Three-dimensional feature spline fitting and optimization method and system based on manifold surface

Abstract

The present application relates to a kind of three-dimensional feature spline fitting and optimization method and system based on manifold curved surface.It is converted into periodic regression problem by the spherical azimuth parameterization of discrete candidate point set, the periodic closed characteristics of fourier series are used to naturally ensure curve closure and fairing, and the return oscillation in cartesian space is avoided.Ridge line intensity field is constructed and dynamic oversampling driven by dihedral angle is introduced, with the decoupling strategy of sparse control point optimization and dense evaluation point constraint, in the feasible direction cone of pseudo-gradient, curvature adsorption, vertex smoothing and logarithmic obstacle three joint objective functions are carried out based on simulated annealing meta-heuristic search, accurate adsorption of curve to real ridge line is realized, and self-intersection and out-of-bound are prevented.Local curvature distribution is offset parameter predicted by neural network, offset profile is generated by mapping geodesic index to fit surface, and the defect that fixed normal offset is separated from surface in undercut area is overcome.

Description

Technical Field

[0001] This invention belongs to the field of three-dimensional geometry processing and computer-aided design technology, specifically relating to a method and system for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces. Background Technology

[0002] In digital dental restoration and CAD / CAM systems, the marginal fit of a restoration depends on the accuracy of identifying and fitting the cervical margin of the prepared prosthesis. Positional deviations directly lead to clinical failures such as steps, overhangs, and poor fit. With the widespread use of intraoral scanners, high-precision three-dimensional manifold triangular meshes have become the standard carrier for characterizing the geometry of prepared prostheses. How to automatically fit a closed, smooth, and non-self-intersecting feature spline on this mesh, and generate an offset profile for the fitting surface accordingly, has become a key step in automated restoration design.

[0003] However, existing technologies suffer from systemic flaws in addressing these requirements. Manual point mapping relies on technician experience and suffers from poor repeatability, easily exhibiting jitter, folding, and self-intersection in areas with dramatic curvature changes or high scanning noise. Curvature threshold edge detection methods are highly sensitive to threshold settings and noise; thresholds that are too high lead to broken lines, while thresholds that are too low produce numerous false edges, making it difficult to simultaneously guarantee the closure and smoothness of the results within a unified framework. Cartesian space direct spline fitting methods fail when there is projection reflection at the neckline, as the same projection coordinate corresponds to multiple curve segments, leading to fitting failure or oscillations, and require additional closure constraints for complex implementation. Fixed-step offset methods along the normal have a single direction; when there are significant differences in slope on both sides or indentations, the offset profile may detach from the surface and penetrate the material. Even if a curvature adsorption optimization framework based on the objective function is used to overcome the above-mentioned closure and smoothness problems, in the metaheuristic search process, the existing scheme implements a point-by-point independent uniform random sampling strategy for each control point. The displacement decisions of each control point are isolated from each other and have no directional coordination. It completely ignores the displacement response coupling relationship between adjacent control points caused by the spatial continuity of the feature ridge along the surface direction. This results in frequent directional conflicts between adjacent points, a large number of invalid rejections of candidate solutions, low search efficiency, and oscillations in the convergence path.

[0004] Therefore, a three-dimensional feature spline fitting and optimization scheme based on manifold surfaces is desired. Summary of the Invention

[0005] This invention application provides a method and system for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces. According to one aspect of this application, a method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces is provided, comprising: S1, based on the manifold surface triangular mesh containing the vertex coordinate set and the triangular patch topology set, the spherical azimuth parameterization is performed on the surface feature candidate point set output by the neural network through vertex-by-vertex reasoning on the manifold surface triangular mesh to obtain the spherical azimuth sample set. S2, perform order-regularized periodic Fourier series fitting and surface back projection on the spherical azimuth angle sample set to obtain the initial closed curve; S3, the curvature of the triangular mesh of the manifold surface is estimated and smoothed to obtain the ridge intensity field, and the initial closed curve is sparsely sampled to obtain a sparse control point set. Dynamic oversampling is performed based on the cumulative dihedral angle of the curve segment between adjacent control points to obtain a dynamic oversampling evaluation point set. S4, based on the gradient direction of the ridge intensity field and the cosine similarity threshold, constructs a feasible direction cone for each point in the sparse control point set, and optimizes the feature spline by using the curvature adsorption term, vertex smoothing term and illegal region logarithmic obstacle term accumulated on the point set as the objective function in the Cartesian product space of the feasible direction cone. S5, based on the local curvature distribution characteristics of each point in the optimized feature spline, uses a pre-trained neural network to predict the offset direction and amplitude probabilistically, and then obtains the surface offset profile through surface geodesic index mapping and closed smoothing processing.

[0006] According to another aspect of this application, a three-dimensional feature spline fitting and optimization system based on manifold surfaces is provided, comprising: The spherical parameterization module is used to perform spherical azimuth parameterization on the set of candidate points of surface features output by the neural network inferring vertex by vertex on the triangular mesh of the manifold surface, which contains the vertex coordinate set and the triangular patch topology set, to obtain the spherical azimuth sample set. The Fourier fitting module is used to perform order-regularized periodic Fourier series fitting and surface back projection on the spherical azimuth sample set to obtain the initial closed curve. The dynamic oversampling module is used to estimate the curvature and smooth the triangular mesh of the manifold surface to obtain the ridge intensity field, and to sparsely sample the initial closed curve to obtain a sparse control point set. It also performs dynamic oversampling based on the cumulative dihedral angle of the curve segment between adjacent control points to obtain a dynamic oversampling evaluation point set. The curvature adsorption optimization module is used to construct feasible direction cones for each point in the sparse control point set based on the gradient direction of the ridge intensity field and the cosine similarity threshold. In the Cartesian product space of the feasible direction cone, the curvature adsorption term, vertex smoothing term and illegal region logarithmic obstacle term accumulated on the dynamic oversampling evaluation point set are used as the objective function to obtain the optimized feature spline. The surface offset module is used to predict the offset direction and magnitude probabilistically through a pre-trained neural network based on the local curvature distribution characteristics of each point in the optimized feature spline, and then obtain the surface offset profile through surface geodesic index mapping and closed smoothing processing.

[0007] Compared with existing technologies, this invention proposes a three-dimensional feature spline fitting and optimization method based on manifold surfaces. First, the discrete candidate point set is transformed into a periodic regression problem with the azimuth angle as the sole independent variable by spherical azimuth parameterization. The periodic closure property of the truncated Fourier series naturally ensures the closure and smoothness of the fitted curve, fundamentally avoiding the difficulties of Cartesian space oscillation and additional closure constraints. Second, a ridge intensity field is constructed, and a dihedral-driven dynamic oversampling mechanism is introduced. A decoupling strategy, with sparse control points participating in optimization and dense evaluation points participating in constraint assessment, is used. Under the constraint of a pseudo-gradient feasible direction cone, a metaheuristic search based on simulated annealing is performed through a joint objective function of curvature adsorption, vertex smoothing, and logarithmic barrier, achieving accurate adsorption of the curve to the real ridge, while preventing self-intersection and boundary crossing. Finally, the offset direction and amplitude are predicted by a neural network using the local curvature distribution as input. An offset profile conforming to the surface is generated by traveling along the surface through geodesic index mapping, overcoming the defect of fixed normal offset detaching from the surface in the concave region. Attached Figure Description

[0008] Figure 1 This is a flowchart of a three-dimensional feature spline fitting and optimization method based on manifold surfaces according to an embodiment of this application; Figure 2 This is a schematic diagram of data flow for a three-dimensional feature spline fitting and optimization method based on manifold surfaces according to an embodiment of this application; Figure 3 This is a flowchart illustrating the process of using a manifold surface triangular mesh, which includes a set of vertex coordinates and a set of triangular patch topologies, to perform spherical azimuth parameterization on the set of candidate points for surface features output by a neural network through vertex-by-vertex reasoning on the manifold surface triangular mesh, in accordance with the embodiments of this application, to obtain a spherical azimuth sample set. Figure 4 This is a flowchart illustrating the process of performing order-regularized periodic Fourier series fitting and surface back-projection on a spherical azimuth angle sample set to obtain an initial closed curve, according to the manifold surface-based three-dimensional feature spline fitting and optimization method of this application. Figure 5The flowchart describes a method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to an embodiment of this application. It describes the process of estimating and smoothing the curvature of triangular meshes on manifold surfaces to obtain the ridge intensity field, sparsely sampling the initial closed curve to obtain a sparse control point set, and performing dynamic oversampling based on the cumulative dihedral angle of the curve segment between adjacent control points to obtain a dynamic oversampling evaluation point set. Figure 6 The flowchart describes a method for fitting and optimizing 3D feature splines based on manifold surfaces according to embodiments of this application. It involves constructing feasible direction cones for each point in a sparse control point set based on the gradient direction of the ridge intensity field and the cosine similarity threshold. The flowchart also describes how to optimize feature splines by using the curvature adsorption term, vertex smoothing term, and illegal region logarithmic obstacle term accumulated on the dynamic oversampling evaluation point set as the objective function. Figure 7 This is a schematic diagram of the pseudo-gradient feasible direction cone and optimization evaluation decoupling mechanism of the three-dimensional feature spline fitting and optimization method based on manifold surface according to an embodiment of this application; Figure 8 The flowchart of the three-dimensional feature spline fitting and optimization method based on manifold surface according to the embodiments of this application is as follows: the Cartesian product of the set of feasible directions of pseudo-gradient is used as the search space, and the candidate displacements of each control point in the sparse control point set are sampled in the corresponding cone and iteratively optimized until convergence according to the acceptance criterion to obtain the optimized control point set. Figure 9 This is a comparison diagram of the curvature adsorption optimization effect before and after the three-dimensional feature spline fitting and optimization method based on manifold surface according to the embodiments of this application; Figure 10 This is a block diagram of a three-dimensional feature spline fitting and optimization system based on manifold surfaces according to an embodiment of this application. Detailed Implementation

[0009] 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. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0010] Figure 1 This is a flowchart of a three-dimensional feature spline fitting and optimization method based on manifold surfaces according to an embodiment of this application. Figure 2 This is a schematic diagram of the data flow of the three-dimensional feature spline fitting and optimization method based on manifold surfaces according to an embodiment of this application. Figure 1 and Figure 2As shown, the method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to an embodiment of this application includes the following steps: S1, based on a manifold surface triangular mesh containing a vertex coordinate set and a triangular patch topology set, spherical azimuth parameterization is performed on the surface feature candidate point set output by the neural network through vertex-by-vertex reasoning on the manifold surface triangular mesh to obtain a spherical azimuth sample set; S2, the spherical azimuth sample set is fitted with an order-regularized periodic Fourier series and back-projected onto the surface to obtain an initial closed curve; S3, the curvature of the manifold surface triangular mesh is estimated and smoothed to obtain a ridge intensity field, and a sparse control point set is obtained by sparse sampling of the initial closed curve, and based on the adjacent control points... S4. Dynamic oversampling is performed on the cumulative dihedral angles of the curve segments between control points to obtain a dynamic oversampling evaluation point set; S5. Based on the gradient direction of the ridge intensity field and the cosine similarity threshold, a feasible direction cone is constructed for each point in the sparse control point set, and optimization is performed in the Cartesian product space of the feasible direction cone using the cumulative curvature adsorption term, vertex smoothing term, and illegal region logarithmic obstacle term on the dynamic oversampling evaluation point set as the objective function to obtain the optimized feature spline; S6. Based on the local curvature distribution characteristics of each point in the optimized feature spline, the probabilistic prediction of the offset direction and amplitude is performed through a pre-trained neural network, and the surface offset profile is obtained through surface geodesic index mapping and closure smoothing.

[0011] Specifically, in step S1, based on the manifold surface triangular mesh containing the vertex coordinate set and the triangular facet topology set, the spherical azimuth parameterization is performed on the surface feature candidate point set output by the neural network through vertex-by-vertex inference on the manifold surface triangular mesh to obtain a spherical azimuth sample set. It should be noted that, given that the candidate point set output by the neural network through vertex-by-vertex inference on the manifold surface triangular mesh is only a discrete positional prior, it contains noise, uneven density, and local missing values. Furthermore, when the neckline has a reflection in Cartesian coordinates, multiple curve segments correspond to the same projected coordinates, and directly fitting with the coordinate axes as independent variables will result in oscillation failure. Based on this, the technical solution of this application first calculates the area-weighted sphere center and principal axis direction of the surface feature candidate point set based on the manifold surface triangular mesh containing the vertex coordinate set and the triangular facet topology set to construct a local orthogonal frame, converting each candidate point into a spherical azimuth sample set with the azimuth around the principal axis as the only independent variable. Through the above processing, even if the neckline folds back in Cartesian space, the radial distance and elevation angle remain single-valued periodic functions with respect to the azimuth angle, thus effectively avoiding the folding oscillation problem in Cartesian space fitting and providing a reasonable parameterization basis for the subsequent closed-loop fitting of the periodic Fourier series.

[0012] Figure 3This is a flowchart illustrating the process of using a manifold surface-based 3D feature spline fitting and optimization method according to an embodiment of this application. The method involves a manifold surface triangular mesh containing vertex coordinates and a triangular patch topology set. The flowchart describes the process of parameterizing the surface feature candidate point set output by a neural network through vertex-by-vertex reasoning on the manifold surface triangular mesh to obtain a spherical azimuth sample set. (See attached flowchart.) Figure 3 As shown, step S1 includes: S11, based on the local area weights of each candidate point determined by the topological set of triangular facets in the triangular mesh of the manifold surface, the spatial coordinates of each point in the candidate point set of surface features are weighted and averaged to obtain the reference sphere center; S12, based on the reference sphere center, the normalized direction vector of each point in the candidate point set of surface features is calculated and the area-weighted covariance matrix is ​​constructed, and the covariance matrix is ​​decomposed into eigenvalues ​​and the eigenvector corresponding to the minimum eigenvalue is extracted as the principal axis direction to construct a local orthogonal frame; S13, based on the reference sphere center and the local orthogonal frame, the spherical coordinate projection transformation is performed on each point in the candidate point set of surface features, and the azimuth angle of each point around the principal axis and the corresponding radial distance and elevation angle are calculated to obtain the spherical azimuth angle sample set.

[0013] In step S11, based on the local area weights of each candidate point determined by the topological set of triangular facets in the manifold surface triangular mesh, the spatial coordinates of each point in the candidate point set of surface features are aggregated by a weighted average to obtain the reference sphere center. It should be noted that since the distribution density of each point in the candidate point set of surface features on the manifold surface is affected by the mesh density rather than being uniform, directly taking the arithmetic mean as the reference center will result in a bias in the candidate points of the mesh-refined region relative to the center position. Therefore, the technical solution of this application first aggregates the spatial coordinates of each point in the candidate point set of surface features based on the local area weights of each candidate point determined by the topological set of triangular facets in the manifold surface triangular mesh to obtain the reference sphere center. Through the above processing, the bias effect of uneven mesh density on the center position can be effectively eliminated, ensuring that the reference sphere center accurately reflects the geometric centroid of the candidate points on the surface, providing a stable coordinate origin for subsequent spherical azimuth parameterization.

[0014] More specifically, in a concrete example of this application, for each candidate point in the set of candidate points for surface features, the triangular facet to which the candidate point belongs is located based on the triangular facet topology set, and the spatial coordinates of the three vertices of the triangular facet in the vertex coordinate set are read. The area of ​​the triangular facet is calculated by the cross product of the two edge vectors, and the obtained area value is used as the local area weight of the candidate point. After obtaining the area weights corresponding to all candidate points, the reference sphere center is calculated by summing the products of the three-dimensional spatial coordinates of each candidate point and its area weight, and then dividing by the sum of all area weights. The calculation expression is as follows: in, For reference center of the ball, For the first candidate point in the surface feature set The three-dimensional spatial coordinate vectors of the candidate points The area weight of the triangle to which the candidate point belongs. The total number of candidate points. This weighting method ensures that in areas with locally refined meshes, the densely distributed candidate points will not excessively shift the center of the sphere due to their numerical advantage, so that the resulting reference center of the sphere is geometrically equivalent to the uniformly sampled center of the candidate points on the surface.

[0015] In step S12, a normalized direction vector is calculated for each point in the candidate point set of surface features based on the reference sphere center, and an area-weighted covariance matrix is ​​constructed. The covariance matrix is ​​then decomposed into eigenvalues, and the eigenvector corresponding to the smallest eigenvalue is extracted as the principal axis direction to construct a local orthogonal frame. It should be noted that since the candidate point set of surface features is distributed around the preparatory body in an approximately annular spatial region, if the normal direction corresponding to this annular distribution is uncertain, a reasonable spherical coordinate system cannot be established for subsequent azimuth projection. Therefore, the technical solution of this application further calculates a normalized direction vector for each point in the candidate point set of surface features based on the reference sphere center, constructs an area-weighted covariance matrix, and decomposes the covariance matrix into eigenvalues, extracting the eigenvector corresponding to the smallest eigenvalue as the principal axis direction to construct a local orthogonal frame. Through the above processing, the normal direction of the plane containing the annular distribution of candidate points can be automatically determined, thereby providing a locally coordinate basis with clear direction and complete orthogonality for subsequent azimuth calculation.

[0016] More specifically, in a concrete example of this application, the direction vector of each candidate point in the set of candidate points for surface features is first calculated relative to the center of the reference sphere and then normalized, i.e., the following is calculated: in, For the first Spatial coordinates of candidate points The reference center of the sphere obtained from the aforementioned steps, Let be the Euclidean distance between the two. After obtaining all normalized direction vectors, calculate the area-weighted directional mean. And construct the area-weighted covariance matrix using the deviations of each direction vector from the mean: in, For the first The area weight of the triangular facet to which each candidate point belongs. This represents the transpose of a vector. The covariance matrix is ​​a third-order real symmetric positive semi-definite matrix. Eigenvalue decomposition yields three eigenvalues ​​and their corresponding orthogonal eigenvectors. Since the candidate points are distributed around the neckline, their direction vectors are much more dispersed within the torus than in the torus normal. Therefore, the eigenvector corresponding to the smallest eigenvalue is the torus normal direction, which is extracted as the principal axis direction. Subsequently, the eigenvector corresponding to the largest eigenvalue is taken as the first frame axis. and through cross product operation The second frame axis is obtained, thereby constructing a locally orthogonal frame. The spindle in this frame The normal direction pointing towards the torus of the neckline, and and The plane in which Zhang Cheng's candidate points are mainly distributed provides a geometrically reasonable coordinate reference for subsequent parameterization using the azimuth angle around the principal axis as the independent variable.

[0017] In step S13, based on the reference sphere center and the local orthogonal frame, a spherical coordinate projection transformation is performed on each point in the candidate point set of surface features, and the azimuth angle and corresponding radial distance and elevation angle of each point around the principal axis are calculated to obtain a spherical azimuth angle sample set. It should be noted that, due to the distribution of the neckline around the preparatory body, there is a reflection phenomenon in some projection directions under the Cartesian coordinate system, that is, the same projected coordinate value corresponds to multiple segments of the curve, causing the fitting relationship with the coordinate axes as independent variables to no longer satisfy the single-valuedness condition. Based on this, the technical solution of this application further performs a spherical coordinate projection transformation on each point in the candidate point set of surface features based on the reference sphere center and the local orthogonal frame, and calculates the azimuth angle and corresponding radial distance and elevation angle of each point around the principal axis to obtain a spherical azimuth angle sample set. Through the above processing, the radial distance and elevation angle are both single-valued periodic functions with respect to the azimuth angle, thus transforming the three-dimensional curve fitting problem into a periodic regression problem with the azimuth angle as the only independent variable.

[0018] More specifically, in a concrete example of this application, for each candidate point in the set of candidate points for surface features, its direction vector relative to the reference sphere center is first calculated. Then, this direction vector is projected onto the three axes of the local orthogonal frame using an inner product to obtain the coordinate components on each axis. The azimuth angle about the principal axis is calculated using the four-quadrant arctangent function, the radial distance is taken as the Euclidean distance between the candidate point and the reference sphere center, and the elevation angle is determined by the projection components of the direction vector on the principal axis. The specific calculation expression is as follows: in, For the first Candidate points around the principal axis azimuth angle, From the candidate point to the reference center radial distance, This is the projection of the candidate point's direction vector onto the principal axis, i.e., the elevation component. For normalized direction vectors, and Let be two basis vectors that span a circular distribution plane in a locally orthogonal frame. This involves vector dot product operations. After performing the above calculations for all candidate points, the azimuth, radial distance, and elevation angle of each point are combined into a sequence of triplets, thus forming the spherical azimuth sample set. Under this parameterized representation, the azimuth angle... In the interval The upper part monotonically increases to mark the angular positions of candidate points along the ring distribution, while and As for Even if the neckline reflects in Cartesian space, the mapping relationship between the two with respect to the azimuth angle remains single-valued and is expressed as... The period is set, which lays the groundwork for subsequent continuous fitting using periodic Fourier series.

[0019] It should be noted that the aforementioned candidate point set for surface features is generated by a vertex-by-vertex classification network based on graph convolution on a manifold surface triangular mesh. This network uses the vertex coordinates, unit normal vectors, and discrete curvature features of the triangular mesh as input feature vectors for each vertex. Through multi-layer graph convolution operations, it aggregates local geometric information layer by layer in the mesh topological neighborhood, ultimately outputting the probability value of each vertex belonging to the feature boundary region. For vertices with output probability values ​​greater than a preset classification threshold, their corresponding 3D spatial coordinates in the vertex coordinate set are extracted to form the candidate point set for surface features. The network is trained using supervised learning, with the neckline vertex labels annotated by professional technicians in historical annotated cases as ground truth. Parameter optimization is performed by minimizing the vertex-by-vertex binary classification cross-entropy loss function. After training, the network parameters are fixed and used for vertex-by-vertex probability prediction during the inference phase.

[0020] Specifically, in step S2, the spherical azimuth angle sample set is subjected to order-regularized periodic Fourier series fitting and surface back-projection to obtain an initial closed curve. It should be noted that, given that the radial distance and elevation angle in the spherical azimuth angle sample set are both periodic functions of the azimuth angle, and that the candidate points themselves contain noise and uneven density, a periodic continuous fitting method that can both naturally guarantee closure and suppress high-frequency noise is needed. Based on this, the technical solution of this application further performs order-regularized periodic Fourier series fitting and surface back-projection on the spherical azimuth angle sample set to obtain an initial closed curve. Through the above processing, the obtained curve, based on the Fourier basis... Constantly satisfied for the periodicity For any order It holds true; it is naturally closed without the need for additional closure constraints, and its derivatives are continuous. Furthermore, the order is truncated through order regularization. It directly controls the trade-off between smoothness and detail preservation, effectively suppressing the interference of candidate point noise on the fitted curve.

[0021] Figure 4 This is a flowchart illustrating the process of performing order-regularized periodic Fourier series fitting and surface back-projection on a spherical azimuth angle sample set to obtain an initial closed curve, according to the manifold surface-based three-dimensional feature spline fitting and optimization method of this application. Figure 4 As shown, step S2 includes: S21, constructing a Fourier basis matrix composed of periodic trigonometric function bases with each azimuth component in the spherical azimuth sample set as the independent variable according to the truncation order, and constructing an order regularization matrix; S22, based on the Fourier basis matrix and the order regularization matrix, performing regularized weighted least squares solutions on the radial distance component and elevation component in the spherical azimuth sample set to obtain the radial Fourier coefficient vector and the elevation Fourier coefficient vector; S23, based on the radial Fourier coefficient vector and the elevation Fourier coefficient vector, and combined with the reference sphere center and the local orthogonal frame, performing an inverse spherical coordinate transformation and projecting it onto the manifold surface triangular mesh through the nearest point to obtain the initial closed curve.

[0022] In step S21, a Fourier basis matrix composed of periodic trigonometric function bases is constructed using each azimuth component in the spherical azimuth sample set as the independent variable and according to the truncation order. A regularization matrix is ​​then constructed. It should be noted that since the radial distance and elevation angle in the spherical azimuth sample set are both based on the truncation order... Since the function is a periodic continuous function, a matching periodic basis function system needs to be selected to construct a linear regression model. Simultaneously, high-frequency noise in the candidate points needs to be suppressed through regularization. Based on this, the technical solution of this application further constructs a Fourier basis matrix composed of periodic trigonometric function bases using each azimuth component in the spherical azimuth sample set as the independent variable, according to the truncation order, and constructs an order-regularization matrix. Through the above processing, the periodic regression problem can be transformed into a standard linear algebraic solution form, and the order-regularization matrix can be used to... Weighting higher-order components is equivalent to imposing constraints on the second-order derivative energy to suppress high-frequency oscillations.

[0023] More specifically, in a particular example of this application, the cutoff order is set. As a hyperparameter for controlling the smoothness, the radial distance Expand to truncate Periodic Fourier series of order: in, The DC component coefficient, and The first The coefficients of the cosine and sine components, This is the truncation order. Elevation angle. Expand in the same form. For the spherical azimuth sample set... Each azimuth angle value Construct basis vectors for each sample Assemble the basis vectors of all samples by arranging them row by row into a form of size [size missing]. Fourier basis matrix Simultaneously construct a regularization matrix of order. : Among them, the diagonal number The penalty weights corresponding to the cosine and sine components are both Setting the DC component to zero indicates that no penalty is imposed on it. This regularization matrix is ​​designed so that higher-order Fourier components experience stronger energy suppression in subsequent solutions. Its physical meaning is equivalent to constraining the second derivative energy of the fitted curve, thus truncating the order... Within the permissible frequency band, the overall morphological characteristics of low frequencies are preferentially preserved, while high-frequency noise fluctuation components are attenuated.

[0024] In step S22, based on the Fourier basis matrix and the order regularization matrix, regularized weighted least squares solutions are performed on the radial distance and elevation components of the spherical azimuth angle sample set to obtain the radial Fourier coefficient vector and the elevation Fourier coefficient vector, respectively. It should be noted that since the Fourier basis matrix and the order regularization matrix have transformed the periodic fitting problem into a standard linear algebraic form, it is necessary to solve for the Fourier coefficients of the two dependent variables, radial distance and elevation angle, respectively, while compensating for the uneven density of candidate points through sample weights. Based on this, the technical solution of this application further performs regularized weighted least squares solutions on the radial distance and elevation components of the spherical azimuth angle sample set based on the Fourier basis matrix and the order regularization matrix to obtain the radial Fourier coefficient vector and the elevation Fourier coefficient vector. Through the above processing, noise suppression and periodic fitting can be completed simultaneously in a single closed-form operation, obtaining complete frequency domain coefficients representing the shape of the closed curve.

[0025] More specifically, in a particular example of this application, a sample weight diagonal matrix is ​​first constructed based on the area weights of the triangular facets to which each candidate point belongs. This is to compensate for the impact of uneven grid density on fitting accuracy. Taking the radial distance component as an example, let the radial distance observation vector be... The coefficient vector to be solved Construct a weighted least squares objective function with regularization to minimize The first term represents the fitting residual under sample weights, and the second term represents the order regularization penalty. Taking the derivative of the objective function and setting it to zero yields the closed-form solution of the normal equation system: in, The Fourier basis matrix constructed in the preceding steps, The sample weight diagonal matrix, For order regularization matrix, This is the hyperparameter for regularization strength. This is the radial distance observation vector. The closed-form solution yields... This is the radial Fourier coefficient vector. The observation vector is replaced with the elevation component in exactly the same way. The solution is then performed to obtain the Fourier coefficient vector of the elevation angle. Regularization intensity. The value of controls the balance between how closely the fitted curve approximates the observed data and how smoothly it smooths the data. When the values ​​are large, higher-order components are strongly suppressed, causing the curve to tend to be smoother. When the value is smaller, the curve is closer to the original candidate point, but may retain noise details.

[0026] In step S23, based on the radial Fourier coefficient vector and the elevation Fourier coefficient vector, and combined with the reference sphere center and the local orthogonal frame, an inverse spherical coordinate transformation is performed, and the result is projected onto the manifold surface triangular mesh through the nearest point to obtain the initial closed curve. It should be noted that since the radial and elevation Fourier coefficient vectors only describe the frequency domain representation of the curve in the spherical coordinate system, they still need to be restored to three-dimensional Cartesian space coordinates and constrained to the manifold surface to obtain a closed curve usable for subsequent optimization. Based on this, the technical solution of this application further uses the radial and elevation Fourier coefficient vectors, combined with the reference sphere center and the local orthogonal frame, to perform an inverse spherical coordinate transformation and project it onto the manifold surface triangular mesh through the nearest point to obtain the initial closed curve. Through the above processing, the frequency domain fitting result can be transformed into an ordered spatial point sequence strictly located on the manifold surface, and because the Fourier basis... The periodicity naturally ensures that the beginning and end coincide, and the resulting curve is a closed curve without the need for additional closure constraints.

[0027] More specifically, in a particular example of this application, firstly in the interval [0, M uniform azimuth angle sampling values ​​are generated at equal intervals within the range. Substitute each sampled value into a periodic Fourier series with radial and elevation Fourier coefficient vectors as parameters to calculate the corresponding fitted radial distance. With fitted elevation angle Subsequently, in conjunction with the reference center of the ball... Local orthogonal frame Perform the inverse transformation from spherical coordinates to Cartesian coordinates to restore each azimuth angle sampling point to three-dimensional spatial coordinates: in, For the first Three-dimensional spatial reconstruction points corresponding to uniformly sampled azimuth angles For reference center of the ball, This is the fitted radial distance at that azimuth angle. This is the fitted elevation angle at that azimuth. and Let be two basis vectors that span a circular distribution plane in a locally orthogonal frame. The principal axis direction vector is obtained from the inverse transformation described above. There are spatial points distributed around the center of the reference sphere, but these points lie on the outer envelope of a smooth surface described by a Fourier series, and do not strictly lie on the surface of the triangular mesh of the manifold surface. Therefore, for each spatially reconstructed point... Perform the nearest point projection operation, which involves searching for distances across all triangular faces of the manifold surface triangular mesh. The nearest surface position yields the projection point: in, For manifold surface triangular mesh The nearest point projection operator on, The first one located on the surface after projection M sampling points. Arrange all M projection points in the order of azimuth angle generation to form the initial closed curve. Where M is much larger than the number of subsequent sparse control points m. This is because uniform azimuth sampling begins at... And terminate at approach The position of the Fourier series, and the position of the Fourier series. and The function value is constant at the given point, and the beginning and end points of the resulting curve naturally coincide in space, thus satisfying geometric closure without introducing any additional constraint equations. The initial closed curve is smooth and self-intersecting overall, but its local position is not yet precisely aligned with the actual neck ridge line, requiring further refinement through subsequent curvature adsorption optimization.

[0028] Specifically, in step S3, the curvature of the manifold surface triangular mesh is estimated and smoothed to obtain the ridge intensity field, and the initial closed curve is sparsely sampled to obtain a sparse control point set. Furthermore, dynamic oversampling is performed based on the cumulative dihedral angle of the curve segments between adjacent control points to obtain a dynamic oversampling evaluation point set. It should be noted that, although the initial closed curve is globally closed and smoothed, it does not precisely fall on the actual neck ridge line locally. Subsequent optimization requires a scalar field representing the ridge line position as an adsorption target. Simultaneously, a balance needs to be struck between computational efficiency and evaluation accuracy; that is, the optimization degrees of freedom should be as low as possible to accelerate iteration, while the curvature constraint evaluation needs to cover the entire curve to prevent segments between control points from deviating from the ridge line. Based on this, the technical solution of this application further estimates and smooths the curvature of the manifold surface triangular mesh to obtain the ridge intensity field, sparsely samples the initial closed curve to obtain a sparse control point set, and performs dynamic oversampling based on the cumulative dihedral angle of the curve segments between adjacent control points to obtain a dynamic oversampling evaluation point set. Through the above processing, the ridge intensity field provides a gradient guide for curvature adsorption for the subsequent objective function. The decoupling design of the sparse control point set and the dynamic oversampling evaluation point set allows only... Each control point participates in the optimization process. Each evaluation point participates in the constraint evaluation, thus balancing computational efficiency and the rationality of curvature throughout the process. Furthermore, dynamic oversampling adaptively densifies the high-curvature ridge section while maintaining the basic density in the straight section based on the dihedral angle, achieving high geometric fidelity at a computational cost close to that of a low sampling rate.

[0029] Figure 5 This document describes a flowchart illustrating the process of a manifold surface-based 3D feature spline fitting and optimization method according to an embodiment of this application. The method involves curvature estimation and smoothing of a triangular mesh to obtain a ridge intensity field, sparse sampling of an initial closed curve to obtain a sparse control point set, and dynamic oversampling based on the cumulative dihedral angle rotation of curve segments between adjacent control points to obtain a dynamic oversampling evaluation point set. Figure 5 As shown, step S3 includes: S31, constructing a ridge intensity field covering the entire mesh based on the vertex coordinate set and triangular patch topology set of the manifold surface triangular mesh; S32, performing equal arc length sparse sampling on the initial closed curve to obtain a sparse control point set; S33, traversing the shared edges crossed by the surface path between each adjacent control point in the sparse control point set on the manifold surface triangular mesh, accumulating the product of the dihedral angle and the transverse weight of each shared edge to obtain a cumulative dihedral angle rotation sequence; S34, based on the cumulative dihedral angle rotation sequence, using the sparse control point set as nodes, performing non-uniform densification through closed cubic spline interpolation and projecting it onto the manifold surface triangular mesh to obtain a dynamic oversampling evaluation point set.

[0030] In steps S31 and S32, a ridge intensity field covering the entire mesh is constructed based on the vertex coordinate set and triangular patch topology set of the manifold surface triangular mesh. Subsequently, sparse sampling of the initial closed curve with equal arc length is performed to obtain a sparse control point set. It should be noted that since subsequent curvature adsorption optimization requires a scalar field covering the entire mesh to indicate the ridge position, it is also necessary to reduce the dimensionality of the sampling points of the initial closed curve to a small number of control points to reduce the optimization degree of freedom. Based on this, the technical solution of this application further constructs a ridge intensity field covering the entire mesh based on the vertex coordinate set and triangular patch topology set of the manifold surface triangular mesh, and then performs sparse sampling of the initial closed curve with equal arc length to obtain a sparse control point set. Through the above processing, point-by-point queryable ridge adsorption intensity information can be provided for the objective function, while reducing the optimization degree of freedom from the initial closed curve... Each sampling point is compressed to at least a few sparse control points, so that the computational complexity of subsequent iterations is proportional to the number of control points rather than the total number of points on the curve.

[0031] More specifically, in a concrete example of this application, the construction process of the ridge intensity field is as follows: First, traverse each vertex of the vertex coordinate set in the triangular mesh of the manifold surface, determine the first-order ring neighborhood patch set of the vertex based on the triangular patch topology set, and estimate the discrete principal curvature at the vertex using the normal and area information of the neighborhood patches. and Subsequently, using the function value of the principal curvature as the initial estimate of the ridge intensity, the ridge intensity field is defined as follows: in, On curved surfaces The ridge intensity value at the point, For a scalar mapping function based on principal curvature, the absolute value of the maximum principal curvature can be taken. The curvature values ​​are expressed in forms such as shape index or second-order metric of normal variation, taking larger values ​​at the actual neck ridge. Due to the unavoidable measurement noise in the triangular mesh scan data, the directly calculated curvature values ​​may exhibit spurious peaks locally. Therefore, anisotropic diffusion smoothing is applied to the initial ridge intensity field to suppress isolated high-frequency peaks caused by noise while preserving the continuous high-value bands at the actual ridge, ultimately yielding a ridge intensity field covering the entire mesh. The gradient of this field along the surface... The direction pointing to the ridge line intensity increases the fastest provides a directional basis for the subsequent construction of the pseudo-gradient feasible direction cone.

[0032] After the ridge intensity field is constructed, sparse sampling is performed on the initial closed curve to obtain a sparse control point set. Specifically, the number of control point samples is set to... One sample is extracted from the ordered sequence of sampling points contained in the initial closed curve at equal arc length intervals, and selected from these. These points are used as a sparse control point set. ,in Equivalent to This reflects a closed topology. Subsequent optimization phases will only consider this... The positions of the control points are iteratively adjusted, while the overall shape of the curve is restored through cubic spline interpolation between the control points. This maintains the continuity of the curve while controlling the degrees of freedom of the optimization problem. Within 3D space.

[0033] In step S33, the shared edges traversed by the surface paths between adjacent control points in the sparse control point set are traversed on the manifold surface triangular mesh. The product of the dihedral angle and the tangential weight of each shared edge is accumulated to obtain a cumulative dihedral angle sequence. It should be noted that since subsequent dynamic oversampling requires a quantitative index to measure the degree of surface bending in each control point segment to determine the density allocation of each segment, and the dihedral angle is a discrete and faithful measure of the degree of surface bending at the edge, it can be calculated cheaply directly from the mesh. Based on this, the technical solution of this application further traverses the shared edges traversed by the surface paths between adjacent control points in the sparse control point set on the manifold surface triangular mesh, and accumulates the product of the dihedral angle and the tangential weight of each shared edge to obtain a cumulative dihedral angle sequence. Through the above processing, a scalar value reflecting the total bending amount of the surface in each control point segment can be obtained, so that the subsequent sampling density allocation has an adaptive basis that adapts to the local surface geometric complexity.

[0034] More specifically, in a particular example of this application, for every two adjacent control points in the sparse control point set and For the curved sections between them, trace the sequence of triangular facets traversed by the section along the triangular mesh of the manifold surface, and record the shared edges that the surface path successively crosses. For each shared edge Its two adjacent triangular facets and Each has its own unit normal direction and The dihedral angle at the edge is defined as: in, It is the inner product of the normal vectors of two adjacent faces. From discrete differential geometry, we know that the edge... The integral mean curvature it carries is The dihedral angle is a direct discrete measure of the degree of curvature of a surface at that edge, with the ridge line located precisely in the region where the dihedral angle is relatively large. When calculating the dihedral angle of each edge, the influence of how the curve crosses the edge on the angle contribution must be considered. Let the curve's direction of travel be parallel to the edge... The angle between the extension directions is Take the absolute value of its sine. As the transverse weight, the curve takes full weight when it orthogonally crosses the edge, and zero weight when it passes parallel to the edge. The product of the dihedral angle of each edge and its corresponding transverse weight is accumulated along the path to obtain the cumulative dihedral angle of that segment: in, For the first The cumulative dihedral angle value of each control point segment. This represents the total number of shared edges traversed by this segment. For the first A dihedral angle sharing a common edge. The direction of travel of the curve and the first The included angle of the edges of the strip. For the sparse control points, all... The above calculations are performed on each adjacent segment one by one, ultimately yielding the cumulative dihedral angle sequence. The elements with larger values ​​in this sequence correspond to high-curvature ridge segments with sharp surface bends, while the elements with smaller values ​​correspond to relatively flat segments of the surface. This provides a segment-by-segment geometric complexity measure for subsequent allocation of oversampling density based on dihedral angles.

[0035] In step S34, based on the cumulative dihedral angle sequence, the sparse control point set is used as nodes to perform non-uniform densification through closed cubic spline interpolation and projected onto the manifold surface triangular mesh to obtain a dynamic oversampled evaluation point set. It should be noted that if evaluation points are uniformly distributed among the control point segments, there is redundant computation in straight segments, while high-curvature ridge segments suffer from underresolution, leading to distorted discrete estimation of the objective function. Therefore, differentiated sampling density allocation is required based on the geometric complexity of each segment. Based on this, the technical solution of this application further uses the cumulative dihedral angle sequence, with the sparse control point set as nodes, to perform non-uniform densification through closed cubic spline interpolation and project onto the manifold surface triangular mesh to obtain a dynamic oversampled evaluation point set. Through the above processing, sampling resources can be concentrated on the high-curvature segments where the objective function changes most drastically within a fixed total evaluation point budget, achieving high geometric fidelity and evaluation accuracy at a computational cost close to that of a low sampling rate.

[0036] More specifically, in a particular example of this application, it is first based on the cumulative dihedral angle sequence. Assign oversampling evaluation points to each section. Calculate the mean of the cumulative dihedral angles across all sections. Using this as a normalization benchmark, the number of evaluation points is allocated to each segment according to the square root power law rule: in, For the first The number of oversampling evaluation points allocated to each segment. The number of basic sampling points guaranteed for each segment. These are the oversampling intensity control parameters. For the first The cumulative dihedral angle of each section, The index is the average of the cumulative dihedral angles across all sections. Based on the conclusion in piecewise linear interpolation approximation theory that the optimal node density is proportional to the half-power of the curvature, the geometric approximation error of each segment tends to be balanced. The effect of this allocation rule is that the straight segments... Approaching zero to obtain only the basic density And the high curvature ridge section It is much larger than the mean, thus being over-divided locally.

[0037] After determining the number of evaluation points for each section, a sparse control point set is used. Construct closed cubic spline interpolation curves for the type value points, and for the th... Each section is at the control point and Between Non-uniform encryption resampling is performed at equal parameter intervals to obtain encrypted spatial points within the segment. The nearest point projection operator is then applied to each of these encrypted spatial points. The points are projected onto the surface of the manifold triangular mesh and finally converge into a dynamic oversampling evaluation point set, with a total number of points. Much larger than the number of sparse control points This set of evaluation points serves as the basis for evaluating constraints throughout the subsequent calculation of the objective function, ensuring that not only the control points themselves but also the curve segments between them satisfy curvature adsorption and boundary protection constraints.

[0038] Specifically, in step S4, based on the gradient direction of the ridge intensity field and the cosine similarity threshold, feasible direction cones are constructed for each point in the sparse control point set. The optimized feature spline is then obtained by using the cumulative curvature adsorption term, vertex smoothing term, and illegal region logarithmic barrier term accumulated on the evaluation point set as the objective function within the Cartesian product space of the feasible direction cones through dynamic oversampling. It should be noted that, given that the initial closed curve has a decoupled framework for optimization and evaluation after sparse sampling and dynamic oversampling, but the control points are not yet precisely aligned with the true ridge position, it is necessary to retain the overall tendency to approach the ridge while allowing degrees of freedom for the metaheuristic search to coordinate with the smoothing term and boundary protection. Furthermore, it is necessary to ensure, within a unified framework, that the curve fits the ridge, is smooth throughout, and does not generate illegal boundaries such as self-intersections or crossing inverted concavities. Based on this, the technical solution of this application further constructs feasible direction cones for each point in the sparse control point set based on the gradient direction of the ridge intensity field and the cosine similarity threshold. Then, within the Cartesian product space of the feasible direction cones, the objective function is jointly optimized using the curvature adsorption term, vertex smoothing term, and illegal region logarithmic obstacle term accumulated on the dynamic oversampling evaluation point set to obtain the optimized feature spline. Through the above processing, the pseudo-gradient feasible direction cone defines the exploration range with the maximum curvature direction as the axis rather than uniquely determining the movement direction. This avoids the risk of being skewed by noise peaks or pseudo-ridges by directly tracking the instantaneous maximum curvature. The three joint objective functions work together within a unified optimization framework to achieve ridge adsorption, smoothing constraints, and boundary protection, ensuring that the final output optimized feature spline is accurately embedded in the high-curvature ridge throughout and is closed, smooth, and without self-intersection.

[0039] Figure 6 This document describes a flowchart illustrating the optimization process for 3D feature spline fitting and optimization based on manifold surfaces, according to an embodiment of this application. The method utilizes the gradient direction of the ridge intensity field and the cosine similarity threshold. It constructs feasible direction cones for each point in a sparse control point set and optimizes the feature spline within the Cartesian product space of these feasible direction cones using a combination of curvature adsorption terms, vertex smoothing terms, and illegal region logarithmic obstacle terms accumulated on the evaluation point set as the objective function. Figure 6As shown, step S4 includes: S41, calculating the local gradient direction of the ridge intensity field along the surface for each control point in the sparse control point set and normalizing it to obtain the dominant direction of maximum curvature, and spanning a pseudo-gradient feasible direction cone set in the tangent plane with the cosine similarity threshold and the adsorption radius; S42, accumulating the negative ridge field intensity on the dynamic oversampling evaluation point set to define the curvature adsorption energy term, accumulating the second-order difference norm of adjacent points on the sparse control point set to define the vertex smoothing energy term, and applying the directed ridge field intensity to the dynamic oversampling evaluation point set. The negative logarithm of the distance defines the logarithmic obstacle energy term for illegal regions. The three terms are combined in a weighted manner to construct a joint objective function. In S43, the Cartesian product of the feasible direction cones of the pseudo-gradient is used as the search space. Candidate displacements are sampled for each control point in the sparse control point set within the corresponding cone, and the optimization is iteratively optimized until convergence according to the acceptance criterion to obtain the optimized control point set. In S44, a closed cubic spline interpolation is constructed using the optimized control point set as nodes, and uniform resampling is performed. The optimized feature spline is obtained by projecting the nearest point onto the triangular mesh of the manifold surface.

[0040] In step S41, the local gradient direction of the ridge intensity field along the surface is calculated for each control point in the sparse control point set and normalized to obtain the dominant direction of maximum curvature. A set of pseudo-gradient feasible directions is then spanned in the tangent plane using a cosine similarity threshold and the adsorption radius. Since this gradient direction originates from the ridge intensity field rather than the three-term joint objective function itself, it is called a pseudo-gradient to distinguish it from the true gradient of the objective function. It should be noted that directly moving the control points along the instantaneous maximum gradient direction of the ridge intensity field is easily swayed by noise peaks or pseudo-ridges and is not conducive to overall coordination with the smoothing term and illegal region term. Therefore, it is necessary to define a range of permissible directions for exploration based on the gradient direction, rather than a uniquely determined direction of movement. Based on this, the technical solution of this application further calculates the local gradient direction of the ridge intensity field along the surface for each control point in the sparse control point set and normalizes it to obtain the dominant direction of maximum curvature. A set of pseudo-gradient feasible directions is then spanned in the tangent plane using a cosine similarity threshold and the adsorption radius. Through the above processing, while preserving the overall tendency to move towards the ridge line, it is possible to leave room for metaheuristic search to freely explore within the cone, so that the search direction can take into account the synergistic needs of smooth constraints and boundary protection.

[0041] More specifically, in a specific example of this application, for each control point in the sparse control point set First, calculate the ridge intensity field. The local gradient along the manifold surface at this point. This gradient belongs to the control point. tangent plane at the location Its direction points towards the direction in which the intensity field of the ridge line increases most rapidly on the curved surface. Normalizing the obtained gradient vector yields the dominant direction of maximum curvature at this control point: in, Control points The unit vector in the direction of maximum curvature at that point. Ridge intensity field exist The gradient vector along the manifold surface at that point. Let be the vector norm 2. This direction indicates the most favorable approach direction towards the ridge line from the current control point position. After determining the dominant direction of maximum curvature, ... Using the axis as the reference, combined with the cosine similarity threshold... With adsorption radius Construct a pseudo-gradient feasible direction cone within the tangent plane: in, Control points The pseudo-gradient feasible direction cone, Let be the candidate displacement vector within the tangent plane. The cosine similarity threshold is used to control the size of the cone's half-angle. The adsorption radius is used to limit the maximum displacement amplitude in a single step. This is a vector dot product operation. The half-angle of the cone is... ,when When the value is large, the cone angle narrows, causing the search to focus near the gradient direction. When the value is small, the cone angle widens, giving the search greater directional freedom. This applies to all sparse control point sets. Perform the above calculations one by one at each control point to obtain the pseudo-gradient feasible direction cone set. Each cone in this cone set has its axis along the dominant direction of the maximum curvature of the corresponding control point, preserving the overall directional tendency toward the ridge. At the same time, the cone angle range defined by the cosine threshold provides a directional exploration space for the metaheuristic algorithm, allowing the search process to take into account the joint optimization needs of smooth terms and illegal region terms within this range, rather than being locked by a single gradient direction.

[0042] Figure 7 This diagram illustrates the decoupling mechanism between the pseudo-gradient feasible direction cone and the optimization evaluation in the manifold surface-based three-dimensional feature spline fitting and optimization method according to an embodiment of this application. Figure 7 As shown, the left side illustrates the construction of a feasible direction cone, at the control point. Within the tangent plane, in the direction of maximum curvature Using the axis as the cosine similarity threshold Zhang Cheng's half-angle is Radius of action is Directional cone The metaheuristic explores only within this cone, preserving the overall tendency to converge towards the ridge while avoiding being swayed by noise; the right side illustrates the decoupling of optimization and evaluation, with the curve controlled by sparse control points. It represents and participates in the optimization process, while the evaluation is performed at the interpolation encryption point. The above is carried out, among which Much larger This forces the curve segment between the two control points to also satisfy the curvature constraint.

[0043] In step S42, a curvature adsorption energy term is defined by accumulating the negative ridge field intensity on the dynamic oversampling evaluation point set; a vertex smoothing energy term is defined by accumulating the second-order difference norm of adjacent points on the sparse control point set; and an illegal region logarithmic barrier energy term is defined by taking the negative logarithm of the directed distance on the dynamic oversampling evaluation point set. These three terms are weighted and combined to construct a joint objective function. It should be noted that since curvature adsorption optimization needs to simultaneously satisfy three constraints—fitting the ridge, maintaining smoothness, and preventing boundary violations—a single energy term cannot address these competing requirements. Therefore, multiple constraints need to be unified into a quantifiable and comparable objective function framework. Based on this, the technical solution of this application further defines a curvature adsorption energy term by accumulating the negative ridge field intensity on the dynamic oversampling evaluation point set, a vertex smoothing energy term by accumulating the second-order difference norm of adjacent points on the sparse control point set, and an illegal region logarithmic barrier energy term by taking the negative logarithm of the directed distance on the dynamic oversampling evaluation point set. These three terms are weighted and combined to construct a joint objective function. Through the above processing, ridge adsorption, directional smoothing, and boundary protection can be achieved collaboratively within a unified scalar optimization framework, enabling the metaheuristic search process to have clear global evaluation criteria.

[0044] More specifically, in a particular example of this application, the three joint objective functions are composed of a weighted combination of the following three energy terms: in, For a sparse control point set Let the three terms be the joint objective function of the variables. , , These are the weighting coefficients for the three energy terms. The first term is the curvature adsorption energy term, which is applied across all points in the dynamic oversampling evaluation point set. The cumulative calculation across all evaluation points encourages the curve to perfectly follow the high-curvature ridge line throughout its entire length. in, For the dynamic oversampling evaluation point set, the first One evaluation point, The value of the ridge intensity field at this evaluation point is determined. For a monotonically decreasing transformation function, we can take... or The form of this term ensures that higher ridge strength corresponds to lower energy values, thus guiding the curve towards higher curvature ridge positions. The second term is the vertex smoothing energy term, which, in the sparse control point set... Accumulated calculations are performed at each control point to suppress abrupt changes in direction between adjacent control points, preventing jagged edges, jitter, and foldback. in, , , The spatial coordinates of three adjacent control points in a sparse control point set. The first term is the norm square of the second-order difference vector of three adjacent points. This value is zero when the three points are collinear, and the value increases with the degree of deviation from the straight line, thus penalizing local bending of the curve. The third term is the logarithmic barrier energy term for illegal regions, which takes the negative logarithm of the directed distance from each evaluation point to the boundary of the illegal region on the dynamic oversampling evaluation point set. in, As evaluation points The directed distance to the boundary of the illegal region is negative inside the illegal region and positive inside the legal region. (Logarithmic barrier function) The definition of is: in, The obstacle strength coefficient, This is the threshold for the distance between the evaluation point and the boundary of the illegal area. Less than the threshold At that time, the logarithmic obstacle term produces a penalty energy that increases sharply with decreasing distance. When the energy is equal to or less than zero, indicating entry into the illegal region, it tends towards infinity, thus mechanistically preventing the curve from entering the illegal region. The illegal region encompasses the self-fertilization risk area, the undercut direction area, the area affected by adjacent tooth positions, and the area crossed by missing teeth and gaps during scanning. The three energies are weighted by a coefficient. , , To achieve balance adjustment, Controlling the strength of ridge line adsorption, Controlling the strictness of smoothness constraints, By controlling the sensitivity of boundary crossing protection, the three factors are combined to minimize the objective function while simultaneously satisfying the comprehensive requirements of conforming to the ridge line, maintaining smoothness, and preventing boundary crossing.

[0045] In step S43, the Cartesian product of the pseudo-gradient feasible direction cone set is used as the search space. Candidate displacements are sampled within the corresponding cone for each control point in the sparse control point set, and iterative optimization is performed according to the acceptance criterion until convergence, resulting in the optimized control point set. It should be noted that, given that the three joint objective functions provide a unified evaluation criterion integrating ridge adhesion, smoothing constraints, and boundary crossing protection, and the pseudo-gradient feasible direction cone set has defined a search range for each control point that considers both gradient tendency and cooperative degrees of freedom, a gradient-independent global search algorithm is needed to jointly optimize all control points within this constraint space. Based on this, the technical solution of this application further uses the Cartesian product of the pseudo-gradient feasible direction cone set as the search space, samples candidate displacements within the corresponding cone for each control point in the sparse control point set, and iteratively optimizes according to the acceptance criterion until convergence, resulting in the optimized control point set. Through the above processing, the metaheuristic search simultaneously optimizes within the Cartesian product space of the feasible direction cones of each control point. By using the probabilistic acceptance criterion, it gradually converges to the global low-energy state of the three joint objective functions while avoiding getting trapped in local minima, so that all control points collaboratively approach the true ridge position.

[0046] Figure 8 The flowchart illustrates the process of optimizing the control point set using the Cartesian product of the set of feasible cones with pseudo-gradient directions as the search space, according to the embodiment of this application. It involves sampling candidate displacements within the corresponding cone for each control point in the sparse control point set and iteratively optimizing according to the acceptance criterion until convergence. Figure 8 As shown, step S43 includes: S431, based on the three joint objective functions, applying micro-step trial displacements along multiple directions to estimate the energy response of each control point in the sparse control point set within the feasible domain defined by the pseudo-gradient feasible direction cone set, and taking the direction with the largest energy decrease to obtain the local energy response direction set; S432, based on the arc-length adjacency topology relationship of the sparse control point set along the closed curve, performing neighborhood weighted fusion and normalization on the directions of each point in the local energy response direction set according to the arc-length distance attenuation weight to obtain the cooperative sampling bias direction set; S433, using the cooperative sampling bias direction set as the sampling density peak direction, sampling candidate displacements of each control point in the sparse control point set in a non-uniform distribution within the pseudo-gradient feasible direction cone set and iteratively optimizing through surface projection until convergence to obtain the optimized control point set.

[0047] In step S431, based on the three joint objective functions, the energy response of each control point in the sparse control point set is estimated by applying micro-step trial displacements along multiple directions within the feasible region defined by the pseudo-gradient feasible direction cone set. The direction with the largest energy decrease is taken as the local energy response direction set. It should be noted that, given that the simulated annealing method in the existing method directly performs uniform random sampling in each iteration, it is completely unaware of the local response characteristics of the objective function in different directions within the feasible cone, and relies entirely on statistical probability to hit the effective direction. This results in a large amount of computational resources being consumed on invalid candidate solutions that are rejected immediately after generation, leading to low search efficiency. Based on this, the technical solution of this application further estimates the energy response of each control point in the sparse control point set by applying micro-step trial displacements along multiple directions within the feasible region defined by the pseudo-gradient feasible direction cone set, based on the three joint objective functions, and takes the direction with the largest energy decrease as the local energy response direction set. Through the above processing, the steepest descent approximate direction of each control point under the geometric constraints of the feasible cone can be detected with limited computational cost, providing a reliable directional prior anchor point for subsequent neighborhood propagation.

[0048] More specifically, in a concrete example of this application, before initiating formal candidate displacement sampling in each iteration, low-cost direction probing is performed on each control point. For each control point in the sparse control point set, within the feasible region defined by the pseudo-gradient feasible direction cone set, small-step trial displacements are applied along several uniformly distributed probe directions. The local energy change of the three joint objective functions is calculated for each direction, and the direction with the largest energy decrease is selected as the local energy response direction of that control point. That is, for control points... In its corresponding feasible direction cone Several unit direction vectors are uniformly selected within the inner region, and an application with a magnitude of a microstep is applied along each direction. The trial displacement is used to calculate the control point from... Move to Energy change of the joint objective function of the last three terms After traversing all detection directions, select the one that makes The direction with the largest negative absolute value is taken as the local energy response direction of that control point. The local energy response directions of all control points are then converged into a local energy response direction set.

[0049] In the cervical margin scenario of the prepared dental prosthesis, since the actual ridge line is continuously distributed around the tooth surface and its geometric direction has local smooth transition characteristics, the local energy response direction obtained by directional probing at a certain control point contains effective information about the ridge line direction at that location. Compared with blind random sampling, this directional prior provides a reliable anchor point basis for subsequent neighborhood propagation, enabling subsequent steps to establish directional coordination relationships between adjacent control points based on this directional information.

[0050] In step S432, based on the arc-length adjacency topology of the sparse control point set along the closed curve, the directions of each point in the local energy response direction set are weighted and normalized according to the arc-length distance attenuation weight to obtain a cooperative sampling bias direction set. It should be noted that, given that after obtaining the independently estimated local energy response direction of each control point, directly using this direction to guide sampling, while superior to uniform randomness, still does not utilize the displacement response coupling relationship between adjacent control points, and since the ridge line has spatial continuity along the surface, there is a positive correlation of smooth transition between the effective adsorption directions of adjacent control points. Ignoring this prior will lead to frequent conflicts in the directions of adjacent points. Therefore, the technical solution of this application further uses the arc-length adjacency topology of the sparse control point set along the closed curve to weight the neighborhood fusion and normalization of the directions of each point in the local energy response direction set according to the arc-length distance attenuation weight to obtain a cooperative sampling bias direction set. Through the above processing, this prior can be incorporated into the search strategy, thereby reducing conflicts in the directions of adjacent points and improving the search of each control point from isolated search to cooperative optimization along the closed curve topology.

[0051] More specifically, in a concrete example of this application, based on the arc-length adjacency topology of points in the sparse control point set along a closed curve, neighborhood coupling propagation is performed on the direction vectors of points in the local energy response direction set: For each control point, its own local energy response direction is weighted and fused with the local energy response directions of control points in its left and right arc-length neighborhoods according to the arc-length distance attenuation weight. The fusion result is projected back to the tangent plane of the point and normalized to obtain the collaborative sampling bias direction that integrates neighborhood direction information. The collaborative sampling bias directions of all control points constitute the collaborative sampling bias direction set. The calculation expression for the arc-length attenuation weight is: in, For the first The energy response direction of the first control point is related to the... The coupling propagation weights of each control point and The first The and the first The arc length coordinates of each control point on the closed curve The propagation bandwidth parameter controls the spatial attenuation scale of the neighborhood coupling. The exponential attenuation ensures that closer neighboring control points contribute more to the direction of the current point, while the influence of farther control points tends to zero. This aligns with the physical fact that the local correlation of the ridge direction weakens with increasing arc length. The value of the propagation bandwidth parameter determines the spatial range of the collaboration—too small, and only the left and right adjacent points participate in the fusion, resulting in limited collaboration; too large, and directional noise from distant, unrelated points is introduced, interfering with local search accuracy. In practice, a value two to three times the average distance between adjacent control points is suitable. The fusion calculation expression for the collaborative sampling bias direction is: in, For the first The unit vector of the cooperative sampling bias direction of each control point after neighborhood coupling For the first The set of control point indices in the arc-length neighborhood of a control point on a closed curve includes itself. For the first The local energy response direction vector of each control point. The normalization operation of the denominator ensures that the output is a unit direction vector, so that the parameterization of the subsequent sampling distribution is not affected by the difference in the magnitude of each vector before fusion.

[0052] In the scenario of recognizing the cervical margin of the oral cavity preparation body, 50 sparse control points are distributed around the tooth body. When several control points located on the buccal side converge first due to the high contrast of the ridge curvature at that location, their energy response direction carries local information about the direction of the buccal ridge. This information is then propagated to the adjacent control points at the buccal-proximal interface that are still being searched, through arc length attenuation weights. This allows the latter to obtain a directional bias that smoothly transitions with the direction of the buccal ridge without having to blindly explore from scratch. This avoids the deterioration of smooth energy and frequent rejection of candidate solutions caused by adjacent control points moving in opposite directions due to their independent sampling.

[0053] In step S433, using the co-sampling bias direction set as the peak direction of the sampling density, candidate displacements of each control point in the sparse control point set are sampled in a non-uniform distribution within the pseudo-gradient feasible direction cone set, and then iteratively optimized through surface projection until convergence, resulting in an optimized control point set. It should be noted that since the co-sampling bias direction set already provides advantageous search directions for each control point by integrating neighborhood information, if uniform random sampling within the cone is still used, this directional prior is not utilized. Therefore, a non-uniform sampling strategy is needed to concentrate the sampling probability density near the bias direction while retaining global exploration capabilities to replace the original uniform sampling. Based on this, the technical solution of this application further uses the co-sampling bias direction set as the peak direction of the sampling density, sampling candidate displacements of each control point in the sparse control point set in a non-uniform distribution within the pseudo-gradient feasible direction cone set, and then iteratively optimized through surface projection until convergence, resulting in an optimized control point set. Through the above processing, search resources can be concentrated on high-probability effective directions while retaining low-probability coverage of other directions within the cone, achieving a balance between accelerating convergence and maintaining global search capabilities, and reducing the risk of getting trapped in local minima.

[0054] More specifically, in a concrete example of this application, after obtaining the set of co-sampling bias directions, instead of the original uniform random sampling, the co-sampling bias directions of each control point are used as the mode direction of the sampling probability density. Candidate displacement sampling is then performed in a non-uniform distribution within the feasible region defined by the set of pseudo-gradient feasible directions. Regions near the bias directions are assigned a higher sampling probability density to concentrate search resources, while regions far from the bias directions retain a lower but not zero sampling probability density to maintain global exploration capability and prevent getting trapped in local minima. The probability density function of the non-uniform sampling within the cone is: in, In the first Candidate displacement vectors within the feasible cone of each control point, conditioned on the cooperative sampling bias direction. The sampling probability density, Let be the candidate displacement vector within the feasible direction cone. For the first Unit vector of cooperative sampling bias direction for each control point For concentration parameters, For vector dot product operators, For vector modulus operators, For the first The pseudo-gradient feasible direction cone for each control point. It should be noted that when the concentration parameter... When zero is taken, the distribution degenerates into a uniform distribution within the cone, which is equivalent to existing methods; as As the bias increases, the sampling distribution becomes more concentrated near the bias direction, resulting in a stronger search directionality. In practice... It can be adaptively adjusted according to the annealing temperature—a smaller value is used in the high-temperature stage to preserve the breadth of exploration, and a larger value is used in the low-temperature stage to accelerate local fine convergence.

[0055] Candidate displacement vectors are sampled from each control point in the sparse control point set according to this non-uniform distribution to form candidate solutions. New curves are generated through closed cubic spline interpolation and projected onto a 3D manifold surface triangular mesh to obtain a new evaluation point set. These are then substituted into the three-term joint objective function to calculate the energy change, and the Metropolis criterion is used to determine acceptance or rejection. After each iteration, the process returns to step one to re-estimate the energy response direction and update the cooperative bias. This process is repeated until energy convergence or the temperature drops below a threshold, outputting the optimized control point set. After each iteration, the direction detection in step S431 and the neighborhood propagation in step S432 are re-executed, enabling the cooperative bias direction to track the dynamic changes in the terrain of the objective function as the control points move, forming a cooperative search closed loop of detection-propagation-sampling-update.

[0056] In the scenario of recognizing the cervical margin of a dental preparation body, taking the optimization of curvature adsorption around the tooth body using 50 control points as an example, during the high-temperature stage of simulated annealing, the concentration parameter... Taking a smaller value makes the sampling distribution within the cone nearly uniform, and each control point maintains a large directional exploration range to escape the local minima near the initial position; as the annealing temperature gradually decreases... Adaptive scaling gradually tightens the sampling distribution to the vicinity of the cooperative bias direction. After the control points located in the region with low curvature contrast of the tongue-side ridge line obtain directional information from the converged neighboring points through neighborhood propagation, their sampling probability density is concentrated in the effective direction consistent with the neighborhood. This avoids the deterioration of smoothing energy caused by directional conflicts with neighboring points due to blind random sampling during the low-temperature fine search stage. As a result, the entire neck edge line is cooperatively and precisely aligned with the true ridge line position along each segment at the end of the convergence period.

[0057] In step S44, a closed cubic spline interpolation is constructed using the optimized control point set as nodes, and uniform resampling is performed. The optimized feature spline is then projected onto the manifold surface triangular mesh through the nearest point. It should be noted that, since the optimized control point set only contains... A discrete set of sparse nodes cannot be directly output as a continuous closed curve. Interpolation is needed to restore the continuous curve shape between control points and constrain it to the manifold surface. Based on this, the technical solution of this application further constructs closed cubic spline interpolation using the optimized control point set as nodes and performs uniform resampling. The optimized feature splines are then projected onto the triangular mesh of the manifold surface through the nearest point. Through this processing, the discrete optimized control point set can be restored as a uniformly sampled continuous closed curve strictly located on the manifold surface, enabling subsequent offset contour generation to produce high-quality input with uniform arc length distribution.

[0058] More specifically, in a particular example of this application, to optimize the control point set For the control points, a closed cubic spline interpolation curve is constructed using closed boundary conditions. This curve satisfies the position interpolation conditions at each control point and possesses second-order continuous differentiability. The first and second derivatives at the first and last control points are matched to ensure a smooth transition at the closure points. After constructing the closed cubic spline interpolation curve, uniform equal-arc-length resampling is performed along the curve's parameter domain to generate a fine spatial point set with a sampling density much greater than the number of control points. The nearest-point projection operator is then applied to each of the resulting fine spatial points. Projecting these points onto the surface of the triangular facets of the manifold surface triangular mesh yields an ordered sequence of sampling points strictly located on the surface. Arranging these projected points in parameter order constitutes the optimized feature spline. The curve is precisely embedded in the high-curvature ridge groove throughout, maintaining a closed and smooth shape while conforming to the ridge line. Furthermore, due to the effect of the logarithmic obstacle term during the optimization process, it does not self-intersect or enter illegal areas.

[0059] Figure 9 This is a comparison of the curvature adsorption optimization effect before and after using the manifold surface-based three-dimensional feature spline fitting and optimization method according to an embodiment of this application. Figure 9 As shown, the left figure (a) is the initial closed curve obtained by fitting a periodic Fourier series with regularized order. The cyan spherical markers represent curve points resampled along a uniform azimuth angle. The curves are generally off-center and do not yet precisely fit the ridge line. Figure (b) on the right shows the optimized feature spline after evaluation by sparse control point optimization, interpolation densification points, pseudo-gradient feasible cones, and metaheuristic co-optimization. The red curve has been tightened and precisely embedded in the high-curvature ridge groove at the junction of the tooth and gum, maintaining a smooth closure while conforming to the ridge throughout.

[0060] Specifically, in step S5, based on the local curvature distribution characteristics of each point in the optimized feature spline, a pre-trained neural network is used to probabilistically predict the offset direction and amplitude, and the offset profile is obtained through surface geodesic index mapping and closure smoothing. It should be noted that, due to the traditional method along the normal... Offsets with a fixed step size and a single direction are prone to deviations. When there are significant differences in slope on both sides of the pre-formed body or when there are indentations, the offset profile may deviate from the surface, penetrate or detach from the material, failing to adaptively adapt to local geometric changes. Therefore, the technical solution of this application further optimizes the local curvature distribution characteristics of each point in the feature spline, uses a pre-trained neural network to probabilistically predict the offset direction and amplitude, and then performs surface geodesic mapping and closure smoothing to obtain the surface offset profile. Through the above processing, the offset direction and amplitude are adaptively predicted using the local curvature distribution as input, and the offset profile travels along the surface using geodesic mapping instead of a straight line offset along the normal, thus ensuring that the offset profile conforms to the manifold surface throughout its entirety without penetrating or detaching from the material.

[0061] More specifically, in a particular example of this application, first along the optimized feature spline... Each sampling point Local curvature distribution features are extracted on the triangular mesh of the manifold surface. For each sampling point, a feature vector is constructed by sampling within its arc-length neighborhood: in, and The principal curvature sequences on both sides of the sampling point. The rate of change of curvature along the arc length, The angle between the two normal directions. For local height difference, This is the cumulative arc length at that point. The total circumference of the curve. For normalized position encoding, the feature vectors of all sampling points are arranged into an ordered sequence along the curve, containing both point-by-point local geometric information and contextual information, forming a local curvature distribution feature sequence. Subsequently, this local curvature distribution feature sequence is input into a pre-trained sequence prediction network. The network outputs the probability distribution parameters of the offset direction and amplitude point by point. The offset direction is defined in the tangent plane as the geodesic lateral angle. The offset magnitude is represented and modeled as a von Mises distribution, and modeled as a Gaussian distribution: in, The mean parameter of the von Mises distribution for the geodesic transverse angle is... The concentration parameter is used to characterize the confidence level of the direction prediction. The Gaussian mean of the offset magnitude. Let the standard deviation be Gaussian. Take the maximum a posteriori estimate. and As the predicted geodesic lateral angle and predicted offset amplitude at each point, the confidence parameter and This can be used for subsequent quality control. The network backbone can adopt sequence modeling structures such as one-dimensional convolutional networks, temporal convolutional networks, or Transformers to capture contextual dependencies along the boundaries. The prediction results of all sampled points constitute the prediction offset parameter set.

[0062] It should be noted that the aforementioned pre-trained sequence prediction network Specifically, a temporal convolutional network is used as the backbone structure, with a sequence of local curvature distribution features arranged in an ordered manner along a closed curve as the one-dimensional temporal input. The input feature vector for each sampling point... The system includes components such as bilateral principal curvature sequences, curvature change rates, bilateral normal angles, neighborhood curvature histograms, local height differences, and normalized arc length position codes. After concatenation, the dimensions are fixed, and each component is standardized to zero mean and unit variance based on the training set statistics before being input into the network. The network output layer consists of two parallel fully connected branches. The first branch outputs the von Mises distribution parameters of the geodesic lateral angle, where the mean... Activation mapping via the inverse tangent function to Interval, concentration The softplus activation ensures a positive value; the second branch outputs the Gaussian distribution parameters of the offset amplitude, where the mean is... Without applying activation constraints, standard deviation The activation value is guaranteed to be positive via softplus activation. The network is trained based on a large-scale dataset of labeled historical cases. The true values ​​for the offset direction and amplitude are obtained by inverse calculation from the geodesic distance and orientation angle between the prosthesis interface boundary and the corresponding neck margin line, as marked by a professional technician in historical cases. The training objective is to minimize the joint loss function of the orientation error, amplitude error, and surface distance error. in, The cosine error term for direction prediction. This is the mean square error term for amplitude prediction. This is the distance along the surface from the offset point to the true profile. , The last term is the balancing weight of each item. The smoothing regularization term for the predicted values ​​of adjacent sampling points is used to constrain the smooth transition of the prediction result along the curve. Here, pre-training refers to the network completing parameter optimization on a training set containing a large number of labeled cases, fixing the weights, and directly performing forward propagation on the newly input neckline feature sequence during the inference phase to output the prediction result without further parameter updates.

[0063] After obtaining the predicted offset parameter set, the predicted geodesic lateral angles at each point are used as a basis. Synthesize the unit offset direction within the tangent plane of each point on the optimized feature spline. Combined with the predicted offset magnitude The offset points are obtained by traveling along the manifold surface using geodesic index mapping: in, For the first The three-dimensional spatial position of each sampling point after offset along the surface For The surface geodesic index mapping operator starting from the point of origin, To predict the magnitude of the offset, i.e., the arc length of the geodesic journey, In order to be in To predict the geodesic lateral angle in the tangent plane A defined unit offset direction vector. Geodesic index mapping ensures the offset path strictly follows the manifold surface rather than traversing space in a straight line, so that offset points remain close to the surface even in concave regions or areas with significant slope differences. Performing the above calculations on all sampled points yields a three-dimensional offset point set.

[0064] Finally, a truncated-order periodic Fourier series closed-loop smoothing fit is performed on the 3D offset point set to suppress point-by-point jitter in the network prediction and ensure the closure and smoothness of the offset profile. The fitted smooth closed curve is then resampled with uniform arc length, and the nearest-point projection operator is applied to each resampled point. Projecting onto the triangular mesh surface of the manifold surface yields a surface offset profile that is strictly located on the surface. The contour conforms to the manifold surface throughout, and does not penetrate or detach from the material in areas with significant differences in slope on both sides or where there are undercuts. It can be directly used to define the boundary of the joint surface of the restoration.

[0065] Furthermore, a three-dimensional feature spline fitting and optimization system based on manifold surfaces is also provided.

[0066] Figure 10 This is a block diagram of a three-dimensional feature spline fitting and optimization system based on manifold surfaces according to an embodiment of this application. Figure 10As shown, a three-dimensional feature spline fitting and optimization system 100 based on a manifold surface according to an embodiment of this application includes: a spherical parameterization module 110, used to perform spherical azimuth parameterization on a set of surface feature candidate points output by a neural network through vertex-by-vertex reasoning on the manifold surface triangular mesh to obtain a spherical azimuth sample set based on a manifold surface triangular mesh containing a vertex coordinate set and a triangular patch topology set; a Fourier fitting module 120, used to perform order-regularized periodic Fourier series fitting and surface back projection on the spherical azimuth sample set to obtain an initial closed curve; and a dynamic oversampling module 130, used to perform curvature estimation and smoothing on the manifold surface triangular mesh to obtain a ridge intensity field, and sparsely sample the initial closed curve to obtain a sparse control point set. Furthermore, dynamic oversampling is performed based on the cumulative dihedral angle of the curve segment between adjacent control points to obtain a dynamic oversampling evaluation point set; the curvature adsorption optimization module 140 is used to construct feasible direction cones for each point in the sparse control point set based on the gradient direction of the ridge intensity field and the cosine similarity threshold, and to perform optimization in the Cartesian product space of the feasible direction cones using the cumulative curvature adsorption term, vertex smoothing term and illegal region logarithmic obstacle term on the dynamic oversampling evaluation point set as the objective function to obtain the optimized feature spline; the surface offset module 150 is used to perform probabilistic prediction of the offset direction and amplitude through a pre-trained neural network based on the local curvature distribution characteristics of each point in the optimized feature spline, and to obtain the surface offset profile through surface geodesic index mapping and closure smoothing processing.

[0067] Those skilled in the art will understand that the steps, measures, and schemes in the various operations, methods, and processes discussed in this application can be alternated, modified, combined, or deleted; furthermore, other steps, measures, and schemes in the various operations, methods, and processes discussed in this application can also be alternated, modified, rearranged, decomposed, combined, or deleted; furthermore, the steps, measures, and schemes in the prior art that are similar to those disclosed in this application can also be alternated, modified, rearranged, decomposed, combined, or deleted. The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as the combination of these technical features does not contradict each other, it should be considered within the scope of this specification. The above-described embodiments are merely illustrative of several implementation methods of this disclosure, and their descriptions are relatively specific and detailed. However, they should not be construed as limiting the scope of the patent for the embodiments of this disclosure. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the embodiments of this disclosure, and these all fall within the protection scope of the embodiments of this disclosure. Therefore, the protection scope of the embodiments of this disclosure should be determined by the appended claims. As described above, although the present invention has been shown and described with reference to specific preferred embodiments, it should not be construed as limiting the present invention itself. Various changes in form and detail can be made without departing from the spirit and scope of the present invention as defined in the appended claims.

[0068] The present invention and its embodiments have been described above. This description is not restrictive, and the accompanying drawings are only one embodiment of the present invention; the actual structure is not limited thereto. In conclusion, if those skilled in the art are inspired by this description and design similar structures and embodiments without departing from the spirit of the present invention, such designs should fall within the protection scope of the present invention.

Claims

1. A method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces, characterized in that, include: S1, based on the manifold surface triangular mesh containing the vertex coordinate set and the triangular patch topology set, the spherical azimuth parameterization is performed on the surface feature candidate point set output by the neural network through vertex-by-vertex reasoning on the manifold surface triangular mesh to obtain the spherical azimuth sample set. S2, perform order-regularized periodic Fourier series fitting and surface back projection on the spherical azimuth angle sample set to obtain the initial closed curve; S3, the curvature of the triangular mesh of the manifold surface is estimated and smoothed to obtain the ridge intensity field, and the initial closed curve is sparsely sampled to obtain a sparse control point set. Dynamic oversampling is performed based on the cumulative dihedral angle of the curve segment between adjacent control points to obtain a dynamic oversampling evaluation point set. S4, based on the gradient direction of the ridge intensity field and the cosine similarity threshold, constructs a feasible direction cone for each point in the sparse control point set, and optimizes the feature spline by using the curvature adsorption term, vertex smoothing term and illegal region logarithmic obstacle term accumulated on the point set as the objective function in the Cartesian product space of the feasible direction cone. S5, based on the local curvature distribution characteristics of each point in the optimized feature spline, uses a pre-trained neural network to predict the offset direction and amplitude probabilistically, and then obtains the surface offset profile through surface geodesic index mapping and closed smoothing processing.

2. The method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to claim 1, characterized in that, Step S1 includes: Based on the local area weights of each candidate point determined by the topological set of triangular facets in the triangular mesh of the manifold surface, the spatial coordinates of each point in the candidate point set of surface features are aggregated by weighted average to obtain the reference sphere center. Based on the reference sphere center, calculate the normalized direction vector for each point in the candidate point set of surface features and construct the area-weighted covariance matrix. Then, perform eigenvalue decomposition on the covariance matrix and extract the eigenvector corresponding to the minimum eigenvalue as the principal axis direction to construct a local orthogonal frame. Based on the reference sphere center and the local orthogonal frame, spherical coordinate projection transformation is performed on each point in the candidate point set of surface features, and the azimuth angle of each point around the principal axis and the corresponding radial distance and elevation angle are calculated to obtain the spherical azimuth angle sample set.

3. The method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to claim 1, characterized in that, Step S2 includes: Using the azimuth components in the spherical azimuth sample set as independent variables, construct a Fourier basis matrix composed of periodic trigonometric function bases according to the truncation order, and construct an order regularization matrix; Based on the Fourier basis matrix and the order regularization matrix, the radial distance component and elevation component of the spherical azimuth angle sample set are solved by regularized weighted least squares to obtain the radial Fourier coefficient vector and the elevation Fourier coefficient vector. Based on the radial Fourier coefficient vector and the elevation Fourier coefficient vector, and combined with the reference sphere center and the local orthogonal frame, an inverse spherical coordinate transformation is performed, and the result is projected onto the manifold surface triangular mesh through the nearest point to obtain the initial closed curve.

4. The method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to claim 1, characterized in that, Step S3 includes: Based on the vertex coordinate set and triangular patch topology set of the manifold surface triangular mesh, a ridge intensity field covering the entire mesh is constructed; The initial closed curve is sparsely sampled with equal arc length to obtain a sparse control point set; On the manifold surface triangular mesh, traverse the shared edges crossed by the surface path between adjacent control points of the sparse control point set, and accumulate the product of the dihedral angle and the transverse weight of each shared edge to obtain the cumulative dihedral angle sequence. Based on the cumulative dihedral angle sequence, the sparse control point set is used as the node and non-uniform densification is performed through closed cubic spline interpolation. The result is then projected onto the manifold surface triangular mesh to obtain the dynamic oversampling evaluation point set.

5. The method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to claim 1, characterized in that, Step S4 includes: For each control point in the sparse control point set, the local gradient direction of the ridge intensity field along the surface is calculated and normalized to obtain the dominant direction of maximum curvature. Then, a set of pseudo-gradient feasible directions is formed in the tangent plane by using the cosine similarity threshold and the adsorption radius. The curvature adsorption energy term is defined by accumulating the negative ridge field intensity on the dynamic oversampling evaluation point set, the vertex smoothing energy term is defined by accumulating the second-order difference norm of adjacent points on the sparse control point set, and the logarithmic barrier energy term for illegal regions is defined by taking the negative logarithm of the directed distance on the dynamic oversampling evaluation point set. The three terms are combined in a weighted manner to construct a joint objective function. Using the Cartesian product of the set of feasible directions of pseudo-gradients as the search space, candidate displacements are sampled for each control point in the sparse control point set within the corresponding cone, and the optimization is iteratively sought until convergence according to the acceptance criterion to obtain the optimized control point set. A closed cubic spline interpolation is constructed using the optimized control point set as nodes, and uniform resampling is performed. The optimized feature spline is then obtained by projecting the nearest point onto the triangular mesh of the manifold surface.

6. The method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to claim 1, characterized in that, Step S5 includes: Local curvature distribution features are extracted at each sampling point along the optimized feature spline on the manifold surface triangular mesh to obtain a local curvature distribution feature sequence; The local curvature distribution feature sequence is input into a pre-trained sequence prediction network to obtain the predicted offset parameter set; Based on the geodesic lateral angles of each point in the predicted offset parameter set, the unit offset direction is synthesized in the tangent plane of each point of the optimized feature spline, and combined with the offset amplitude, it travels along the manifold surface through the geodesic index mapping to obtain the three-dimensional offset point set; The truncated order periodic Fourier series closed smooth fitting is performed on the 3D offset point set, and then uniformly resampled and projected onto the nearest point onto the manifold surface triangular mesh to obtain the surface offset profile.

7. The method for fitting and optimizing three-dimensional feature spline lines based on manifold surfaces according to claim 5, characterized in that, Using the Cartesian product of the set of feasible directions of pseudo-gradients as the search space, candidate displacements are sampled within the corresponding cone for each control point in the sparse control point set, and iterative optimization is performed according to the acceptance criterion until convergence, resulting in the optimized control point set, including: Based on the three joint objective functions, the energy response of each control point in the sparse control point set is estimated by applying micro-step trial displacements in multiple directions within the feasible domain defined by the pseudo-gradient feasible direction cone set. The direction with the largest energy decrease is taken to obtain the local energy response direction set. Based on the arc-length adjacency topology of the sparse control point set along the closed curve, the directions of each point in the local energy response direction set are weighted and normalized according to the arc-length distance attenuation weight to obtain the cooperative sampling bias direction set. Using the cooperative sampling bias direction set as the peak direction of the sampling density, the candidate displacements of each control point in the sparse control point set are sampled in a non-uniform distribution within the pseudo-gradient feasible direction cone set, and then iteratively optimized through surface projection until convergence, thus obtaining the optimized control point set.

8. A three-dimensional feature spline fitting and optimization system based on manifold surfaces, characterized in that, include: The spherical parameterization module is used to perform spherical azimuth parameterization on the set of candidate points of surface features output by the neural network inferring vertex by vertex on the triangular mesh of the manifold surface, which contains the vertex coordinate set and the triangular patch topology set, to obtain the spherical azimuth sample set. The Fourier fitting module is used to perform order-regularized periodic Fourier series fitting and surface back projection on the spherical azimuth sample set to obtain the initial closed curve. The dynamic oversampling module is used to estimate the curvature and smooth the triangular mesh of the manifold surface to obtain the ridge intensity field, and to sparsely sample the initial closed curve to obtain a sparse control point set. It also performs dynamic oversampling based on the cumulative dihedral angle of the curve segment between adjacent control points to obtain a dynamic oversampling evaluation point set. The curvature adsorption optimization module is used to construct feasible direction cones for each point in the sparse control point set based on the gradient direction of the ridge intensity field and the cosine similarity threshold. In the Cartesian product space of the feasible direction cone, the curvature adsorption term, vertex smoothing term and illegal region logarithmic obstacle term accumulated on the dynamic oversampling evaluation point set are used as the objective function to obtain the optimized feature spline. The surface offset module is used to predict the offset direction and magnitude probabilistically through a pre-trained neural network based on the local curvature distribution characteristics of each point in the optimized feature spline, and then obtain the surface offset profile through surface geodesic index mapping and closed smoothing processing.

9. The three-dimensional feature spline fitting and optimization system based on manifold surfaces according to claim 8, characterized in that, The dynamic oversampling module includes: The ridge intensity field construction unit is used to construct a ridge intensity field covering the entire mesh based on the vertex coordinate set and triangular patch topology set of the manifold surface triangular mesh. Sparse sampling unit is used to perform equal arc length sparse sampling on the initial closed curve to obtain a sparse control point set; The dihedral angle accumulation unit is used to traverse the shared edges crossed by the surface path between adjacent control points in the sparse control point set on the manifold surface triangular mesh, and accumulate the product of the dihedral angle of each shared edge and the transverse weight to obtain the cumulative dihedral angle sequence. The dynamic oversampling unit is used to obtain the dynamic oversampling evaluation point set by non-uniform densification through closed cubic spline interpolation based on the cumulative dihedral angle sequence, with sparse control point set as nodes, and projected onto the manifold surface triangular mesh.

10. The three-dimensional feature spline fitting and optimization system based on manifold surfaces according to claim 8, characterized in that, The curvature adsorption optimization module includes: The feasible direction cone construction unit is used to calculate the local gradient direction of the ridge intensity field along the surface for each control point in the sparse control point set and normalize it to obtain the dominant direction of maximum curvature. It is then used to stretch the pseudo-gradient feasible direction cone set in the tangent plane with the cosine similarity threshold and the adsorption radius. The objective function construction unit is used to define the curvature adsorption energy term by accumulating the negative ridge field intensity on the dynamic oversampling evaluation point set, to define the vertex smoothing energy term by accumulating the second-order difference norm of adjacent points on the sparse control point set, and to define the illegal region logarithmic barrier energy term by taking the negative logarithm of the directed distance on the dynamic oversampling evaluation point set. The three terms are combined in a weighted manner to construct the three joint objective functions. The metaheuristic optimization unit is used to sample candidate displacements of each control point in the sparse control point set within the corresponding cone and iteratively optimizes according to the acceptance criterion until convergence, using the Cartesian product of the set of feasible directions of pseudo-gradients as the search space, to obtain the optimized control point set. The spline reconstruction unit is used to construct closed cubic spline interpolation with the optimized control point set as nodes and perform uniform resampling. The optimized feature spline is then projected onto the manifold surface triangular mesh through the nearest point.

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