High-frequency ultrasound image reconstruction method, storage medium, and computer equipment

Abstract

The present invention discloses a high-frequency ultrasonic image reconstruction method, storage medium, and computer device. The method comprises the following steps: 1) converting the high-frequency ultrasonic image reconstruction problem into an ill-posed inverse problem to simulate the ultrasonic image imaging mechanism equation; 2) introducing a point spread function to update the ultrasonic image imaging mechanism equation; 3) introducing a regularization constraint term to update the ultrasonic image imaging mechanism equation using a maximum a posteriori estimation expression; 4) deriving an alternating direction multiplier equation using an augmented Lagrangian equation; and 5) iteratively solving the alternating direction multiplier equation to obtain a reconstruction result of the high-frequency ultrasonic image. The ultra-high-frequency ultrasonic fast reconstruction method proposed in the present invention grasps and utilizes the characteristics of the high-frequency ultrasonic imaging system. By performing inverse deduction based on the noise characterization equation within the system, the method can recover and reconstruct information between ultrasonic image pixels that is lost due to various factors such as the size of the ultrasonic probe and the system sampling frequency, thereby improving the resolution of the ultrasonic image.

Description

Technical Field

[0001] The present invention relates to the field of image processing, and in particular to a high-frequency ultrasonic image reconstruction method, a storage medium and a computer device. Background Art

[0002] The operating frequency of high-frequency ultrasound systems exceeds 50MHz and can even reach 200-500MHz. A higher operating frequency means higher image resolution and more detailed information can be obtained, so higher image resolution is a goal that has been pursued. According to the Nyquist sampling theorem, in order to completely restore the sampled ultrasonic radio frequency signal, the sampling frequency must be greater than twice the highest frequency of the signal. Therefore, a sampling frequency as high as 1GHz places extremely high demands on chip circuits, and its cost also increases exponentially, and highly sensitive sampling circuits are more susceptible to working environment images. Correspondingly, high sampling rates also bring about the problem of huge data transmission volume, which not only increases the operating cost of the data transmission module, but also increases the workload of the back-end signal image processing. Furthermore, higher operating frequencies also increase the difficulty of preparing ultrasonic transducers and increase costs.

[0003] Therefore, without improving existing hardware conditions, it is of great significance to reconstruct the detailed information of ultrasound images through back-end algorithms to achieve higher resolution image reconstruction under high-frequency working conditions.

[0004] Reference Figure 6 a represents the image obtained by the traditional high-frequency ultrasound system, and its size is (j, k); ultra-high frequency ultrasound image reconstruction aims to reconstruct Figure 6 The missing information between adjacent pixels in a, such as Figure 6 As shown in the blue area in b (i.e. the area marked with “?” that meets the requirements for filling); Figure 6 c is the reconstructed high-resolution image, whose size is (n*j, n*k). Figure 6 d is Figure 6 The result of directly scaling a to (n*j, n*k) dimensions, where n represents a positive integer multiple. As can be seen from the figure, despite the same dimensions, the reconstructed image contains richer spatial information, effectively increasing the amount of information contained within a unit pixel and improving the resolution of ultrahigh-frequency ultrasound images.

[0005] Traditional methods for increasing image information content include interpolation methods such as nearest neighbor, bilinear interpolation, cubic interpolation, and other higher-order interpolation methods. Their basic principle is to perform interpolation by analyzing the characteristic relationships between the pixel to be interpolated and its neighbors or several other pixels in the neighborhood, both in the spatial domain and in the frequency domain. These methods have significant advantages, namely, their wide applicability, requiring only pixel values within the image, and can be applied to both natural and medical images. In practical applications, images generated by different imaging systems possess unique characteristics. For example, electron microscopes, optical microscopes, and ultrasound microscopes, due to their respective imaging methods, produce images with a unique style. Therefore, traditional methods for increasing image information content have the disadvantage of relying solely on pixel values and grayscale values, without considering other factors that may affect resolution, such as signal acquisition and signal conversion processing. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to address the deficiencies in the above-mentioned prior art and provide a high-frequency ultrasound image reconstruction method, storage medium, and computer device. To solve the above-mentioned technical problem, the technical solution adopted by the present invention is: a high-frequency ultrasound image reconstruction method, comprising the following steps:

[0007] 1) Convert the high-frequency ultrasound image reconstruction problem into an ill-posed inverse problem and simulate the imaging mechanism equation of the ultrasound image;

[0008] 2) Introducing the point spread function to update the imaging mechanism equation of ultrasound images;

[0009] 3) Introducing regularization constraints, the ill-posed inverse problem is transformed into a constrained optimization problem, and the imaging mechanism equation of ultrasound images is updated using the maximum a posteriori estimation expression;

[0010] 4) Based on the results of step 3), the alternating direction multiplier equation is derived using the augmented Lagrangian equation;

[0011] 5) Iteratively solve the alternating direction multiplier equation and stop the iteration when convergence to obtain the reconstructed result of the high-frequency ultrasound image.

[0012] Preferably, the imaging mechanism equation of the ultrasound image simulated in step 1) is:

[0013] y=Ax+n (1);

[0014] Where x represents the input original image, y represents the final output image of the system, n represents noise, and A represents the linear or nonlinear operation applied by the imaging system to the original image.

[0015] Preferably, the imaging mechanism equation updated in step 2) is:

[0016] y=SHx+n (2);

[0017] Where x represents the original input image, y represents the final output image of the system, n represents noise, H represents the convolution operation of the point spread function and the input signal, and S represents the downsampling operation.

[0018] Preferably, the point spread function is a type of Bessel equation or a variant thereof, and the point spread function includes at least the following parameters: sound velocity, ultrasonic wavelength, center frequency, and probe aperture.

[0019] Preferably, the imaging mechanism equation obtained by updating in step 3) is:

[0020]

[0021]

[0022] in, Represents output The data fidelity between y and the noise n, y represents the final output image of the system; φ(Ax) is the regularization constraint; λ is the weight coefficient, which balances the weight between data fidelity and regularization constraint; A represents the different image properties of different imaging systems;

[0023] Among them, the regular constraint term φ(Ax) is the l1 norm, l2 norm, total variation or non-convex penalty term.

[0024] Preferably, the alternating direction multiplier equation obtained in step 4) is:

[0025]

[0026] Formula (5) replaces Ax in Formula (4) with v, with the purpose of decomposing the constrained optimization problem into two independent parts: and φ(v); thus the alternating multiplier equation is used to iteratively solve these two parts simultaneously.

[0027] Preferably, the step 5) specifically includes:

[0028] 5-1) Rewrite equation (5) as the augmented Lagrangian equation:

[0029] Ax = v (6)

[0030] 5-2) Translate equation (6) into three equations:

[0031]

[0032] Where k represents the number of iterations, u is the Lagrange multiplier, and η is the weight parameter;

[0033] 5-3) Rewrite equation (7) as:

[0034]

[0035]

[0036] Formula (8-2) can be regarded as a denoising operation, because after the action of the constraint term φ(v), v k+1 By v k It is deduced that each round of iteration updates the denoised image v k+1 and the image v before denoising k The data fidelity between the two images is maintained until convergence is achieved, then the update is stopped and the reconstruction result of the high-frequency ultrasound image is obtained.

[0037] Preferably, the iterative steps in step 5-3) are:

[0038] S1, input noisy low-resolution image y, downsampling (decimation) operation S, system point spread function H, constraint weight parameters λ and η;

[0039] Based on (8-1) and x k =v k -u k , we get the following formula (9), and update x according to the following formula (9):

[0040]

[0041] S2, based on (8-2) and x k =v k -u k , we get the following formula (10), and update v according to the following formula (10):

[0042]

[0043] S3, according to y k+1 and x k+1 Compute the Lagrange multiplier u:

[0044] u k+1 =u k +η(x k+1 -v k+1 ) (11)

[0045] S4, when max{||x k+1 -x k ||2,||v k+1 -v k ||2,||u k+1 -u k The iteration stops when ||2}≤γ and the output is:

[0046] The present invention also provides a storage medium storing a computer program, which is used to implement the method described above when executed.

[0047] The present invention also provides a computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-mentioned method when executing the computer program.

[0048] The beneficial effects of the present invention are:

[0049] The ultra-high frequency ultrasound rapid reconstruction method proposed in the present invention grasps and utilizes the characteristics of the high frequency ultrasound imaging system. By performing inverse deduction based on the noise characterization equation within the system, it can recover and reconstruct the information between the pixels of the ultrasound image that is missing due to many reasons such as the size of the ultrasound probe and the system sampling frequency, and improve the resolution of the ultrasound image.

[0050] The optimization cost of each component in ultra-high frequency ultrasound is very high. The algorithm process proposed in this invention can obtain higher-quality ultra-high frequency ultrasound images based on existing hardware, reducing noise and improving resolution.

[0051] The architecture adopted by the present invention is well applicable to systems with different operating frequencies and different ultrasound probes, and is used to improve the quality of one-dimensional signals and two-dimensional images;

[0052] The super-resolution ultrasound image reconstruction method proposed in the present invention can recover twice or even higher data from less original data, thereby ensuring that the same or even better image quality can be obtained with a smaller data transmission volume, which can also help to realize efficient remote processing functions. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Schematic diagram of the imaging process of ultrasound images;

[0054] Figure 2 Schematic diagram of the effect of the point spread equation on imaging;

[0055] Figure 3-5 These are the results of different solution methods in three examples;

[0056] Figure 6 Schematic diagram of traditional high-frequency ultrasound image reconstruction. DETAILED DESCRIPTION

[0057] The present invention is further described in detail below with reference to the embodiments so that those skilled in the art can implement the invention with reference to the description.

[0058] It should be understood that terms such as “having”, “including” and “comprising” used herein do not preclude the existence or addition of one or more other elements or combinations thereof.

[0059] Example 1

[0060] This embodiment provides a high-frequency ultrasound image reconstruction method, comprising the following steps:

[0061] 1) The high-frequency ultrasound image reconstruction problem is transformed into an ill-posed inverse problem, and the imaging mechanism equation of the ultrasound image is simulated; specifically:

[0062] y=Ax+n (1);

[0063] Where x represents the input original image, y represents the final output image of the system, n represents the cumulative noise introduced by units such as ultrasonic transducers or lenses, and A represents the linear or nonlinear operation applied by the imaging system to the original image. In most ultrasonic imaging systems, operation A includes downsampling (decimation) and aliasing. Figure 1 The imaging process of ultrasound images.

[0064] One of the key techniques of this invention is to utilize the system's accumulated noise n and a linear operation A to recover a lossless, high-resolution image dataset x from a lossy image dataset y. This process of recovering a more complete image from an incomplete dataset is known as an ill-posed inverse problem. Regularization using a cost function is an effective method for inverting ill-posed inverse problems. By adding a regularization term to the cost function, the ill-posed inverse problem can be transformed into a constrained optimization problem.

[0065] 2) The point spread function is introduced to update the imaging mechanism equation of the ultrasound image, specifically:

[0066] y=SHx+n (2);

[0067] Where x represents the original input image, y represents the final output image of the system, n represents noise, H represents the convolution operation of the point spread function and the input signal, and S represents the downsampling operation.

[0068] Regarding the point spread function: In actual ultrasound imaging, due to the physical properties of the ultrasound beam, the shape of the echo signal captured from the target object may differ from its actual shape, resulting in slight distortion in the final imaging result. When the sound beam is scattered or reflected by the target, the echo signal produces an approximate impulse response. To characterize the imaging system's response to point targets, the concept of the point spread function (PSF) was proposed. The characteristics of different ultrasound probes have a certain impact on the system's point spread function. Figure 2Demonstrates the effect of the point spread equation on imaging, Figure 2 Left one: actual point target, Figure 2 Middle: point spread function; Figure 2 Right: The final result after convolution. The point spread equation is equivalent to a convolution process. After the real object is convolved with the point spread equation, a distorted image is obtained.

[0069] The point spread function is a type of Bessel equation or its variants, and the point spread function includes at least the following parameters: sound velocity, ultrasonic wavelength, center frequency, probe aperture, etc.

[0070] In this embodiment, a type of Bessel equation is selected:

[0071] Where α represents the order.

[0072] 3) By introducing a regularization constraint term, the ill-posed inverse problem is transformed into a constrained optimization problem, and the imaging mechanism equation of the ultrasound image is updated using the maximum a posteriori estimation (Maximum-a-posteriori) expression; the updated imaging mechanism equation is:

[0073]

[0074]

[0075] in, Represents output The data fidelity between y and the noise n, y represents the final output image of the system; φ(Ax) is the regularization constraint term, also known as the penalty term to introduce sparsity; λ is the weight coefficient to balance the weight between data fidelity and regularization constraint term; A represents the different image properties of different imaging systems;

[0076] Among them, the regular constraint term φ(Ax) is the l1 norm, l2 norm, total variation or non-convex penalty term.

[0077] 4) Based on the results of step 3), the alternating direction multiplier equation is derived using the augmented Lagrangian equation. There are several methods for solving constrained optimization problems. The augmented Lagrangian equation is preferably used in the present invention. The augmented Lagrangian equation can be used to derive the alternating direction multiplier equation (Alternating direction method of multipliers). Specifically, the alternating direction multiplier equation is:

[0078]

[0079] Formula (5) replaces Ax in Formula (4) with v, with the purpose of decomposing the constrained optimization problem into two independent parts: and φ(v); thus, the alternating multiplier equation is used to solve the two parts simultaneously. After the k+1th iteration, k and v k Update to get x k+1 , v k+1 , until the result converges, stop the iteration. The advantage of this method is that compared with direct deconvolution operation, converting the deconvolution problem into an optimization problem reduces the computational dimension and complexity.

[0080] 5) Iteratively solving the alternating direction multiplier equation, stopping the iteration when convergence, and obtaining the reconstruction result of the high-frequency ultrasound image, specifically including:

[0081] 5-1) Rewrite equation (5) as the augmented Lagrangian equation:

[0082] Ax = v (6)

[0083] 5-2) To find the saddle point where Equation (6) converges, rewrite Equation (6) into three equations:

[0084]

[0085] Where k represents the number of iterations, u is the Lagrange multiplier, and η is the weight parameter;

[0086] 5-3) Rewrite equation (7) as:

[0087]

[0088]

[0089] Formula (8-2) can be regarded as a denoising operation, because after the action of the constraint term φ(v), v k+1 By v k It is deduced that each round of iteration updates the denoised image v k+1 and the image v before denoising k The data fidelity between the two images is maintained until convergence is reached, and the update is stopped. At this time, it means that the image noise has been removed as much as possible, and the reconstruction result of the high-frequency ultrasound image is obtained.

[0090] The constraint term φ(v) in Equation (8) can be viewed as a flexible denoising module. Replacing φ(v) does not affect the solution of the entire optimization function. Therefore, by substituting different φ(v), high-resolution image reconstruction problems for various imaging systems can be easily implemented. The choice of φ(v) can be determined based on the actual application requirements and the characteristics of the imaging system.

[0091] When solving the alternating direction multiplier equation, the regularization constraint term φ(Ax) can be derived using the 11-norm, 12-norm, total variation, or non-convex penalty term. In addition to solving the constrained optimization problem in the spatial or frequency domain, it can also be derived in the wavelet domain. The following example demonstrates the solution process for step 5-3) using the 11-norm constraint term φ(v). The specific iterative steps are:

[0092] S1, input noisy low-resolution image y, downsampling (decimation) operation S, system point spread function H, constraint weight parameters λ and η;

[0093] Based on (8-1) and x k =v k -u k , we get the following formula (9), and update x according to the following formula (9):

[0094]

[0095] S2, based on (8-2) and x k =v k -u k , we get the following formula (10), and update v according to the following formula (10):

[0096]

[0097] S3, according to v k+1 and x k+1 Compute the Lagrange multiplier u:

[0098]

[0099] S4, when max{||x k+1 -x k ||2,||v k+1 -v k ||2,||u k+1 -u k The iteration stops when ||2}≤γ and the output is:

[0100] In this embodiment, the derivation process of other constraints in the wavelet domain and the frequency domain is not described in detail.

[0101] In the present invention, the process of outputting images from a high-frequency ultrasonic imaging system is characterized as the convolution of an input signal and a system function, and random noise is superimposed, making the image signal degradation process more intuitive and easier to solve.

[0102] In the present invention, based on the physical properties of ultrasonic imaging, the characteristic point spread function of the high-frequency ultrasonic system is taken as the focus of denoising and high-resolution reconstruction. This fully utilizes the specificity of ultrasonic imaging and makes the reconstruction results more accurate than traditional general methods.

[0103] Image super-resolution reconstruction is an ill-posed inverse problem. In the present invention, regularization constraints are introduced to transform it into a constrained optimization problem for solution.

[0104] In the present invention, alternating direction multiplier equations are used to transform the above problem into several sub-equations, thereby reducing the computational dimension and achieving the simultaneous solution of multiple variables in each iteration, thereby improving the computational speed and shortening the time it takes for the equation group to iterate to the convergence condition.

[0105] In this invention, during the solution of the alternating direction multiplier equation, the insertion of a constraint term in one of the sub-equations can be approximately considered a denoising operation. Furthermore, this constraint term can be flexibly replaced without affecting the overall computational architecture, facilitating the insertion of different denoising operators to reconstruct the image. This demonstrates the flexibility and robustness of this architecture.

[0106] In the present invention, the solution of the alternating direction multiplier equation can be performed in the time domain, can be completed in the time domain by Fourier transform, and can also be completed in the wavelet domain, with a high degree of freedom, and can be selected according to the characteristics of the signal. The derivation process is not detailed here.

[0107] In this paper, the methods for solving the alternative multiplier equations all employ convex functions (such as the l1 norm, l2 norm, and total variation) as constraints to solve convex optimization problems. This is because convex optimization problems facilitate iterative convergence, allowing for optimal solutions. In practice, ill-posed inverse problems can also be solved using non-convex constraints, further introducing sparsity and thus further reducing computational complexity.

[0108] Reference Figure 3 , in one example, the results of different constraints and their solution methods are shown in order: A: low-resolution noisy image output by a high-frequency ultrasound imaging system; B: optimization result of a traditional interpolation method; C: optimization result of the present invention under the l1-norm constraint in the time domain; D: optimization result of the present invention under the total variation constraint in the time domain; E: optimization result of the present invention under the l1-norm constraint in the wavelet domain; and F: optimization result of the present invention under the non-convex constraint. Parameters such as the signal-to-noise ratio (SNR) and root mean square error (RMSE) are significantly improved compared to the original image. The specific parameters are shown in Table 1 below:

[0109] Table 1

[0110] PSNR ISNR RMSE TIME / S B 35.236 / 0.0371 / C 37.513 2.362 0.0133 2.541 D 39.985 4.132 0.0103 2.999 E 35.635 0.923 0.0233 1.757 F 42.194 6.356 0.0071 4.971

[0111] Reference Figure 4In another example, the results of different solution methods are shown in order: A: original high-frequency ultrasound image; B: optimization result of a traditional interpolation method; C: result of the present invention under the l1-norm constraint in the time domain; D: result of the present invention under the total variation constraint in the time domain; E: result of the present invention under the l1-norm constraint in the frequency domain; F: result of the present invention under the non-convex constraint. The results of parameters such as the signal-to-noise ratio (SNR) and root mean square error (RMSE) are shown in Table 2 below:

[0112] Table 2

[0113]

[0114]

[0115] It can be seen that the signal-to-noise ratio (SNR) of the method of the present invention is improved from 26.75 to 33.66, an increase of 10-15%, compared with the traditional interpolation method; the root mean square error is reduced from 0.0382 to 0.0216; and the calculation speed is also improved by 15-20%.

[0116] Reference Figure 5 In another example, the results of different solution methods are shown in order: A: original high-frequency ultrasound image; B: optimization result of a traditional interpolation method; C: result of the present invention under the l1-norm constraint in the time domain; D: result of the present invention under the total variation constraint in the time domain; E: result of the present invention under the l1-norm constraint in the frequency domain; F: result of the present invention under the non-convex constraint. The results of parameters such as signal-to-noise ratio (SNR) and root mean square error (RMSE) are shown in Table 3 below:

[0117] The results of parameters such as signal-to-noise ratio (SNR) and root mean square error (RMSE) are shown in Table 1 below:

[0118] Table 3

[0119] PSNR ISNR RMSE TIME / S B 28.595 / 0.0372 / C 31.534 2.939 0.0265 0.327 D 29.631 1.036 0.0330 1.032 E 29.620 1.025 0.0341 2.757 F 33.923 5.328 0.0239 0.471

[0120] It can also be seen that by adopting the method of the present invention, both the signal-to-noise ratio (SNR) and the root mean square error (RMSER) are significantly improved.

[0121] Example 2

[0122] A storage medium stores a computer program, which is used to implement the method of embodiment 1 when executed.

[0123] Example 3

[0124] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the method of embodiment 1 is implemented when the processor executes the computer program.

[0125] This embodiment further provides a cloud server, on which a computer program is stored, and when the program is executed, it is used to implement the method of embodiment 1.

[0126] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the description and implementation methods. They can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily implemented. Therefore, without departing from the general concept defined by the claims and the scope of equivalents, the present invention is not limited to specific details.

Claims

1. A high-frequency ultrasound image reconstruction method, characterized in that: The following steps are involved: 1) Convert the high-frequency ultrasound image reconstruction problem into an ill-posed inverse problem and simulate the imaging mechanism equation of the ultrasound image; 2) Introducing the point spread function to update the imaging mechanism equation of ultrasound images; 3) Introducing regularization constraints, the ill-posed inverse problem is transformed into a constrained optimization problem, and the imaging mechanism equation of ultrasound images is updated using the maximum a posteriori estimation expression; 4) Based on the results of step 3), the alternating direction multiplier equation is derived using the augmented Lagrangian equation; 5) Iteratively solving the alternating direction multiplier equation, stopping the iteration when convergence, and obtaining the reconstructed result of the high-frequency ultrasound image; The imaging mechanism equation obtained by updating in step 3) is: in, Represents output The data fidelity between y and the noise n, y represents the final output image of the system; φ(Ax) is the regularization constraint; λ is the weight coefficient, which balances the weight between data fidelity and regularization constraint; A represents the different image properties of different imaging systems; Among them, the regular constraint term φ(Ax) is the l1 norm, l2 norm, total variation or non-convex penalty term.

2. The high-frequency ultrasonic image reconstruction method according to claim 1, characterized in that: The imaging mechanism equation of the ultrasound image simulated in step 1) is: y=Ax+n (1); Where x represents the input original image, y represents the final output image of the system, n represents noise, and A represents the linear or nonlinear operation applied by the imaging system to the original image.

3. The high-frequency ultrasonic image reconstruction method according to claim 2, characterized in that: The imaging mechanism equation updated in step 2) is: y=SHx+n (2); Where x represents the original input image, y represents the final output image of the system, n represents noise, H represents the convolution operation of the point spread function and the input signal, and S represents the downsampling operation.

4. The high-frequency ultrasonic image reconstruction method according to claim 3, characterized in that: The point spread function is a type of Bessel equation or a variant thereof, and the point spread function includes at least the following parameters: sound velocity, ultrasonic wavelength, center frequency, and probe aperture.

5. The high-frequency ultrasonic image reconstruction method according to claim 4, characterized in that: The alternating direction multiplier equation obtained in step 4) is: Formula (5) replaces Ax in Formula (4) with v, with the purpose of decomposing the constrained optimization problem into two independent parts: and φ(v); thus the alternating multiplier equation is used to iteratively solve these two parts simultaneously.

6. The high-frequency ultrasonic image reconstruction method according to claim 5, characterized in that: The step 5) specifically includes: 5-1) Rewrite equation (5) as the augmented Lagrangian equation: 5-2) Translate equation (6) into three equations: Where k represents the number of iterations, u is the Lagrange multiplier, and η is the weight parameter; 5-3) Rewrite equation (7) as: Formula (8-2) can be regarded as a denoising operation, because after the action of the constraint term φ(v), v k+1 By v k It is deduced that each round of iteration updates the denoised image v k+1 and the image v before denoising k The data fidelity between the two images is maintained until convergence is achieved, then the update is stopped and the reconstruction result of the high-frequency ultrasound image is obtained.

7. The high-frequency ultrasonic image reconstruction method according to claim 6, characterized in that: The iterative steps in step 5-3) are: S1, input noisy low-resolution image y, downsampling (decimation) operation S, system point spread function H, constraint weight parameters λ and η; Based on (8-1) and x k =v k -u k , we get the following formula (9), and update x according to the following formula (9): S2, based on (8-2) and x k =v k -u k , we get the following formula (10), and update v according to the following formula (10): S3, according to v k+1 and x k+1 Compute the Lagrange multiplier u: u k+1 =u k + η(x k+1 -v k+1 ) (11); S4, when max{‖x k+1 -x k ‖2,‖v k+1 -v k ‖2,‖u k+1 -u k The iteration stops when ‖2}≤γ and the output is:

8. A storage medium having a computer program stored thereon, characterized in that: When a processor executes the computer program, the method according to any one of claims 1 to 7 is implemented.

9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the computer program, the method according to any one of claims 1 to 7 is implemented.

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