A fast phase unwrapping method based on tail non-convex regularization

Abstract

The application provides a fast phase unwrapping method based on tail non-convex regularization, and aims at solving the following two technical problems of the existing phase unwrapping algorithm: one is that details are lost due to the discontinuity of the image edge; and the other is that it is difficult to accurately realize phase unwrapping when facing high complexity or large phase jumps. The steps of the application are as follows: based on the wrapped image, a non-convex tail optimization model with box constraint is constructed; the non-convex tail optimization model is solved by using a completely split primal-dual algorithm under the condition that the support set is empty, so that an initial phase is obtained; errors in the initial phase estimation are corrected through tail minimization, and the support set is dynamically updated, so that an unwrapped image is obtained. Compared with the traditional phase unwrapping algorithm, the application combines the non-convex regularization with the tail minimization strategy, so that the accuracy and robustness of the phase unwrapping are improved, and the application has significant superiority in key evaluation indexes such as signal-to-noise ratio and structural similarity.

Description

Technical Field

[0001] This invention relates to the field of digital image processing, specifically to the field of phase unwinding, and particularly to a fast phase unwinding method. Background Technology

[0002] Two-dimensional phase unwrapping (PU) is a crucial step in phase imaging technology, widely used in fields such as interferometric synthetic aperture radar and sonar (InSAR / InSAS), magnetic resonance imaging (MRI), and optical interferometry. In these techniques, due to limitations of the acquisition system, only the wrapped phase can be measured, inevitably introducing discontinuities and jumps in the phase distribution, thus limiting the integrity of the phase information. The purpose of phase unwrapping is to recover the continuous phase distribution behind these jumps, thereby obtaining complete information about the sample. Due to the irreversibility and complexity of the phase wrapping process, phase unwrapping is an extremely challenging task, attracting extensive research and exploration.

[0003] Phase unwinding methods mainly fall into the following categories: path tracking, minimum... Norms, Bayesian / regularization, and parametric modeling are employed. Path-tracking methods, as heuristics, achieve phase unwrapping by integrating the phase gradients of neighboring pixels along specific paths, typically defined by branch-cutting techniques or quality maps. However, this method is susceptible to noise and path selection, and prone to errors when handling complex phase transitions, leading to unwrapping results that deviate from reality. Minimum Normative methods seek optimal solutions to phase unwrapping by constructing a norm function representing the difference between the true phase and the wrapped phase. This method effectively mitigates abrupt edge changes but sacrifices important details in the original image. Especially when processing large-scale data or complex phase transitions, the computational cost is high, affecting the accuracy and efficiency of unwrapping. Bayesian / regularized methods perform statistical inference to unwrap the phase by combining sampled phase signal data and prior knowledge. This method is highly dependent on prior information and has high computational complexity. Parametric modeling algorithms seek optimal solutions by constraining the phase unwrapping problem to a parametric surface. However, this method is susceptible to inaccurate model estimations and has high computational costs, limiting its application.

[0004] In the two-dimensional phase untangling problem, deep learning, especially residual neural networks, has shown significant potential, effectively approximating the complex mapping between tangled and untangled phases through a supervised learning framework. Such methods, combining residual networks with objective optimization functions, have been successfully applied to the untangling of quantitative phase images of biological cells. However, their heavy reliance on large amounts of labeled data for training limits their generalization ability in practical applications due to data scarcity.

[0005] Variational regularization methods have demonstrated superior interpretability and stability in various image processing and inverse problems. In the field of phase unwinding based on variational regularization, the design of iterative algorithms typically revolves around convex optimization models. While these methods provide stability and ensure solvability, they often struggle to effectively capture the inherent piecewise structural features in the phase, especially the local variability of the entanglement count. To improve solution accuracy, existing research has proposed models based on global weighting strategies, but these increase computational overhead during iterative solutions and may affect solution quality due to insufficient consideration of local details. This limitation is particularly pronounced when dealing with highly complex phase transitions or significant amplitude variations. By strengthening sparsity constraints, non-convex regularization techniques exhibit superior recovery performance compared to traditional convex regularization methods, with a significant characteristic being their ability to significantly improve image contrast and enrich detail representation. Summary of the Invention

[0006] To address two shortcomings of existing phase unwinding algorithms in reconstruction results: firstly, detail loss due to discontinuities at image edges; and secondly, difficulty in accurately achieving phase unwinding when facing high complexity or significant phase jumps, this invention proposes a fast phase unwinding method based on tail-based non-convex regularization. A non-convex tail optimization model with box constraints is proposed to achieve rapid phase unwinding solutions. This method combines anisotropic and isotropic total variation regularization and introduces non-convex regularization with tail optimization. Through a fully split primal-dual hybrid gradient algorithm, the dual form of the non-convex tail optimization model is obtained, significantly improving the accuracy and robustness of phase unwinding. Simultaneously, this method achieves high-precision, low-computational-cost fast phase unwinding processing.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows: a fast phase unwinding method based on tail non-convex regularization, the steps of which are as follows:

[0008] Step 1: Based on the entanglement image, construct a non-convex tail optimization model with box constraints;

[0009] Step 2: With the support set empty, solve the non-convex tail optimization model using the fully split primal dual algorithm to obtain the initial phase;

[0010] Step 3: Correct the error in the initial phase estimation by minimizing the tail, and dynamically update the support set to obtain the phase-unwrapped image.

[0011] Preferably, the non-convex tail optimization model with box constraints is:

[0012]

[0013] in, The pixel value of pixel (i,j) representing the pixel of the piecewise variable z. For discrete gradient operators, For the entangled phase The amount of gradient winding, It is the regularization weight parameter. For penalty parameters, It is the entanglement operator, a constant. The upper bound of the envelope. It is for variables The support set, and .

[0014] Preferably, the winding phase Phase of a two-dimensional image get: N represents the length and width of the two-dimensional image;

[0015] The phase winding process is represented as:

[0016]

[0017] in, Phase The pixel value at pixel (i,j) Indicates the entanglement phase The pixel value at pixel (i,j).

[0018] Preferably, the variable with fragmentation and phase ;

[0019] The amount of winding Then we get the variable: .

[0020] Preferably, the method for solving the non-convex tail optimization model with box constraints is as follows:

[0021] based on The original dual form of the saddle point problem for a non-convex tail optimization model with box constraints is as follows:

[0022]

[0023] in, , It is an index function on the closed set Z; They are respectively from and The dual variable introduced by the dual form, and They are closed sets and closed set Indicator functions;

[0024] Introducing parameters into the saddle point problem The quadratic term yields the penalty model:

[0025]

[0026] The penalty model can be transformed into an optimization problem of saddle points:

[0027]

[0028] Among them, the intermediate function intermediate variables , , Representation function convex conjugate;

[0029] Solve the optimization saddle point problem using the fully split primal-dual hybrid gradient algorithm.

[0030] Preferably, the index function for:

[0031]

[0032] Closed set

[0033] Closed set

[0034] Closed set ;

[0035] in, , , , , Representing variables respectively and dual variables and exist or In the direction of The pixel value at that location.

[0036] Preferably, the implementation method of the fully split primal-dual hybrid gradient algorithm is as follows:

[0037] make For variables Dual variables Dual variables The The solution of the iteration, variables Dual variables Dual variables The subproblem in the (k+1)th iteration is as follows:

[0038] ;

[0039] ;

[0040] ;

[0041] ;

[0042] in, This represents the extrapolation step of the acceleration algorithm, with step size. All are positive numbers.

[0043] Preferably, the first In the next iteration, the variables Dual variables Dual variables The closed-form solutions to the subproblems are as follows:

[0044] ;

[0045] ;

[0046] in, Represents the projection operator. Represented as:

[0047]

[0048] To support the set.

[0049] Preferably, the method for obtaining the phase-unwrapped image in step three is as follows:

[0050] (1). Initial value selection: Select appropriate parameters for regularization weights. ,parameter Step length Step length Step length Penalty parameters and let the variable Number of iterations Support set ;

[0051] (2). Iteration:

[0052] Calculate closed-form solutions for piecewise variables. ;

[0053] Calculate the closed-form solution of the dual variable. ; Calculate the closed-form solution of the dual variable. ;

[0054] Based on prior information, set initial iteration conditions. When these initial conditions are met: update the support set.

[0055] (3) The termination condition is met: , The iteration terminates when the positive constant is given empirically.

[0056] The phase of the image after phase unwinding is obtained. .

[0057] Preferably, the support set estimation The setup method is as follows:

[0058] 1) Calculate the residuals: in, This indicates taking the absolute value of each element of the variable;

[0059] 2) The residual The elements in the array are arranged in descending order. Find the first... The positions of the elements with the largest absolute values ​​are identified, and the set of their indices is set as the support set. ;

[0060] 3) Support set Substitute variables with partitioning Closed-form solution ;renew (Return to step 1). This indicates the step size for updating the support set.

[0061] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0062] Because the recovery of unstable or significantly discontinuous regions in the phase unwinding solution is poor, this invention introduces a tail minimization strategy that focuses on the complement of the signal support set, aiming to further optimize the solution quality and enhance the model's performance. Therefore, combining non-convex regularization with a tail minimization strategy to improve the accuracy and robustness of phase unwinding is not only a necessary technical innovation but also has significant practical implications and broad application prospects. The specific beneficial effects of this invention are as follows:

[0063] 1) This invention combines non-convex regularization terms with compatibility constraint norms, and also incorporates box constraints to enhance the sparsity of the solution, which is beneficial for the adaptive balance of discontinuous regions, thereby significantly improving the accuracy and robustness of phase unwinding.

[0064] 2) This invention designs an innovative tail optimization mechanism that intelligently identifies and focuses on unstable or significantly discontinuous regions in the phase untangling solution. Through an iterative optimization process, this mechanism gradually corrects the initial estimation error and dynamically excludes stable pixels, thereby greatly reducing unnecessary computational burden and effectively suppressing error accumulation.

[0065] 3) This invention proposes an accelerated fully split primordial-dual hybrid gradient algorithm, aiming to transform non-convex optimization problems into their dual form. In this form, this invention not only derives and proves the convergence of the algorithm, but also demonstrates that the algorithm can obtain closed-form solutions; through this method, the solution efficiency of the algorithm is effectively improved, and it is particularly suitable for optimization scenarios of complex non-convex problems.

[0066] 4) To comprehensively evaluate the performance of the proposed algorithm, this invention designed and implemented a series of simulations to assess convergence, parameter influence, and overall performance. Compared with traditional phase unwinding algorithms, this invention demonstrates significant superiority in key evaluation metrics such as signal-to-noise ratio (SNR) and structural similarity (SSIM). Attached Figure Description

[0067] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0068] Figure 1 This is a flowchart of the present invention.

[0069] Figure 2 This is a comparison of the reconstruction results of the "Lake" image by the present invention and existing methods. (a) is the true phase image, (b) is the tangled phase image, (c) is the result of untangling using GA, (d) is the result of untangling using LS, (e) is the result of untangling using RPU, (f) is the result of untangling using IRTV, (g) is the result of untangling using PUMA, (h) is the result of untangling using a non-convex algorithm with tailless optimization, and (i) is the result of untangling using a tailed optimization algorithm.

[0070] Figure 3This is a comparison of the reconstruction results of the "Man" image by the present invention and existing methods. (a) is the true phase image, (b) is the entangled phase image, (c) is the result of unwinding using GA, (d) is the result of unwinding using LS, (e) is the result of unwinding using RPU, (f) is the result of unwinding using IRTV, (g) is the result of unwinding using PUMA, (h) is the result of unwinding using a non-convex algorithm with tailless optimization, and (i) is the result of unwinding using a tailed optimization algorithm.

[0071] Figure 4 This is a comparison of the reconstruction results of the "pepper" image by the present invention and existing methods. (a) is the true phase image, (b) is the entangled phase image, (c) is the result of unwinding using GA, (d) is the result of unwinding using LS, (e) is the result of unwinding using RPU, (f) is the result of unwinding using IRTV, (g) is the result of unwinding using PUMA, (h) is the result of unwinding using a non-convex algorithm with tailless optimization, and (i) is the result of unwinding using a tailed optimization algorithm. Detailed Implementation

[0072] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0073] like Figure 1 As shown, this invention proposes a fast phase unwinding method based on tail-based non-convex regularization, utilizing a tail-optimized non-convex regularization mechanism to achieve efficient phase unwinding. First, a non-convex regularized model with box constraints is constructed based on the anisotropic and isotropic total variation (AITV). Then, a fully split primal-dual hybrid gradient algorithm is used to solve this model. After iterating to a predetermined number of times, the support set is estimated based on the previous iteration results, and the accuracy of the support set is continuously optimized using a designed tail-minimization mechanism. This process gradually corrects errors in the initial phase estimation and dynamically excludes stable pixels from the computational support, effectively curbing error accumulation and further improving the accuracy of the solution. This method significantly improves the accuracy and robustness of phase unwinding. The specific steps are as follows:

[0074] Step 1: Based on the entanglement image, construct a non-convex tail optimization model with box constraints.

[0075] Discrete two-dimensional winding phase Phase of a two-dimensional image get:

[0076] (1)

[0077] in, Let N represent the set of real numbers, and let N represent the length and width of the image. This corresponds to the number of pixels in the image. It is a entanglement operator, and the phase entanglement process is represented as:

[0078] (2)

[0079] in, Phase exist Pixel value of direction, Indicates the entanglement phase The pixel value in the i-th row and j-th column. Due to the entanglement operator... It's a surjective problem, not a single-shot problem, so the entanglement process is irreversible. Therefore, the two-dimensional phase unentanglement problem is inherently ill-posed, and it's usually difficult to solve from a single entangled phase. The untangled phase is obtained directly. .

[0080] Assumption For discrete gradient operators, i.e.

[0081]

[0082] in, These represent finite differences in the horizontal and vertical directions, respectively. Assume phase... For a two-dimensional extension where each pixel satisfies the Itō continuity condition, i.e.

[0083] (3)

[0084] Then, the following relationship exists:

[0085] (4)

[0086] Phase unwinding is essentially a discontinuous piecewise problem. To address this problem and guarantee the piecewise nature of the solution, this invention introduces variables with piecewise definitions. At this point, formula (2) can be rewritten as:

[0087] (5)

[0088] Wrapping phase The amount of gradient winding is expressed as From formula (5), we can obtain the variables:

[0089] (6)

[0090] Therefore, solving for the phase The task can be transformed into solving variables. . use Norms to strengthen The sparsity of the non-convex tail optimization model with box constraints proposed in this invention can be written as:

[0091] (7)

[0092] in, This represents the pixel value corresponding to each pixel point of the variable z. It is the regularization weight parameter, which has been experimentally tested. It has better reconstruction effect. The penalty parameter is a constant. The upper bound of the envelope. It is for variables support set, To enhance the consistency of solutions, we utilize... Punishment Equation Because the grayscale values ​​of a digital image are finite, box constraints... This is reasonable, thus ensuring that the iterative solution is physically reasonable and numerically stable.

[0093] Step 2: In the support set Given an empty set, the non-convex tail optimization model is solved using the fully split primal dual algorithm (FS-PDHG algorithm) to obtain the initial phase.

[0094] based on The original dual form of the saddle point problem related to the nonconvex regularized model, i.e., equation (7), can be expressed as:

[0095] (8)

[0096] in, This additional tail norm can significantly reduce artifacts caused by boundary diffusion, thereby improving the accuracy of the algorithm. It is a closed set An index function on the can be expressed as:

[0097]

[0098] They are respectively from and The dual variable introduced by the dual form, and They are closed sets and closed set The indicator function is expressed as

[0099] ;

[0100]

[0101] in, , , , , Representing variables respectively and dual variables and exist or In the direction of The pixel value at that location.

[0102] To ensure the convergence of the alternating minimization algorithm for the non-convex tail optimization model, a parameterized algorithm is introduced into equation (8) of the saddle point problem. The quadratic term yields the following penalty model:

[0103] (9)

[0104] The purpose of introducing the above penalty term is to enhance the smoothness of the normalized image gradient, preserve important structural information, and make... The subproblems become strongly convex problems, a property that ensures the stability and fast convergence of the algorithm. Let the intermediate variables be:

[0105]

[0106]

[0107]

[0108] in, , , Representation function convex conjugate.

[0109] This allows the penalty model of the saddle point problem in equation (4), i.e., equation (9), to be written as:

[0110]

[0111] Among them, the intermediate function The saddle point problem described above is solved using the fully split primal-dual hybrid gradient algorithm (FS-PDHG). Let... For the first The iterative solution for each step has the following iterative format:

[0112] ;

[0113] ;

[0114] ;

[0115] ;

[0116] in, This represents the extrapolation step of the accelerated algorithm, with parameters... All are positive numbers, representing the step size of the algorithm. Regarding variables... , , These subproblems are strictly convex and have closed-form solutions.

[0117] No. In this iteration, the proposed untangling algorithm can be described as follows:

[0118] (10);

[0119] (11);

[0120] (12).

[0121] in, Represents the projection operator. This represents the discrete gradient operator.

[0122] Step 3: By minimizing the tail, the error in the initial phase estimation is gradually corrected, and the support set is dynamically updated.

[0123] In the steps described above, we set the support set to an empty set and update each pixel in the image. After a specified number of iterations, we initiate a tail optimization mechanism. By minimizing the tail, we gradually correct the errors in the initial phase estimation and dynamically exclude stable pixels from the calculated support set, thereby effectively curbing error accumulation, significantly improving the algorithm's convergence speed, and ensuring that the predetermined stopping condition is reached quickly.

[0124] Because of the support set With minimizing variables Related, therefore it is difficult to obtain an accurate support set. It can be iteratively refined by solving equation (7) (the non-convex tail optimization model) or its dual form (8) (the saddle point problem), where the support set is estimated. The definition follows a clear and strict set of rules that precisely define the support set. The range of elements covered and their attributes. Support set estimation. The setup process is as follows:

[0125] 1. Calculate the residuals:

[0126] in, This indicates taking the absolute value of each element of the variable.

[0127] 2. The residual The elements in the array are arranged in descending order. Find the first... The set of the indices of the elements with the largest absolute values ​​is set as the support set. .

[0128] 3. Support set Substitute into equation (10) to update variables .

[0129] 4. Update Go back to step one and start over. This indicates the step size for updating the support set.

[0130] Specifically, this method aims to focus resources on regions crucial to phase continuity—those areas where phase continuity may exist despite initial inaccuracies. During iteration, the number of elements in the support set is gradually increased to expand its scope. Simultaneously, pixels that have been stably resolved are sequentially excluded from the support set, and the phase estimates of previously inaccurate pixels are gradually corrected, thereby reducing unnecessary computational overhead and effectively mitigating the impact of error accumulation. The non-convex model algorithm with tail optimization is summarized below:

[0131] 1. Initial value selection: Select appropriate parameters. , , , , , and order Input the winding phase. Initialize the number of iterations , .

[0132] 2. Iteration:

[0133] Calculate the target image using equation (10) ;

[0134]

[0135] Calculate the dual variables using equation (11) ;

[0136] Calculate the dual variables using equation (12) ;

[0137] Based on prior information, initial iteration conditions are set. When these conditions are met:

[0138] Update support set

[0139] Termination conditions are met: , The iteration terminates when a positive constant is given empirically. Output The image after phase unwinding is obtained.

[0140] All simulation data were derived from a selection of grayscale images from the USC-SIPI database. Each reconstructed image has a resolution of 256×256 pixels. The amplitude of the simulation images was normalized to the range. ( Within ) . Subsequently, different wrapping phase images are generated according to formula (2). It is worth noting that as the image amplitude increases, the true phase The edges become sharper, making phase unwrapping increasingly challenging. For the following simulation, the obtained wrapped phase is used as the initial value for the algorithm, and a threshold for the termination condition is set. .

[0141] The experimental platform for this invention is a personal laptop with 16GB of memory, and the program is designed using Matlab R2021a.

[0142] Specific experiments

[0143] A comparison of reconstructions of the "Lake" image using various models is shown in the figure. Figure 2 . Figure 2 (a) is the true phase image. Figure 2 (b) is the entangled phase image. Figure 2 (c) shows the result of untangling using GA (path tracing method). Figure 2 (d) represents the result of untangling using LS (least squares) method. Figure 2 (e) represents the result of unwinding using RPU (phase unwinding based on the intensity transport equation). Figure 2 (f) represents the result of untangling using IRTV (Isotropic Regularization of Weighted Least Squares). Figure 2 (g) represents the result of unwinding using PUMA (Phase Unwinding Maximum Flow / Minimum Cut Method). Figure 2 (h) represents the unwinding result of the non-convex algorithm without tail optimization. Figure 2 (i) The result of untangling using the tail optimization algorithm of the present invention.

[0144] Figure 2This indicates that when dealing with complex scenarios involving continuous phase jumps, such as Figure 2 As shown in the green boxes, the non-convex algorithm without tail optimization can accurately identify and segment these regions. Compared with traditional methods, the non-convex model effectively highlights the "backbone" structure in the image, thereby enhancing its structural information. Furthermore, as shown in the red and blue boxes, adding tail optimization further achieves phase continuity and reduces the staircase effect commonly found in regions of significant phase changes. This results in smoother phase transitions and improves overall image quality.

[0145] A comparison of reconstructions of the "Man" image using various models is shown in the figure. Figure 3 . Figure 3 (a) is the true phase image. Figure 3 (b) is the entangled phase image. Figure 3 (c) shows the result of unwinding using GA. Figure 3 (d) represents the result of unwinding using LS. Figure 3 (e) represents the result of unwinding using RPU. Figure 3 (f) represents the result of unwinding using IRTV. Figure 3 (g) represents the result of unwinding using PUMA. Figure 3 (h) represents the unwinding result of the non-convex algorithm without tail optimization. Figure 3 (i) represents the result of untangling using the tail optimization algorithm of this invention.

[0146] Figure 3 This indicates that the GA and RPU algorithms have significant limitations when handling complex entangled phase scenes. In contrast, the LS, IRTV, and PUMA algorithms can untangle most areas of the image to a certain extent, but significant phase entanglement artifacts still exist in specific regions, such as the area shown in the red box in the sub-image. These artifacts cause discontinuities and breaks in the phase distribution. Notably, the tail-free non-convex algorithm, while inheriting the advantages of the above algorithms, significantly improves the phase unwrapping performance, resulting in a smoother and more continuous reconstruction with reduced phase discontinuities. Tail-optimized algorithms further enhance the reconstruction of entangled regions by continuously optimizing the support set, making the reconstruction quality visually close to the real image.

[0147] A comparison of reconstructions of the "pepper" image using various models is shown in the figure. Figure 4 . Figure 4 (a) is the true phase image. Figure 4 (b) is the entangled phase image. Figure 4 (c) shows the result of unwinding using GA. Figure 4 (d) represents the result of unwinding using LS. Figure 4(e) represents the result of unwinding using RPU. Figure 4 (f) represents the result of unwinding using IRTV. Figure 4 (g) represents the result of unwinding using PUMA. Figure 4 (h) represents the unwinding result of the non-convex algorithm without tail optimization. Figure 4 (i) represents the result of untangling using the tail optimization algorithm of this invention.

[0148] Figure 4 The results show that, in the area marked by the white circle, the reconstruction results of the first five methods exhibit significant discontinuities at the texture transition edges of the chili pepper. These discontinuities are particularly pronounced with the LS, RPU, and IRTV algorithms, leading to extensive grayscale diffusion during phase unwinding. In contrast, the two algorithms proposed in this invention demonstrate superior performance in addressing these issues. Both effectively avoid edge discontinuities, ensuring that edge information in detailed areas like the chili pepper is clearly and accurately preserved in the reconstructed image.

[0149] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A fast phase unwinding method based on tail non-convex regularization, characterized in that, The steps are as follows: Step 1: Based on the entanglement image, construct a non-convex tail optimization model with box constraints; Step 2: With the support set empty, solve the non-convex tail optimization model using the fully split primal dual algorithm to obtain the initial phase; Step 3: Correct the error in the initial phase estimation by minimizing the tail, and dynamically update the support set to obtain the phase-unwrapped image; The non-convex tail optimization model with box constraints is as follows: ； in, The pixel value of pixel (i,j) representing the pixel of the piecewise variable z. For discrete gradient operators, For the entangled phase The amount of gradient winding, It is the regularization weight parameter. For penalty parameters, It is the entanglement operator, a constant. The upper bound of the envelope. It is for variables The support set, and .

2. The fast phase unwinding method based on tail non-convex regularization according to claim 1, characterized in that, The winding phase Phase of a two-dimensional image get: N represents the length and width of the two-dimensional image; The phase winding process is represented as: ； in, Phase The pixel value at pixel (i,j) Indicates the entanglement phase The pixel value at pixel (i,j).

3. The fast phase unwinding method based on tail non-convex regularization according to claim 2, characterized in that, The variable with partitioning and phase ; The amount of winding Then we get the variable: .

4. The fast phase unwinding method based on tail non-convex regularization according to any one of claims 1-3, characterized in that, The method for solving the non-convex tail optimization model with box constraints is as follows: based on The original dual form of the saddle point problem for a non-convex tail optimization model with box constraints is as follows: ； in, , It is an index function on the closed set Z; They are respectively from and The dual variable introduced by the dual form, and They are closed sets and closed set Indicator functions; Introducing parameters into the saddle point problem The quadratic term yields the penalty model: ； The penalty model is transformed into an optimization saddle point problem: ； Among them, the intermediate function intermediate variables , , Representation function convex conjugate; Solve the optimization saddle point problem using the fully split primal-dual hybrid gradient algorithm.

5. The fast phase unwinding method based on tail non-convex regularization according to claim 4, characterized in that, The index function for: ； Closed set ; Closed set ; Closed set ; in, , , , , Representing variables respectively and dual variables and exist or In the direction of The pixel value at that location.

6. The fast phase unwinding method based on tail non-convex regularization according to claim 5, characterized in that, The implementation method of the fully split primal-dual hybrid gradient algorithm is as follows: make For variables Dual variables Dual variables The The solution of the iteration, variables Dual variables Dual variables The subproblem in the (k+1)th iteration is as follows: ； ； ； ； in, This represents the extrapolation step of the acceleration algorithm, with step size. All are positive numbers.

7. The fast phase unwinding method based on tail non-convex regularization according to claim 6, characterized in that, No. In the next iteration, the variables Dual variables Dual variables The closed-form solutions to the subproblems are as follows: ； ； ； in, Represents the projection operator. Represented as: ； To support the set.

8. The fast phase unwinding method based on tail non-convex regularization according to claim 7, characterized in that, The method for obtaining the phase-unwrapped image in step three is as follows: (1). Initial value selection: Select appropriate parameters for regularization weights. ,parameter Step length Step length Step length Penalty parameters and let the variable Number of iterations Support set ; (2). Iteration: Calculate closed-form solutions for piecewise variables. ; Calculate the closed-form solution of the dual variable. ; Calculate the closed-form solution of the dual variable. ; Based on prior information, set initial iteration conditions. When these initial conditions are met: update the support set. ; (3) The termination condition is met: , The iteration terminates when the positive constant is given empirically. The phase of the image after phase unwinding is obtained. .

9. The fast phase unwinding method based on tail non-convex regularization according to claim 8, characterized in that, The support set estimation The setup method is as follows: 1) Calculate the residuals: ;in, This indicates taking the absolute value of each element of the variable; 2) The residual The elements in the array are arranged in descending order. Find the first... The positions of the elements with the largest absolute values ​​are identified, and the set of their indices is set as the support set. ; 3) Support set Substitute variables with partitioning Closed-form solution ;renew (Return to step 1). This indicates the step size for updating the support set.

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