SAR sequence datasets navigating through the same imaging region in an observed object, and sequentially constructing high-latitude signals one by pixels in each SAR image and arranging the high-latitude signals into a matrix form;

s2: reconstructing image information by using a blind compressed sensing frame by utilizing the correlation of the elevation directions of adjacent azimuth-distance units of an observation object, and modeling an image signal into a product of a sparse matrix and a dictionary matrix; establishing a tomography SAR imaging model based on blind compressed sensing; the image signal is an image signal in a data set after preprocessing;

s3: continuously transforming and solving the tomography SAR imaging model based on blind compressed sensing; and solving the target optimization problem by using an alternative multiplier method, solving the target optimization problem through variable alternative circulation, and solving the sub-problem in a circulating manner to obtain a high-resolution tomography SAR imaging result.

2. The method for tomographic SAR three-dimensional imaging based on blind compressed sensing as claimed in claim 1, wherein the preprocessing comprises: single-vision complex image sequence registration, pathogen elimination, phase compensation and baseline estimation.

3. The method as claimed in claim 1, wherein the expression of the blind compressed sensing-based SAR imaging model is as follows:

in the formula (1), U is a data fidelity term and a sparse coefficient, V is a dictionary matrix, A represents a chromatography operator of a chromatography SAR, y is echo data, lambda is a regularization parameter, and s.t. is a constraint condition; the dictionary matrix V is added with a unit F-norm;

in formula (1)

For ensuring data consistency;

by using non-convex terms l on U _P (p<1) To promote sparsity of spatial coefficients; u and V are decoupled by introducing the constraint S = UV, where S is an auxiliary variable of UV.

4. The method according to claim 3, wherein the step of solving the blind compressed sensing-based tomographic SAR imaging model comprises:

s301: convergence of the decoupling acceleration equation is performed through variable decomposition, and a constraint Z = S is introduced, wherein Z is an auxiliary variable, and the constraint optimization problem is written as:

approximating the p penalty of U in equation (2) as

Where L is an auxiliary variable and β is a regularization parameter;

s302: the constraint in S301 is enforced using an enhanced lagrangian framework, the expression of the AL function being:

in which Q is an auxiliary variable of V, Λ _S 、Λ _V 、Λ _Z Is an augmented Lagrangian multiplier, Λ' _S 、Λ′ _V 、Λ′ _Z Is the inverse of the Lagrange multiplier, beta _S 、β _V 、β _U 、β _Z Is a penalty parameter, and solves the variables U, V, Q, L by using a strategy of alternately learning minimization,S、Z；

S303: resolving into six subproblems, resolving all the subproblems through a minimization formula (3), resolving one variable at a time, and keeping other variables fixed; the six sub-problems include: an L sub-problem, a U sub-problem, a Q sub-problem, a V sub-problem, an S sub-problem, and a Z sub-problem.

5. The method for tomographic SAR three-dimensional imaging based on blind compressed sensing according to claim 4, wherein the method for analyzing L variable is as follows:

l sub-problem: ignoring all terms that are independent of the L variable, equation (3) is written as:

solving using a contraction rule:

wherein "+" is defined as (τ) ₊ Convergence operator of = max {0, τ }.

6. The method for tomographic SAR three-dimensional imaging based on blind compressed sensing according to claim 5, wherein the method for analyzing the U variable is as follows:

problem of the U son: minimizing equation (3) yields a quadratic problem with the U variable:

the quadratic sub-problem solution is as follows:

in the formula, S _n 、V _n 、V′ _n To solve for the process variable of U, I is the identity matrix,

is a penalty parameter.

7. The method for tomographic SAR three-dimensional imaging based on blind compressed sensing according to claim 6, wherein the method for analyzing Q variable is as follows:

q, problem: obtained by minimizing equation (3) with respect to the Q variable:

equation (8) is solved using the projection method specified in the following equation;

the projection equation is as follows:

if the F-norm of the Q variable is less than 1, Q = V; otherwise, the variable V is a unit F-norm; due to Q _n Is through V _n Scaled so that the F-norm is uniform.

8. The method for tomographic SAR three-dimensional imaging based on blind compressed sensing of claim 7, wherein the method for analyzing V variables is as follows:

problem V: the quadratic problem with the V variable is found from equation (3):

minimizing equation (10) with respect to V yields the following closed form solution:

V _n+1 ＝(β _S U′ _n+1 U _n+1 +β _V I) ^-1 (β _S U′ _n+1 S _n +U′ _n+1 Λ _x +β _V Q _n+1 -Λ _V ) (11)。

9. the method for tomographic SAR three-dimensional imaging based on blind compressed sensing of claim 8, wherein the method for analyzing S variable is as follows:

s, a sub-problem: removing terms that are independent of the S variable according to equation (3) yields:

minimizing equation (12) with respect to S yields the following closed form solution:

S _n+1 ＝(β _S I+β _Z ) ^-1 (β _S U _n+1 V _n+1 -Λ _s +β _Z Z _n +Λ _Z ) (13)。

10. the method for tomographic SAR three-dimensional imaging based on blind compressed sensing according to claim 9, wherein the method for analyzing Z variable is as follows:

problem Z: ignoring constants that are independent of the Z variable according to equation (3) yields:

equation (14) is a fourier domain replacement problem, which is resolved as follows:

in the formula (15), a' is an inverse operation of a, and y is echo data;

the calculation is performed by loop iteration of six subproblems to obtain the solution of equation (1).

Images