Rapid phase unwrapping method based on tail non-convex regularization

Abstract

The invention provides a rapid phase unwrapping method based on tail non-convex regularization. The rapid phase unwrapping method is used for overcoming the defects of an existing phase unwrapping algorithm in two aspects in a reconstruction result: firstly, details are lost due to discontinuity of image edges; and secondly, the technical problem that phase unwrapping is difficult to realize accurately when the complexity is relatively high or large-amplitude phase jump exists is solved. The method comprises the following steps: constructing a non-convex tail optimization model with box constraint based on a wound image; when the support set is an empty set, solving the non-convex tail optimization model by using a completely split primal dual algorithm to obtain an initial phase; and correcting an error in initial phase estimation through tail minimization, and dynamically updating the support set to obtain an image after phase unwrapping. Compared with a traditional phase unwrapping algorithm, the non-convex regularization and tail minimization strategies are combined to improve the precision and robustness of phase unwrapping, and the method has remarkable superiority in key evaluation indexes such as the signal-to-noise ratio and the structural similarity.

Description

Technical Field

[0001] The present invention relates to the technical field of digital image processing, in particular to the technical field of phase unwrapping, and in particular to a fast phase unwrapping method. Background Art

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

[0003] Phase unwrapping methods mainly belong to the following categories: path tracking, minimum L p norm, Bayesian / regularization, and parameterized modeling. Path tracking methods are heuristic methods that achieve phase unwrapping by integrating the phase gradients of adjacent pixels along specific paths, which are usually defined by branch-cut techniques or quality maps. However, this method is susceptible to noise and path selection, and is prone to errors when dealing with complex phase jumps, resulting in unwrapping results that deviate from reality. Minimum L p The norm method seeks the optimal solution to achieve phase unwrapping by constructing the norm function of the difference between the true phase and the wrapped phase. This method effectively alleviates the abrupt changes in the edge, but also sacrifices the important details of the original image. Especially when processing large-scale data or complex phase jumps, the computational overhead is high, affecting the accuracy and efficiency of unwrapping. The Bayesian / regularization method combines the sampling data and prior knowledge of the phase signal to perform statistical inference to unwrap the phase. This method has a strong reliance on prior information and high computational complexity. The parametric modeling algorithm seeks the optimal solution by constraining the phase unwrapping problem to a parameterized surface for optimization. When implementing phase unwrapping, this method is easily affected by inaccurate model estimation and has a high computational cost, which limits its scope of application.

[0004] In the two-dimensional phase unwrapping problem, deep learning, especially residual neural networks, has shown significant potential, effectively approximating the complex mapping between the wrapped phase and the unwrapped phase through a supervised learning framework. This type of method combines residual networks with target optimization functions and has been successfully applied to the unwrapping of quantitative phase images of biological cells. However, it is highly dependent on a large amount of labeled data for training, which limits its 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 unwrapping based on variational regularization, the design of iterative algorithms usually revolves around convex optimization models. Although these methods can provide stability and ensure solvability, they often have difficulty in effectively capturing the inherent piecewise structural characteristics in the phase, especially the local variability of the winding count. To improve the accuracy of the solution, some studies have proposed models based on global weighting strategies, but they increase the computational overhead in the iterative solution process and may affect the quality of the solution due to insufficient consideration of local details. This limitation is particularly evident when dealing with highly complex phase changes or significant amplitude changes. By strengthening the sparsity constraint, non-convex regularization techniques have shown more excellent restoration performance than traditional convex regularization methods. Its notable feature is that it can significantly improve the contrast of the image and enrich the detail expression. Summary of the invention

[0006] In view of the two shortcomings of the existing phase unwrapping algorithm in the reconstruction results: one is the loss of details due to the discontinuity of the image edge; the other is the technical problem that it is difficult to accurately achieve phase unwrapping when faced with high complexity or large phase jumps. The present invention proposes a fast phase unwrapping method based on tail non-convex regularization, and proposes a non-convex tail optimization model with box constraints to achieve fast phase unwrapping solution. This method combines anisotropic and isotropic total variation regularization, and introduces non-convex regularization with tail optimization. Through the fully split primal-dual hybrid gradient algorithm, the dual form of the non-convex tail optimization model is obtained, which significantly improves the accuracy and robustness of phase unwrapping. At the same time, this method achieves fast phase unwrapping processing with high precision and low computational cost.

[0007] In order to achieve the above object, the technical solution of the present invention is implemented as follows: a fast phase unwrapping method based on tail non-convex regularization, the steps of which are as follows:

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

[0009] Step 2: When the support set is an empty set, the primal-dual algorithm with complete splitting is used to solve the non-convex tail optimization model and obtain the initial phase;

[0010] Step 3: Correct the error in the initial phase estimation through tail minimization, 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] Among them, (z) i,j represents the pixel value of the pixel point (i, j) of the variable z with slices, is the discrete gradient operator, r Ψ is the winding amount of the winding phase Ψ gradient, α is the regularization weight parameter, β is the penalty parameter, is the winding operator, the constant c is the upper bound of the envelope, and T is the variable The support set of

[0014] Preferably, the wrapping phase Ψ is obtained by wrapping the phase of the two-dimensional image get: N represents the length and width of the two-dimensional image;

[0015] The phase winding process is expressed as:

[0016]

[0017] Among them, (Φ) i,j is the pixel value of phase Φ at pixel (i, j), Represents the pixel value of the winding phase Ψ at the pixel point (i, j).

[0018] Preferably, the variable with slices And the phase Φ = Ψ-z;

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

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

[0021] Based on ||·|| 2,1 The saddle point problem of the non-convex tail optimization model with box constraints is:

[0022]

[0023] in, Ⅱ Z (z) is the index function on the closed set Z; Respectively by and The dual variable introduced by the dual form of S (p) and II Q (q) are the indicator functions of the closed set S and the closed set Q respectively;

[0024] Introducing a quadratic term with parameter μ into the saddle point problem leads to the penalty model:

[0025]

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

[0027]

[0028] Among them, the intermediate function Intermediate variables f* represents the convex conjugate of function f;

[0029] Solving optimization saddle point problems using a fully split primal-dual hybrid gradient algorithm.

[0030] Preferably, the indicator function II Z (z) is:

[0031]

[0032] Closed Set

[0033] Closed Set

[0034] Closed Set

[0035] Among them, z i,j ,(q x ) i,j ,(q y ) i,j 、(p x ) i,j 、(p y ) i,j denote the pixel value of the variable z and the dual variables p and q at (x, y) in the x or y direction respectively.

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

[0037] Let (z k ,p k ,q k ) is the k-th iteration solution of variable z, dual variable q, and dual variable p. The k+1-th iteration process of the subproblem of variable z, dual variable q, and dual variable p is:

[0038]

[0039]

[0040] in, Represents the extrapolation step of the acceleration algorithm, and the step sizes η, τ, and γ are all positive numbers.

[0041] Preferably, in the k+1th iteration, the closed-form solutions of the subproblems of variable z, dual variable q, and dual variable p are respectively:

[0042]

[0043] Among them, Proj(.;.) represents the projection operator, It is expressed as:

[0044]

[0045] T k For the support set.

[0046] Preferably, the method for obtaining the phase unwrapped image in step three is:

[0047] (1) Initial value selection: Select appropriate parameter regularization weight parameter α, parameter μ, step size η, step size γ, step size τ, penalty parameter β, and set the variable z 0 =q 0 =p 0 =0; number of iterations k = 1, support set

[0048] (2) Iteration:

[0049] Compute the closed-form solution z for a variable with slices k+1 ;

[0050] Compute the closed-form solution q for the dual variable k+1 ; Compute the closed-form solution p of the dual variable k+1 ;

[0051] Set the initial conditions for iteration according to the prior information. When the initial conditions for iteration are met: update the support set

[0052] (3) Termination conditions are met: tol is a positive constant given by experience, and the iteration is terminated;

[0053] Get the phase Ψ-z of the image after phase unwrapping k+1 .

[0054] Preferably, the support set estimation The setting method is:

[0055] 1) Calculate the residual: Among them, |·| means taking the element-by-element absolute value of the variable;

[0056] 2) Arrange the elements in the residual A in descending order, find the positions of the first m elements with the largest absolute values, and set the set of their indices as the support set T k ;

[0057] 3) The support set T k Substitute the closed-form solution z with the sliced ​​variable z k+1 ; Update m←m+m 0 , return to step 1); m 0 represents the step size of the support set update.

[0058] Compared with the prior art, the present invention has the following beneficial effects:

[0059] Since the recovery effect of unstable or obviously discontinuous regions in the phase unwrapping solution is poor, the present invention introduces a tail minimization strategy to focus on the complement of the signal support set, aiming to further optimize the quality of the solution and enhance the effectiveness of the model. Therefore, combining non-convex regularization with the tail minimization strategy to improve the accuracy and robustness of phase unwrapping is not only a necessary technical innovation, but also has important practical significance and broad application prospects. The beneficial effects of the present invention are specifically manifested in:

[0060] 1) The present invention combines a non-convex regularization term with a compatibility constraint norm and a box constraint to enhance the sparsity of the solution and facilitate the adaptive balance of discontinuous regions, thereby significantly improving the accuracy and robustness of phase unwrapping.

[0061] 2) The present invention designs an innovative tail optimization mechanism to intelligently identify and focus on unstable or obviously discontinuous regions in the phase unwrapping solution. This mechanism gradually corrects the estimation error in the initial stage through an iterative optimization process, dynamically excludes stable pixels, thereby greatly reducing unnecessary computational burden and effectively suppressing error accumulation.

[0062] 3) This paper proposes an accelerated full-splitting primal-dual hybrid gradient algorithm, which aims to transform non-convex optimization problems into their dual form. In this form, this paper not only derives and proves the convergence of the algorithm, but also shows that the algorithm can obtain a closed-form solution; through this method, the algorithm's solution efficiency is effectively improved, which is particularly suitable for optimization solutions to complex non-convex problems.

[0063] 4) In order to comprehensively evaluate the performance of the proposed algorithm, the present invention designs and implements a series of simulations to evaluate aspects such as convergence, parameter influence and overall performance. Compared with the traditional phase unwrapping algorithm, the present invention has significant advantages in key evaluation indicators such as signal-to-noise ratio (SNR) and structural similarity (SSIM). BRIEF DESCRIPTION OF THE DRAWINGS

[0064] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0065] Figure 1 It is a flow chart of the present invention.

[0066] Figure 2 The comparison of the reconstruction results of the "Lake" image by the present invention and the existing method. Among them, (a) the real phase image, (b) the entangled phase image, (c) the result of GA detangling, (d) the result of LS detangling, (e) the result of RPU detangling, (f) the result of IRTV detangling, (g) the result of PUMA detangling, (h) the result of non-convex algorithm detangling without tail optimization, (i) the result of detangling using tail optimization algorithm.

[0067] Figure 3 Comparison of the reconstruction results of the "Man" image by the present invention and the existing method. (a) True phase image, (b) Wrapped phase image, (c) Unwrapping result using GA, (d) Unwrapping result using LS, (e) Unwrapping result using RPU, (f) Unwrapping result using IRTV, (g) Unwrapping result using PUMA, (h) Unwrapping result using non-convex algorithm without tail optimization, (i) Unwrapping result using tail optimization algorithm.

[0068] Figure 4 Comparison of the reconstruction results of the "pepper" image by the present invention and the existing method. Among them, (a) the real phase image, (b) the entangled phase image, (c) the result of GA detangling, (d) the result of LS detangling, (e) the result of RPU detangling, (f) the result of IRTV detangling, (g) the result of PUMA detangling, (h) the result of non-convex algorithm detangling without tail optimization, (i) the result of detangling using tail optimization algorithm. DETAILED DESCRIPTION

[0069] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0070] like Figure 1 As shown, the present invention proposes a fast phase unwrapping method based on tail non-convex regularization, and uses non-convex regularization with a tail optimization mechanism to achieve efficient phase unwrapping. First, a non-convex regularized model with box constraints is constructed based on anisotropic and isotropic total variation (AITV). Subsequently, the model is solved using a fully split primal-dual mixed gradient algorithm. 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 the designed tail minimization mechanism. This process gradually corrects the errors in the initial phase estimation, and dynamically excludes stable pixels from the calculation support, effectively curbing the accumulation of errors, thereby further improving the accuracy of the solution. This method significantly improves the accuracy and robustness of phase unwrapping, and the specific steps are as follows:

[0071] Step 1: Based on the winding image, a non-convex tail optimization model with box constraints is constructed.

[0072] Discrete 2D winding phase By wrapping the phase of the two-dimensional image get:

[0073]

[0074] in, represents a set of real numbers, N represents the length and width of the image, and N×N corresponds to the number of image pixels. is the winding operator, and the phase winding process is expressed as:

[0075]

[0076] Among them, (Φ) i,j is the pixel value of phase Φ in the (i, j) direction, represents the pixel value of the i-th row and j-th column of the winding phase Ψ. Since the winding operator It is a surjective rather than an injective one, so the winding process is irreversible. Therefore, the two-dimensional phase unwrapping problem is inherently ill-posed, and it is usually difficult to directly obtain the unwrapped phase Φ from a single wrapped phase Ψ.

[0077] Assumptions is a discrete gradient operator, that is

[0078]

[0079] in, denote the finite differences in the horizontal and vertical directions respectively. Assume that the phase Φ satisfies the two-dimensional extension of the Ito continuity condition for each pixel, that is,

[0080]

[0081] Then, there is the following relationship:

[0082]

[0083] Phase unwrapping is essentially a discontinuous piecewise problem. In order to solve this problem and ensure the piecewise nature of the solution, the present invention introduces variables with piecewise definitions. At this time, formula (2) can be rewritten as:

[0084] Φ=Ψ-z; (5)

[0085] The winding amount of the winding phase Ψ gradient is expressed as From formula (5), we can get the variables:

[0086]

[0087] Therefore, the task of solving the phase Φ can be transformed into solving the variable z. 1 -αL 2 norm to strengthen The non-convex tail optimization model with box constraints proposed in this invention can be written as:

[0088]

[0089] Among them, (z) i,j represents the pixel value corresponding to each pixel point of the variable z, α is the regularization weight parameter. According to experimental tests, when α=0.5, the reconstruction effect is better. β is the penalty parameter, the constant c is the upper bound of the envelope, and T is the value of The support set of In order to enhance the consistency of the solution, we use Penalty Equation Since the grayscale values ​​of digital images are finite, the box constraint |z|≤c is reasonable, thus ensuring that the iterative solution is physically reasonable and numerically stable.

[0090] Step 2: When the support set T is an empty set, the completely split primal-dual algorithm (FS-PDHG algorithm) is used to solve the non-convex tail optimization model and obtain the initial phase.

[0091] Based on ||·|| 2,1 The saddle point problem associated with the non-convex regularized model, i.e., equation (7), can be expressed as:

[0092]

[0093] in, This additional tail norm can significantly reduce the artifacts caused by boundary diffusion, thereby improving the accuracy of the algorithm.Z (z) is a closed set An indicator function on can be expressed as:

[0094]

[0095] Respectively by and The dual variable introduced by the dual form of S (p) and II Q (q) are the indicator functions of the closed set S and the closed set Q, respectively, expressed as

[0096]

[0097] Among them, z i,j ,(q x ) i,j ,(q y ) i,j 、(p x ) i,j 、(p y ) i,j denote the pixel value of the variable z and the dual variables p and q at (x, y) in the x or y direction respectively.

[0098] In order to ensure the convergence of the alternating minimization algorithm for the non-convex tail optimization model, a quadratic term with parameter μ is introduced into Equation (8) of the saddle point problem, and the following penalty model is obtained:

[0099]

[0100] The purpose of introducing the above penalty term is to enhance the smoothness of the normalized image gradient, retain important structural information, and make the p-subproblem a strong convex problem. This property ensures the stability and rapid convergence of the algorithm. Let the intermediate variable:

[0101] G(z)=Ⅱ Z (z);

[0102]

[0103] g * (q)=Ⅱ Q (q);

[0104] Where f(·)=||·|| 1 , g(·)=||·|| 2,1 , f* represents the convex conjugate of function f.

[0105] The saddle point problem of formula (4), i.e., the penalty model of formula (9), can be written as:

[0106]

[0107] Among them, the intermediate function The fully split primal-dual hybrid gradient algorithm (FS-PDHG) is used to solve the above saddle point problem. Let (z k ,p k ,q k ) is the iterative solution of the kth step, and the iterative format is as follows:

[0108]

[0109] in, represents the extrapolation step of the acceleration algorithm, and the parameters η, τ, and γ are all positive numbers, indicating the step size of the algorithm. These subproblems about the variables z, q, and p are strictly convex and have closed-form solutions.

[0110] In the k+1th iteration, the proposed disentanglement algorithm can be described as:

[0111]

[0112] Among them, Proj(.;.) represents the projection operator, Represents the discrete gradient operator.

[0113] Step 3: Through tail minimization, the errors in the initial phase estimate are gradually corrected and the support set is dynamically updated.

[0114] In the above steps, we set the support set to an empty set, update each pixel in the image, and start the tail optimization mechanism when the specified number of iterations is reached. Through tail minimization, the error in the initial phase estimation is gradually corrected, and stable pixels are dynamically excluded from the calculated support set, thereby effectively curbing the accumulation of errors, significantly improving the convergence speed of the algorithm, and ensuring that the predetermined stop condition is quickly reached.

[0115] Because the support set T and the minimization variable It is difficult to obtain the exact support set T. It can be iteratively refined by solving equation (7) (non-convex tail optimization model) or its dual form (8) (saddle point problem), where the definition of the estimated support set T follows a set of clear and strict rules that precisely define the range of elements covered by the support set T and its properties. Support set estimation The setup process is as follows:

[0116] 1. Calculate the residual:

[0117] Here, |·| means taking the element-by-element absolute value of the variable.

[0118] 2. Arrange the elements in the residual A in descending order, find the position of the first m elements with the largest absolute value, and set the set of their indexes as the support set T k .

[0119] 3. Set the support set T k Substitute into equation (10) to update the variable z k+1 .

[0120] 4. Update m←m+m 0 , return to the first step and start again. 0 Indicates the step size for updating the support set.

[0121] Specifically, the method aims to focus resources on areas that are critical for phase continuity, that is, those areas where phase continuity may exist despite inaccurate initial estimates. In the iterative process, the number of elements in the support set is gradually increased so as to gradually expand the range of the support set. At the same time, pixels that have been stably solved will be excluded from the support set in turn, and the phase estimates of pixels that were previously inaccurately estimated will be 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 as follows:

[0122] 1. Initial value selection: Select appropriate parameters α, μ, η, γ, τ, β, and set z 0 =q 0 =p 0 = 0. Input winding phase Ψ, initialize the number of iterations k = 1,

[0123] 2. Iteration:

[0124] Calculate the target image z using formula (10) k+1 ;

[0125]

[0126] Using formula (11) to calculate the dual variable q k+1 ;

[0127] Using formula (12) to calculate the dual variable p k+1 ;

[0128] The initial conditions for iteration are set according to the prior information. When the conditions are met:

[0129] Update support set

[0130] Termination conditions are met: tol is a positive constant given by experience, and the iteration terminates. Output Ψ-z k , and get the phase unwrapped image.

[0131] All simulated data are derived from selected grayscale images from the USC-SIPI database, which can be accessed at http: / / sipi.usc.edu / database / . The resolution of each reconstructed image is 256×256 pixels. The amplitude of the simulated image is normalized to the range [0,θ] (θ∈{9,10,11,12}). Subsequently, different wrapped phase images are generated according to formula (2). It is worth noting that as the image amplitude increases, the edge of the true phase Φ becomes sharper, which makes the phase unwrapping task more challenging. For the following simulations, the obtained wrapped phase is used as the initial value of the algorithm, and the threshold tol of the termination condition is set to 1e -4 .

[0132] The experimental platform of the present invention: a personal notebook with 16G memory, and Matlab R2021a is used for program design.

[0133] Specific experiments

[0134] Comparison of reconstruction of the “Lake” image using various models, see Figure 2 . Figure 2 (a) is the real phase image, Figure 2 (b) is the entangled phase image, Figure 2 (c) is the result of untangling using GA (path tracing method), Figure 2 (d) is the result of LS (least squares method) unwrapping. Figure 2 (e) is the result of unwrapping using RPU (phase unwrapping based on intensity transfer equation), Figure 2 (f) is the result of unwrapping using IRTV (isotropic regularization of weighted least squares), Figure 2 (g) is the result of unwrapping using PUMA (Phase Unwrapping Maximum Flow / Minimum Cut Method), Figure 2 (h) is the result of unwinding by a non-convex algorithm without tail optimization. Figure 2 (i) The result of unwinding using the tail optimization algorithm of the present invention.

[0135] Figure 2 It is shown that when dealing with complex scenarios involving continuous phase jumps, e.g. Figure 2 As shown in the green box in the figure, the non-convex algorithm without tail optimization can accurately identify and segment these areas. Compared with traditional methods, the non-convex model can effectively highlight the "trunk" structure in the image, thereby enhancing its structural information. In addition, as shown in the red and blue boxes in the figure, the addition of tail optimization can further achieve phase continuity and reduce the step effect that is common in areas with significant phase changes, which makes the phase transition smoother and improves the overall image quality.

[0136] Comparison of reconstruction of the "Man" image using various models, see Figure 3 . Figure 3 (a) is the real phase image, Figure 3 (b) is the entangled phase image, Figure 3 (c) is the result of detangling using GA, Figure 3 (d) is the result of LS detangling. Figure 3 (e) is the result of using RPU detangling, Figure 3 (f) is the result of IRTV unwinding. Figure 3 (g) is the result of unwinding using PUMA. Figure 3 (h) is the result of unwinding by a non-convex algorithm without tail optimization. Figure 3 (i) is the result of unwinding using the tail optimization algorithm of the present invention.

[0137] Figure 3 It shows that the GA and RPU algorithms have obvious limitations when dealing with complex entangled phase scenes. In contrast, the LS, IRTV and PUMA algorithms can achieve unwrapping for most areas of the image to a certain extent, but there are still obvious phase entanglement artifacts in specific areas, such as the area shown in the red box in the sub-image. These artifacts lead to discontinuity and breakage of the phase distribution. It is worth noting that the non-convex algorithm without tail optimization inherits the advantages of the above algorithms while significantly improving the phase unwrapping performance. The reconstructed results are smoother and continuous, and the occurrence of phase discontinuity is reduced. On this basis, the tail optimization algorithm further enhances the reconstruction of the entangled area by continuously optimizing the support set, so that the reconstruction quality is visually close to the real image.

[0138] Comparison of reconstruction of the "pepper" image using various models, see Figure 4 . Figure 4 (a) is the real phase image, Figure 4 (b) is the entangled phase image, Figure 4 (c) is the result of detangling using GA, Figure 4 (d) is the result of LS detangling. Figure 4 (e) is the result of using RPU detangling, Figure 4 (f) is the result of IRTV unwinding. Figure 4 (g) is the result of unwinding using PUMA. Figure 4 (h) is the result of unwinding by a non-convex algorithm without tail optimization. Figure 4 (i) is the result of unwinding using the tail optimization algorithm of the present invention.

[0139] Figure 4The results show that in the area marked by the enlarged white circle, the reconstruction results of the first five methods show obvious discontinuity at the texture transition edge of the pepper. In particular, under the LS, RPU and IRTV algorithms, the edge discontinuity is particularly prominent, resulting in a large range of grayscale diffusion during the phase unwrapping process. In contrast, the two algorithms proposed in the present invention show excellent performance in solving these problems. Both of them effectively avoid edge discontinuity and ensure that the edge information of detail areas such as peppers in the reconstructed image is clearly and accurately preserved.

[0140] 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 principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A fast phase unwrapping method based on tail non-convex regularization, characterized in that: The steps are as follows: Step 1: Based on the winding image, a non-convex tail optimization model with box constraints is constructed; Step 2: When the support set is an empty set, the primal-dual algorithm with complete splitting is used to solve the non-convex tail optimization model and obtain the initial phase; Step 3: Correct the error in the initial phase estimation through tail minimization and dynamically update the support set to obtain the phase unwrapped image.

2. The fast phase unwrapping method based on tail non-convex regularization according to claim 1, characterized in that: The non-convex tail optimization model with box constraints is: Among them, (z) i,j represents the pixel value of the pixel point (i, j) of the variable z with slices, ▽ is the discrete gradient operator, r Ψ is the winding amount of the winding phase Ψ gradient, α is the regularization weight parameter, β is the penalty parameter, is the winding operator, the constant c is the upper bound of the envelope, and T is the variable ▽zr Ψ The support set of 3. The fast phase unwrapping method based on tail non-convex regularization according to claim 2, characterized in that: The winding phase Ψ is obtained by winding the phase of the two-dimensional image get: N represents the length and width of the two-dimensional image; The phase winding process is expressed as: Among them, (Φ) i,j is the pixel value of phase Φ at pixel (i, j), Represents the pixel value of the winding phase Ψ at the pixel point (i, j).

4. The fast phase unwrapping method based on tail non-convex regularization according to claim 3, characterized in that: The variable with slice And the phase Φ = Ψ-z; The winding amount Then we get the variable:

5. The fast phase unwrapping method based on tail non-convex regularization according to any one of claims 2 to 4, characterized in that: The method for solving the non-convex tail optimization model with box constraints is: Based on ||·|| 2,1 The saddle point problem of the non-convex tail optimization model with box constraints is: in, Ⅱ Z (z) is the index function on the closed set Z; Respectively by and The dual variable introduced by the dual form of S (p) and II Q (q) are the indicator functions of the closed set S and the closed set Q respectively; Introducing a quadratic term with parameter μ into the saddle point problem leads to the penalty model: The penalty model is transformed into an optimization saddle point problem: Among them, the intermediate function Intermediate variable G(z) = II Z (z); g * (q)=Ⅱ Q (q); f(·)=||·||1, g(·)=||·|| 2,1 , f* represents the convex conjugate of function f; Solving optimization saddle point problems using a fully split primal-dual hybrid gradient algorithm.

6. The fast phase unwrapping method based on tail non-convex regularization according to claim 5, characterized in that: The index function II Z (z) is: Closed Set Closed Set Closed Set Among them, z i,j ,(q x ) i,j ,(q y ) i,j 、(p x ) i,j 、(p y ) i,j denote the pixel value of the variable z and the dual variables p and q at (x, y) in the x or y direction respectively.

7. The fast phase unwrapping method based on tail non-convex regularization according to claim 6, characterized in that: The implementation method of the completely split primal-dual hybrid gradient algorithm is: Let (z k ,p k ,q k ) is the k-th iteration solution of variable z, dual variable q, and dual variable p. The k+1-th iteration process of the subproblem of variable z, dual variable q, and dual variable p is: in, Represents the extrapolation step of the acceleration algorithm, and the step sizes η, τ, and γ are all positive numbers.

8. The fast phase unwrapping method based on tail non-convex regularization according to claim 7, characterized in that: In the k+1th iteration, the closed-form solutions of the subproblems of variable z, dual variable q, and dual variable p are: Among them, Proj(.;.) represents the projection operator, It is expressed as: T k For the support set.

9. The fast phase unwrapping method based on tail non-convex regularization according to claim 8, characterized in that: The method for obtaining the phase unwrapped image in step 3 is: (1) Initial value selection: Select appropriate parameter regularization weight parameter α, parameter μ, step size η, step size γ, step size τ, penalty parameter β, and set the variable z 0 =q 0 =p 0 =0; number of iterations k = 1, support set (2) Iteration: Compute the closed-form solution z for a variable with slices k+1 ; Compute the closed-form solution q for the dual variable k+1 ; Compute the closed-form solution p of the dual variable k+1 ; Set the initial conditions for iteration according to the prior information. When the initial conditions for iteration are met: update the support set T k =E(▽z k+1 -r Ψ ); (3) Termination conditions are met: tol is a positive constant given by experience, and the iteration is terminated; Get the phase Ψ-z of the image after phase unwrapping k+1 .

10. The fast phase unwrapping method based on tail non-convex regularization according to claim 9, characterized in that: The support set estimate E(▽z k+1 -r Ψ ) is set as follows: 1) Calculate the residual: A = |▽z k -r Ψ |; where |·| means taking the element-by-element absolute value of the variable; 2) Arrange the elements in the residual A in descending order, find the positions of the first m elements with the largest absolute values, and set the set of their indices as the support set T k ; 3) The support set T k Substitute the closed-form solution z with the sliced ​​variable z k+1 ; Update m←m+m0, return to step 1); m0 represents the step length of the support set update.

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