A multi-objective constrained adaptive triangulated net discrete phase unwrapping method and system

Abstract

The present application relates to the technical field of phase field expansion, in particular to a multi-target constraint adaptive triangular net discrete phase expansion method and system; comprising the following steps: phase quality evaluation is carried out on discrete sampling points, the local signal-to-noise ratio, phase gradient amplitude, local phase consistency measure and boundary distance factor of each sampling point are calculated to comprehensively evaluate the phase quality, the local signal-to-noise ratio and phase gradient amplitude index of the discrete sampling point are calculated to quantify the phase reliability, which is used as the basis for the subsequent network construction to provide quality basis.The multi-target constraint adaptive triangular net discrete phase expansion method provided by the present application realizes high-precision and high-robustness expansion of discrete phase data through the organic combination of three core technologies, namely adaptive triangular net construction guided by phase quality, multi-target constraint MCF framework and intelligent path planning guided by quality.

Description

Technical Field

[0001] This invention relates to the field of phase field expansion technology, specifically to an adaptive triangular mesh discrete phase expansion method and system with multi-objective constraints. Background Technology

[0002] In recent years, deep learning-based methods have been used in phase unwrapping, primarily employing the natural grid layout of data over continuous regions. Without relying on physical data structures, traditional deep learning methods depend heavily on large amounts of high-quality data during the learning process. However, in practical applications, problems such as scarce data, data with significant noise, and complex stripes often arise. In such cases, relying solely on data-driven models makes it difficult to obtain accurate and reliable predictions.

[0003] Therefore, traditional path tracing methods generally rely heavily on the choice of unfolding path. Different paths can lead to different results, especially when noise causes the local phase gradient to exceed π, resulting in incorrect 2π transition judgments. Secondly, excessively large phase gradients and insufficient sampling violate the Nyquist sampling theorem, which can also lead to abnormal phase unfolding. All phase unfolding methods are extremely sensitive to noise, but the sensitivity varies. In path tracing methods, noise can cause incorrect 2π transition judgments, and the error accumulates and propagates along the path. In least squares methods, noise affects the accuracy of gradient calculation and spreads to the entire solution through global optimization. Phase discontinuities exist at object boundaries and occluded areas in discontinuous regions, in the phase field of multi-connected regions containing holes or separated regions, in regions with high slope and extremely large phase gradients, and in regions with rich texture and drastic changes in local details. The default assumption is that the solution is continuous and smooth, which can lead to "missed solutions" in discontinuous regions.

[0004] Based on this, the present invention provides an adaptive triangulation discrete phase expansion method and system with multi-objective constraints to solve the above-mentioned technical problems. Summary of the Invention

[0005] The purpose of this invention is to provide an adaptive triangular mesh discrete phase expansion method and system with multi-objective constraints, thereby solving the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] First aspect of the invention:

[0008] This invention proposes an adaptive triangulated network discrete phase expansion method with multi-objective constraints, comprising the following steps:

[0009] S1. Perform phase quality assessment on discrete sampling points, calculate the local signal-to-noise ratio, phase gradient magnitude, local phase consistency measure, and boundary distance factor for each sampling point to comprehensively evaluate its phase quality. By calculating the local signal-to-noise ratio and phase gradient magnitude of discrete sampling points, the phase reliability is quantified and used as a basic input to provide a quality basis for subsequent network construction.

[0010] S2. The mesh density is dynamically adjusted based on the quality assessment results, with high-quality areas being denser and low-quality areas being sparser, and the triangular mesh structure is optimized through quality-guided edge weights;

[0011] S3. Extend the MCF model based on the triangulation network, integrate phase gradient reliability, spatial continuity and boundary protection constraints, construct a multi-objective optimization function, and define the optimization objective using the triangulation network as the carrier;

[0012] S4. Select high-quality integration paths based on signal-to-noise ratio, gradient magnitude, and distance factor; determine the initial expansion starting point and optimize flow allocation; independent of MCF but providing it with an efficient initial solution.

[0013] S5. Solve the global optimal flow allocation by using a dynamic tree acceleration algorithm, combine the path planning results to achieve high-precision phase expansion, integrate the path planning output, and complete the core expansion calculation;

[0014] S6. Evaluate the effectiveness based on accuracy and noise resistance indicators, iterate backward to adjust the quality evaluation threshold or constraint weights, and optimize the parameters of the entire process through closed-loop feedback.

[0015] Preferably, the implementation steps of step S2 are as follows:

[0016] S21. Based on the quality assessment results of step S1, an adaptive triangular network construction method guided by phase quality is adopted to dynamically adjust the network structure according to the phase quality characteristics of discrete sampling points;

[0017] S22. Introduce a composite quality evaluation function and adopt a quality-adaptive importance sampling mechanism to make the sampling probability positively correlated with the quality score, ensuring that the high-quality areas evaluated in step S1 are sampled densely and the low-quality areas are appropriately sparsed.

[0018] S23. Use the boundary distance factor from step S1 to identify boundary points, identify and specially process the sampling points in the boundary region, and enhance the sampling probability of boundary points through the boundary detection operator to improve the processing accuracy of the boundary region;

[0019] S24. An improved constrained Delaunay triangulation algorithm is adopted, and quality-guided edge weights are introduced. The weight parameters are derived from the phase consistency metric in step S1, and finally an optimized triangular mesh structure is obtained.

[0020] Preferably, the implementation steps of step S3 are as follows:

[0021] S31. Based on the triangular network optimized in step S2, extend the MCF model, extend the traditional MCF minimum cost flow solution framework, introduce a multi-objective constraint optimization function, and integrate the edge weight definition cost function from step S24;

[0022] S32. The three types of constraints—phase gradient reliability constraint, spatial continuity constraint, and boundary protection constraint—are integrated and balanced to evaluate the gradient reliability between discrete points, reduce the impact of unreliable gradients, and promote phase consistency between adjacent triangles. At the same time, the structural edges and discontinuous regions are protected to avoid excessive smoothing. The gradient reliability constraint depends on the mesh density in step S21, the spatial continuity constraint depends on the edge weight in step S24, and the boundary protection constraint depends on the boundary point processing in step S23.

[0023] S33. Define the MCF cost function for multi-objective optimization, and adaptively configure the weight coefficients of the MCF cost function according to the local signal-to-noise ratio distribution in step S1.

[0024] Preferably, the implementation steps of step S4 are as follows:

[0025] S41. To further improve the unfolding accuracy, a quality-guided integral path selection algorithm is proposed, which directly uses the quality assessment results of step S1 to screen reliable paths, replacing the traditional path planning method;

[0026] S42. Construct a point-pair quality metric function to evaluate path quality based on factors such as signal-to-noise ratio, phase gradient magnitude, and Euclidean distance between points;

[0027] S43. Divide the path quality level into high-quality path, medium-quality path and low-quality path, and optimize the path according to the quality metric. Combine the signal-to-noise ratio, gradient magnitude and distance between points in step S1 to classify the path quality level.

[0028] S44. Select the most reliable point evaluated in step S1 as the integration starting point, integrate along the quality path to obtain the initial unfolded phase, and construct the initial flow distribution of the MCF.

[0029] Preferably, the implementation steps of step S5 are as follows:

[0030] S51. Combine the triangular network structure of step S2 to perform multi-scale decomposition. When the approximate solution of the scale layer is mapped to the fine scale layer, load the initial flow allocation of step S44, use the dynamic tree data structure to accelerate the network simplex algorithm, construct the hierarchical representation of the problem, quickly solve the approximate solution at the coarse scale layer, and then map the result to the fine scale layer as the initial solution for progressively refined optimization.

[0031] S52. Under the conditions of satisfying capacity constraints and node balance constraints, the global optimal solution is obtained by solving the multi-objective cost function in step S3, thereby achieving high-precision unfolding of discrete phase data.

[0032] Preferably, the implementation steps of step S6 are as follows:

[0033] S61. Verify the unfolding results, evaluate the unfolding accuracy, noise resistance and processing speed indicators, and the verification results are directly fed back to the quality assessment threshold adjustment in step S1 and the weight coefficient optimization in step S33.

[0034] S62. Based on the verification results, optimize and adjust the method to further improve the performance and stability of phase expansion.

[0035] On the other hand, the present invention also proposes an adaptive triangulation discrete phase expansion system for high-precision expansion processing of discrete sampled phase data. The system includes a computing device and a phase acquisition array module. The phase acquisition array module is deployed according to non-uniform spatial distribution rules to form a multi-level sampling network. The computing device is used to perform an adaptive triangulation discrete phase expansion method with multi-objective constraints.

[0036] Preferably, the computing device includes:

[0037] A phase quality assessment module, which is based on local signal-to-noise ratio, phase gradient magnitude, local phase consistency metric, and boundary distance factor, is used to generate a multidimensional quality score map of discrete sampling points;

[0038] An adaptive triangulation generation module, wherein the adaptive triangulation generation module regulates the sampling density by combining a quality-driven dynamic partitioning strategy with a quality evaluation function, performs enhanced sampling on the boundary region, and constructs a quality-guided weighted triangulation structure;

[0039] A multi-objective constraint optimization module is used to extend the minimum cost flow framework, integrate phase gradient reliability constraints, spatial continuity constraints, and boundary protection constraints, and define a multi-objective cost function with adaptive weight configuration.

[0040] A quality-guided path planning module is provided, which constructs a path quality metric function based on signal-to-noise ratio, gradient magnitude, and distance factor to screen the optimal integration path and determine the initial unfolding phase.

[0041] A hierarchical MCF solver, which uses a dynamic tree acceleration algorithm to achieve multi-scale optimization, is used to generate an approximate solution at the coarse-scale layer and then map it to the fine-scale layer for fine-tuning.

[0042] The verification and feedback optimization module evaluates the results based on expansion accuracy, noise suppression capability, and computational efficiency to form a closed-loop parameter optimization mechanism.

[0043] The computing device is used to generate a quality map by sequentially passing discrete phase data through the phase quality assessment module. The adaptive triangulation generation module dynamically adjusts the mesh topology according to the quality map. The multi-objective constraint optimization module constructs a constraint flow network based on the triangulation. The quality-guided path planning module generates an initial phase distribution. Finally, the hierarchical MCF solver completes the global optimal phase expansion. The verification and feedback optimization module continuously optimizes the parameter configuration of each module.

[0044] Compared with the prior art, the beneficial effects of the present invention are:

[0045] The proposed multi-objective constrained adaptive triangulation discrete phase unfolding method achieves high-precision and robust unfolding of discrete phase data by organically combining three core technologies: phase quality-guided adaptive triangulation construction, multi-objective constrained MCF framework, and quality-guided intelligent path planning. Compared with traditional methods, this invention offers higher phase unfolding accuracy, stronger noise resistance, and faster processing speed, demonstrating significant technical advantages and application prospects. It is particularly suitable for phase field unfolding with discrete sampling point distributions and can effectively handle complex scenarios such as noise, discontinuous regions, and high gradient regions, providing strong technical support for fields such as optical measurement and medical imaging. Attached Figure Description

[0046] Figure 1 This diagram shows an overview of the system architecture of the present invention.

[0047] Figure 2 A diagram illustrating the construction of the adaptive triangular network of the present invention is shown;

[0048] Figure 3 A schematic diagram of the multi-objective constraint function of the present invention is shown;

[0049] Figure 4 The simulation of the package according to the present invention is shown in the comparison diagram before and after unfolding;

[0050] Figure 5 The following diagram shows a comparison of the simulation results of discrete wrapping phase unfolding according to the present invention;

[0051] Figure 6 The diagram shows the error effect of the simulation of discrete wrapping phase expansion according to the present invention;

[0052] Figure 7 A 2D comparison diagram of the discrete packaged phase before and after the present invention is shown;

[0053] Figure 8 A 3D comparison diagram of the present invention before and after the actual discrete package phase unfolding is shown. Detailed Implementation

[0054] The technical solutions of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. 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 of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0055] Example 1, please refer to Figures 1 to 8 This invention proposes an adaptive triangulated network discrete phase expansion method with multi-objective constraints, comprising the following steps:

[0056] S1. Perform phase quality assessment on discrete sampling points, calculate the local signal-to-noise ratio, phase gradient magnitude, local phase consistency measure, and boundary distance factor for each sampling point to comprehensively evaluate its phase quality. By calculating the local signal-to-noise ratio and phase gradient magnitude of discrete sampling points, the phase reliability is quantified and used as a basic input to provide a quality basis for subsequent network construction.

[0057] S2. The mesh density is dynamically adjusted based on the quality assessment results, with high-quality areas being denser and low-quality areas being sparser, and the triangular mesh structure is optimized through quality-guided edge weights;

[0058] S3. Extend the MCF model based on the triangulation network, integrate phase gradient reliability, spatial continuity and boundary protection constraints, construct a multi-objective optimization function, and define the optimization objective using the triangulation network as the carrier;

[0059] S4. Select high-quality integration paths based on signal-to-noise ratio, gradient magnitude, and distance factor; determine the initial expansion starting point and optimize flow allocation; independent of MCF but providing it with an efficient initial solution.

[0060] S5. Solve the global optimal flow allocation by using a dynamic tree acceleration algorithm, combine the path planning results to achieve high-precision phase expansion, integrate the path planning output, and complete the core expansion calculation;

[0061] S6. Evaluate the effectiveness based on accuracy and noise resistance indicators, iterate backward to adjust the quality evaluation threshold or constraint weights, and optimize the parameters of the entire process through closed-loop feedback.

[0062] It should also be noted that the implementation steps of step S2 are as follows:

[0063] S21. Based on the quality assessment results of step S1, an adaptive triangular network construction method guided by phase quality is adopted to dynamically adjust the network structure according to the phase quality characteristics of discrete sampling points;

[0064] S22. Introduce a composite quality evaluation function and adopt a quality-adaptive importance sampling mechanism to make the sampling probability positively correlated with the quality score, ensuring that the high-quality areas evaluated in step S1 are sampled densely and the low-quality areas are appropriately sparsed.

[0065] S23. Use the boundary distance factor from step S1 to identify boundary points, identify and specially process the sampling points in the boundary region, and enhance the sampling probability of boundary points through the boundary detection operator to improve the processing accuracy of the boundary region;

[0066] S24. An improved constrained Delaunay triangulation algorithm is adopted, and quality-guided edge weights are introduced. The weight parameters are derived from the phase consistency metric in step S1, and finally an optimized triangular mesh structure is obtained.

[0067] It should also be noted that the implementation steps of step S3 are as follows:

[0068] S31. Based on the triangular network optimized in step S2, extend the MCF model, extend the traditional MCF minimum cost flow solution framework, introduce a multi-objective constraint optimization function, and integrate the edge weight definition cost function from step S24;

[0069] S32. The three types of constraints—phase gradient reliability constraint, spatial continuity constraint, and boundary protection constraint—are integrated and balanced to evaluate the gradient reliability between discrete points, reduce the impact of unreliable gradients, and promote phase consistency between adjacent triangles. At the same time, the structural edges and discontinuous regions are protected to avoid excessive smoothing. The gradient reliability constraint depends on the mesh density in step S21, the spatial continuity constraint depends on the edge weight in step S24, and the boundary protection constraint depends on the boundary point processing in step S23.

[0070] S33. Define the MCF cost function for multi-objective optimization, and adaptively configure the weight coefficients of the MCF cost function according to the local signal-to-noise ratio distribution in step S1.

[0071] It should also be noted that the implementation steps of step S4 are as follows:

[0072] S41. To further improve the unfolding accuracy, a quality-guided integral path selection algorithm is proposed, which directly uses the quality assessment results of step S1 to screen reliable paths, replacing the traditional path planning method;

[0073] S42. Construct a point-pair quality metric function to evaluate path quality based on factors such as signal-to-noise ratio, phase gradient magnitude, and Euclidean distance between points;

[0074] S43. Divide the path quality level into high-quality path, medium-quality path and low-quality path, and optimize the path according to the quality metric. Combine the signal-to-noise ratio, gradient magnitude and distance between points in step S1 to classify the path quality level.

[0075] S44. Select the most reliable point evaluated in step S1 as the integration starting point, integrate along the quality path to obtain the initial unfolded phase, and construct the initial flow distribution of the MCF.

[0076] It should also be noted that the implementation steps of step S5 are as follows:

[0077] S51. Combine the triangular network structure of step S2 to perform multi-scale decomposition. When the approximate solution of the scale layer is mapped to the fine scale layer, load the initial flow allocation of step S44, use the dynamic tree data structure to accelerate the network simplex algorithm, construct the hierarchical representation of the problem, quickly solve the approximate solution at the coarse scale layer, and then map the result to the fine scale layer as the initial solution for progressively refined optimization.

[0078] S52. Under the conditions of satisfying capacity constraints and node balance constraints, the global optimal solution is obtained by solving the multi-objective cost function in step S3, thereby achieving high-precision unfolding of discrete phase data.

[0079] It should also be noted that the implementation steps of step S6 are as follows:

[0080] S61. Verify the unfolding results, evaluate the unfolding accuracy, noise resistance and processing speed indicators, and the verification results are directly fed back to the quality assessment threshold adjustment in step S1 and the weight coefficient optimization in step S33.

[0081] S62. Based on the verification results, optimize and adjust the method to further improve the performance and stability of phase expansion.

[0082] This invention also proposes an adaptive triangulation discrete phase expansion system for high-precision expansion processing of discrete sampled phase data. The system includes a computing device and a phase acquisition array module. The phase acquisition array module is deployed according to non-uniform spatial distribution rules to form a multi-level sampling network. The computing device is used to execute a method for adaptive triangulation discrete phase expansion with multi-objective constraints.

[0083] It should also be noted that the computing device includes:

[0084] A phase quality assessment module, which is based on local signal-to-noise ratio, phase gradient magnitude, local phase consistency metric, and boundary distance factor, is used to generate a multidimensional quality score map of discrete sampling points;

[0085] An adaptive triangulation generation module, wherein the adaptive triangulation generation module regulates the sampling density by combining a quality-driven dynamic partitioning strategy with a quality evaluation function, performs enhanced sampling on the boundary region, and constructs a quality-guided weighted triangulation structure;

[0086] A multi-objective constraint optimization module is used to extend the minimum cost flow framework, integrate phase gradient reliability constraints, spatial continuity constraints, and boundary protection constraints, and define a multi-objective cost function with adaptive weight configuration.

[0087] A quality-guided path planning module is provided, which constructs a path quality metric function based on signal-to-noise ratio, gradient magnitude, and distance factor to screen the optimal integration path and determine the initial unfolding phase.

[0088] A hierarchical MCF solver, which uses a dynamic tree acceleration algorithm to achieve multi-scale optimization, is used to generate an approximate solution at the coarse-scale layer and then map it to the fine-scale layer for fine-tuning.

[0089] The verification and feedback optimization module evaluates the results based on expansion accuracy, noise suppression capability, and computational efficiency to form a closed-loop parameter optimization mechanism.

[0090] The computing device is used to generate a quality map by sequentially passing discrete phase data through the phase quality assessment module. The adaptive triangulation generation module dynamically adjusts the mesh topology according to the quality map. The multi-objective constraint optimization module constructs a constraint flow network based on the triangulation. The quality-guided path planning module generates an initial phase distribution. Finally, the hierarchical MCF solver completes the global optimal phase expansion. The verification and feedback optimization module continuously optimizes the parameter configuration of each module.

[0091] Example 2, please refer to Figures 1 to 8 In practical applications, the adaptive triangular mesh discrete phase expansion method based on the multi-objective constraints of the above system specifically includes the following steps:

[0092] (1) Dynamically adjust the network structure based on the phase quality characteristics of discrete sampling points;

[0093] Specifically, this includes the following technical means: introducing a composite quality evaluation function Q(x,y), and comprehensively considering the following factors:

[0094] Q(x,y)=w1·SNR(x,y)+w2·G -1 (x,y)+w3·C(x,y)+w4·D(x,y)

[0095] Where SNR(x,y): local signal-to-noise ratio evaluation, G(x,y): phase gradient magnitude. C(x,y): Local phase consistency measure. D(x,y) is the boundary distance factor, and w1, w2, w3, w4 are adaptive weight coefficients.

[0096] Based on the aforementioned quality evaluation function, this method employs a quality-adaptive importance sampling mechanism, ensuring a positive correlation between sampling probability and quality score:

[0097] P(x,y)=Q(x,y) / ∑∑Q(i,j)

[0098] Select point sets using Monte Carlo sampling or importance sampling methods to ensure sufficient sampling density in high-quality areas and appropriate sparsification in low-quality areas;

[0099] This sampling strategy enables the triangular mesh structure to adaptively reflect the complex distribution of the phase field;

[0100] To further improve the processing accuracy of boundary regions, this method identifies and specially processes sampling points in boundary regions, enhancing the sampling probability of boundary points through boundary detection operators:

[0101]

[0102] Where E(x,y) is the edge detection function, based on local variance or the Canny operator;

[0103] The sampling probability in the boundary region is additionally enhanced:

[0104] P'(x,y)=P(x,y)+α·B(x,y)

[0105] Where α is the boundary enhancement coefficient, with a value range of [0.5, 2.0];

[0106] In the process of constructing the adaptive triangulation network, an improved constrained Delaunay triangulation (CDT) algorithm is adopted, and quality-guided edge weights are introduced:

[0107]

[0108] Where β is the phase difference sensitivity parameter;

[0109] Finally, the objective function for triangular mesh optimization can be obtained:

[0110] F(T)=∑ e w(e)L(e)+λ∑tA(t)[1–Q(t)]

[0111] Where T is the triangulation network, e is the edge in the triangulation network, t is the triangular element, L(e) is the edge length, A(t) is the area of ​​the triangle, Q(t) is the mass measure of the triangle, and λ is the balance coefficient.

[0112] (2) MCF solution framework with multi-objective constraints:

[0113] Based on the construction of adaptive triangulation networks, this method innovatively extends the traditional MCF solution framework by introducing a multi-objective constraint optimization function to comprehensively balance three types of constraints: phase gradient reliability constraint, spatial continuity constraint, and boundary protection constraint.

[0114] To evaluate the reliability of gradients between discrete points and reduce the impact of unreliable gradients, the first type of constraint term is defined as:

[0115]

[0116] in The actual phase gradient between points i and j. The phase gradient is estimated for local smoothing, σ 2 This is for noise variance estimation, where α is the sensitivity adjustment parameter;

[0117] This constraint effectively identifies anomalous gradients, reduces their weight during the unfolding process, and improves the robustness of the algorithm.

[0118] To promote phase consistency between adjacent triangles, a second type of constraint is added:

[0119] C2(t1,t2)=exp(-β·d(t1,t2) 2 )·(1-|cos(θ 12 )|)

[0120] Where t1 and t2 are adjacent triangles, d(t1,t2) is the distance between the centers of the triangles, and θ 12 β is the angle between the normal vectors of the triangle, and β is the distance sensitivity parameter.

[0121] This constraint can effectively ensure the smoothness and continuity of the phase field, especially in regions where phase changes are relatively gentle.

[0122] To protect structural edges and discontinuous regions and avoid excessive smoothing, this method adds a third type of constraint term C3(i,j):

[0123] C3(i,j)=γ[1-exp(-λ·E(i,j) 2 )]

[0124] Where e(i,j) is the edge intensity function;

[0125] γ and λ are adjustment parameters;

[0126] This constraint provides additional protection at the boundaries, preventing excessive smoothing from causing a loss of detail;

[0127] Based on the above three types of constraints, the MCF cost function for multi-objective optimization is defined as follows:

[0128] Cost(i,j)=w1·C1(i,j)+w2·C2(t(i),t(j))+w3C3(i,j)+ε;

[0129] Where w1, w2, w3 are adaptive weight coefficients, t(i), t(j) are triangles containing points i and j, and ε is a small positive number to ensure that the cost function is non-negative;

[0130] The weighting coefficients employ a dynamic adjustment mechanism, adaptively configured based on the characteristics of the local phase field.

[0131] w1=k1exp(-SNR -1 )

[0132]

[0133] w3=k3E avg

[0134] Where SNR is the local signal-to-noise ratio. E represents the local phase variance. avg The average edge intensity is given by k1, k2, and k3, which are positive constants.

[0135] This method effectively maps the relationships between discrete sampling points to a network flow model: each triangle represents a network node, the shared edges between adjacent triangles constitute network edges, and residual points are identified by the following formula:

[0136]

[0137] The capacity limit is set based on edge quality:

[0138]

[0139] Cost assignment uses the multi-objective constrained cost function described above;

[0140] In the solution process, this method uses a dynamic tree data structure to accelerate the network simplex algorithm, constructs a hierarchical representation of the problem, quickly solves the approximate solution at the coarse-scale layer, and then maps the result to the fine-scale layer as the initial solution for progressive optimization.

[0141] The entire solution process satisfies Kirchhoff's law and network flow balance constraints:

[0142] ∑ kflow(k,j)-∑ i flow(j,i)=supply(j)

[0143] The final objective function is:

[0144] min∑ e cost(e)flow(e)

[0145] Solve for the globally optimal flow allocation under the conditions of satisfying capacity constraints and node balance constraints;

[0146] (3) Quality-guided path planning:

[0147] To further improve the unfolding accuracy, this method innovatively proposes a quality-guided integral path selection algorithm to replace the traditional BFS path;

[0148] First, construct the point-to-point quality metric function:

[0149]

[0150] Where: SNR(i) and SNR(j) are the signal-to-noise ratios at points i and j, respectively. Let d(i,j) be the phase gradient magnitude, d(i,j) be the Euclidean distance between points, and α and β be the adjustment parameters.

[0151] Path quality levels are classified as: High-Quality Path (HQ) (Q path >T1); Medium Quality Path (MQ)(T2) path ≤T1); Low-quality path (LQ)(Q path ≤T1) where T1 and T2 are phase quality thresholds, which are dynamically adjusted according to the application scenario;

[0152] This method improves upon the traditional Dijkstra algorithm by introducing a quality metric for path optimization:

[0153] d(s,v)=min{{u∈pred(v)}{d(s,u)+[1-Q path(u,v) ]}}

[0154] Where d(s,v) is the shortest path cost from the starting point s to the point v, and pred(v) is the set of predecessor nodes of v;

[0155] To ensure the reliability of the integration path's starting point, this method selects the most reliable point in the phase field as the integration starting point:

[0156]

[0157] Where V is the set of all points, and N(v) is the set of neighborhood points of v;

[0158] ​The initial solution generation method for MCF based on the mass path is to obtain the initial expansion phase φ0 by integrating along the mass path;

[0159] Calculate the initial integer field:

[0160]

[0161] Constructing the initial flow allocation for MCF:

[0162] flow(e)=k0(head(e))-k0(tail(e))

[0163] Where k0(head(e)) and k0(tail(e)) represent the initial integer field values ​​of the head and tail nodes of edge e, respectively, and flow(e) is the initial flow allocated to edge e. Based on the initial flow allocation results, the algorithm performs a global balance check on the flow in the triangular network to ensure that the inflow and outflow of each node satisfy the network's conservation constraints. For nodes that do not meet the flow balance requirements, a flow adjustment strategy is adopted to achieve overall network flow balance by modifying the flow of some edges.

[0164] This high-quality initial solution significantly improves the convergence speed and solution quality of the MCF algorithm, and greatly enhances the expansion accuracy and computational efficiency.

[0165] The multi-objective constrained adaptive triangulation discrete phase expansion method proposed in this invention achieves high-precision and high-robust expansion of discrete phase data by organically combining three core technologies: phase quality-guided adaptive triangulation construction, multi-objective constrained MCF framework, and quality-guided intelligent path planning.

[0166] Compared with traditional methods, this method improves phase unwrapping accuracy by 45-60%, noise immunity by 65%, and processing speed by 3-5 times, demonstrating significant technical advantages and application prospects.

[0167] This method is particularly suitable for phase field expansion with discrete sampling point distributions. It can effectively handle complex scenarios such as noise, discontinuous regions, and high gradient regions, providing strong technical support for fields such as optical measurement and medical imaging.

[0168] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0169] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A multi-objective constrained adaptive triangular mesh discrete phase expansion method, characterized in that, Includes the following steps: S1. Perform phase quality assessment on discrete sampling points, calculate the local signal-to-noise ratio, phase gradient magnitude, local phase consistency measure, and boundary distance factor for each sampling point to comprehensively evaluate its phase quality. By calculating the local signal-to-noise ratio and phase gradient magnitude of discrete sampling points, the phase reliability is quantified and used as a basic input to provide a quality basis for subsequent network construction. S2. Dynamically adjust the mesh density based on the quality assessment results, densifying high-quality areas and sparsening low-quality areas, and optimize the triangular mesh structure through quality-guided edge weights: The implementation steps of step S2 are as follows: S21. Based on the quality assessment results of step S1, an adaptive triangular network construction method guided by phase quality is adopted to dynamically adjust the network structure according to the phase quality characteristics of discrete sampling points; S22. Introduce a composite quality evaluation function and adopt a quality-adaptive importance sampling mechanism to make the sampling probability positively correlated with the quality score, ensuring that the high-quality areas evaluated in step S1 are sampled densely and the low-quality areas are appropriately sparsed. S23. Use the boundary distance factor from step S1 to identify boundary points, identify and specially process the sampling points in the boundary region, and enhance the sampling probability of boundary points through the boundary detection operator to improve the processing accuracy of the boundary region; S24. An improved constrained Delaunay triangulation algorithm is adopted, and quality-guided edge weights are introduced. The weight parameters are derived from the phase consistency metric in step S1, and finally an optimized triangular mesh structure is obtained. S3. Extend the MCF model based on the triangulation network, integrate phase gradient reliability, spatial continuity and boundary protection constraints, construct a multi-objective optimization function, and define the optimization objective using the triangulation network as the carrier; S4. Select high-quality integration paths based on signal-to-noise ratio, gradient magnitude, and distance factor; determine the initial expansion starting point and optimize flow allocation; independent of MCF but providing it with an efficient initial solution. S5. Solve the global optimal flow allocation by using a dynamic tree acceleration algorithm, combine the path planning results to achieve high-precision phase expansion, integrate the path planning output, and complete the core expansion calculation; S6. Evaluate the effectiveness based on accuracy and noise resistance indicators, iterate backward to adjust the quality evaluation threshold or constraint weights, and optimize the parameters of the entire process through closed-loop feedback.

2. The adaptive triangular mesh discrete phase expansion method with multi-objective constraints according to claim 1, characterized in that, The implementation steps of step S3 are as follows: S31. Based on the triangular network optimized in step S2, extend the MCF model, extend the traditional MCF minimum cost flow solution framework, introduce a multi-objective constraint optimization function, and integrate the edge weight definition cost function from step S24; S32. The three types of constraints—phase gradient reliability constraint, spatial continuity constraint, and boundary protection constraint—are integrated and balanced to evaluate the gradient reliability between discrete points, reduce the impact of unreliable gradients, and promote phase consistency between adjacent triangles. At the same time, the structural edges and discontinuous regions are protected to avoid excessive smoothing. The gradient reliability constraint depends on the mesh density in step S21, the spatial continuity constraint depends on the edge weight in step S24, and the boundary protection constraint depends on the boundary point processing in step S23. S33. Define the MCF cost function for multi-objective optimization, and adaptively configure the weight coefficients of the MCF cost function according to the local signal-to-noise ratio distribution in step S1.

3. The adaptive triangular mesh discrete phase expansion method with multi-objective constraints according to claim 2, characterized in that, The implementation steps of step S4 are as follows: S41. To further improve the unfolding accuracy, a quality-guided integral path selection algorithm is proposed, which directly uses the quality assessment results of step S1 to screen reliable paths, replacing the traditional path planning method; S42. Construct a point-pair quality metric function to evaluate path quality based on factors such as signal-to-noise ratio, phase gradient magnitude, and Euclidean distance between points; S43. Divide the path quality level into high-quality path, medium-quality path and low-quality path, and optimize the path according to the quality metric. Combine the signal-to-noise ratio, gradient magnitude and distance between points in step S1 to classify the path quality level. S44. Select the most reliable point evaluated in step S1 as the integration starting point, integrate along the quality path to obtain the initial unfolded phase, and construct the initial flow distribution of the MCF.

4. The adaptive triangular mesh discrete phase expansion method with multi-objective constraints according to claim 3, characterized in that, The implementation steps of step S5 are as follows: S51. Combine the triangular network structure of step S2 to perform multi-scale decomposition. When the approximate solution of the scale layer is mapped to the fine scale layer, load the initial flow allocation of step S44, use the dynamic tree data structure to accelerate the network simplex algorithm, construct the hierarchical representation of the problem, quickly solve the approximate solution at the coarse scale layer, and then map the result to the fine scale layer as the initial solution for progressively refined optimization. S52. Under the conditions of satisfying capacity constraints and node balance constraints, the global optimal solution is obtained by solving the multi-objective cost function in step S3, thereby achieving high-precision unfolding of discrete phase data.

5. The adaptive triangular mesh discrete phase expansion method with multi-objective constraints according to claim 4, characterized in that, The implementation steps of step S6 are as follows: S61. Verify the unfolding results, evaluate the unfolding accuracy, noise resistance and processing speed indicators, and the verification results are directly fed back to the quality assessment threshold adjustment in step S1 and the weight coefficient optimization in step S33. S62. Based on the verification results, optimize and adjust the method to further improve the performance and stability of phase expansion.

6. An adaptive triangulated network discrete phase expansion system for high-precision expansion processing of discrete sampled phase data, characterized in that, The system includes a computing device and a phase acquisition array module. The phase acquisition array module is deployed according to a non-uniform spatial distribution rule to form a multi-level sampling network. The computing device is used to execute the method of multi-objective constrained adaptive triangular network discrete phase expansion as described in any one of claims 1-5.

7. The system of claim 6, wherein the computing device comprises: A phase quality assessment module, which is based on local signal-to-noise ratio, phase gradient magnitude, local phase consistency metric, and boundary distance factor, is used to generate a multidimensional quality score map of discrete sampling points; An adaptive triangulation generation module, wherein the adaptive triangulation generation module regulates the sampling density by combining a quality-driven dynamic partitioning strategy with a quality evaluation function, performs enhanced sampling on the boundary region, and constructs a quality-guided weighted triangulation structure; A multi-objective constraint optimization module is used to extend the minimum cost flow framework, integrate phase gradient reliability constraints, spatial continuity constraints, and boundary protection constraints, and define a multi-objective cost function with adaptive weight configuration. A quality-guided path planning module is provided, which constructs a path quality metric function based on signal-to-noise ratio, gradient magnitude, and distance factor to screen the optimal integration path and determine the initial unfolding phase. A hierarchical MCF solver, which uses a dynamic tree acceleration algorithm to achieve multi-scale optimization, is used to generate an approximate solution at the coarse-scale layer and then map it to the fine-scale layer for fine-tuning. The verification and feedback optimization module evaluates the results based on expansion accuracy, noise suppression capability, and computational efficiency indicators to form a closed-loop parameter optimization mechanism. The computing device is used to generate a quality map by sequentially passing discrete phase data through the phase quality assessment module. The adaptive triangulation generation module dynamically adjusts the mesh topology according to the quality map. The multi-objective constraint optimization module constructs a constraint flow network based on the triangulation. The quality-guided path planning module generates an initial phase distribution. Finally, the hierarchical MCF solver completes the global optimal phase expansion. The verification and feedback optimization module continuously optimizes the parameter configuration of each module.

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