A planar parameterization method applicable to complex geometries and possessing G1 smoothness properties

Abstract

This invention proposes a planar parameterization method applicable to complex geometric regions and possessing G1 smoothness properties. The technique first generates the meshed structure of a given region based on its central axis information; then, by applying G1 continuity constraints to adjacent patches, it constructs an initial parameterization that satisfies G1 continuity overall; finally, it uses quasi-conformal mapping techniques to further optimize the parameterization quality of each patch. This invention simultaneously uses rectangular and triangular Bezier patches to represent the geometric region, making it effective for arbitrarily topologically complex domains. Compared with other existing methods, it not only enhances design flexibility but also achieves significant improvements in the generation and parameterization quality of multi-patch structures, resulting in greater efficiency.

Description

Technical Field

[0001] This invention relates to the field of iso-geometric analysis, specifically a planar parameterization method applicable to complex geometric regions and possessing G1 smoothness characteristics. Background Technology

[0002] Isogeometric analysis (IgA) aims to bridge the gap between Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE). Its core concept lies in using unified basis functions with high smoothness, typically employing non-uniform rational B-splines (NURBS) to simultaneously represent the physics field in CAE and the geometric model in CAD. Compared to traditional finite element methods, IgA exhibits several significant advantages, including but not limited to accurate representation of geometry, high-order smoothness of basis functions, and improved accuracy per degree of freedom. These advantages make isogeometric analysis uniquely valuable and promising for handling complex geometries and engineering problems requiring high-order accuracy.

[0003] A fundamental task within the IgA framework is to construct analytically perceptual parameterizations from a given boundary representation of a physical domain. Basic requirements include injectivity, low distortion, and applicability to complex domains. Despite extensive research in academia and engineering, most methods are limited to handling geometries with simple topologies.

[0004] While multi-patch parametric techniques can theoretically handle arbitrary topologies, most existing techniques still face challenges in dealing with complex domains in practical applications. For example, in research based on multi-cubes, singularities within regions are unavoidable, significantly impacting patch layout. Furthermore, skeleton-based methods have limitations in obtaining high-quality parametric results for complex shapes. Some methods attempting to construct planes from complex boundaries through domain segmentation and global optimization, while theoretically feasible, often lead to a significant increase in the number of patches in the parametric structure in practice. The parametric results achieved by these methods typically only reach global C0. 0 The lack of continuity cannot meet the requirements of high-precision analysis. Therefore, it is necessary to seek better new methods to overcome the above shortcomings. Summary of the Invention

[0005] To address the problems of existing technologies, this invention provides a planar parameterization method applicable to complex geometric regions and possessing G1 smoothness properties, effective for any topologically complex domain. Compared to other existing methods, it not only enhances design flexibility but also achieves significant improvements in the generation and parameterization quality of multi-faceted structures, resulting in greater efficiency.

[0006] To achieve the above objectives, the present invention provides a method for dealing with arbitrary topologically complex domains with G... 1The specific steps of the planar parameterization method for smoothness properties are as follows:

[0007] 1) Automatic region decomposition with progressive refinement, using triangular and rectangular Bezier patches to represent complex domains, specifically including:

[0008] 1.1) The skeleton is extracted based on the region boundary using a sparse optimization-based method; the skeleton extraction process is as follows:

[0009] A sparse optimization-based method is used to extract the skeleton, and a compact median transformation is computed for any 2D object represented by a point cloud containing noisy or missing data. Then, the branch points, branches, the maximum inscribed circle centered at each branch point, and the corresponding tangent points on the skeleton are determined.

[0010] Assume c t (u) is the terminal branch spline curve connected to the branch point b, r b It is the radius of the largest incircle centered at b. Then delete the terminal branches that satisfy the following conditions.

[0011]

[0012] The left-hand portion represents the arc length of the terminal branch, r. max It is the maximum value of the radius of the circle centered at each branch point, where α and β are positive parameters.

[0013] 1.2) The skeleton guides the initial decomposition process, connects the corresponding tangent points on the domain boundary, and automatically generates triangles and rectangular patches to construct the basic geometric structure;

[0014] 1.3) Based on the initial decomposition, further identify the shapes of the remaining facets and select suitable corner points for each facet; the process of obtaining the corner points is as follows:

[0015] After completing the initial decomposition of region Ω, further analysis and identification of the remaining faces are performed to determine whether there are faces that can be approximated as triangles. For a given face... for One of the vertices, Boundary (edge) )for An edge that lies inside Ω is also called an internal boundary. Represents a piece of dough The perimeter, L p Let δ represent the perimeter of Δpp′p″, and let δ>0 as a threshold. If the facet satisfy

[0016]

[0017] Then it is believed It is a triangle. At this point, point p and the two endpoints p′ and p″ of the corresponding internal boundary are selected as the three corner points required for subsequent parameterization.

[0018] After identifying and distinguishing the triangular facets, the remaining facets are classified as quadrilateral facets according to default rules. For each quadrilateral facet, four corner points need to be selected on the domain boundary. Two tangent points have already been determined and can be used as corner points. For the remaining two corner points, an automated method based on the optimal transmission principle is used for selection.

[0019] 1.4) Continue to refine the decomposition of the region, and construct a mathematical model of the resulting patch boundary segments using Bezier curves; the process of creating the mathematical representation of the Bezier curve is as follows:

[0020] For a given set of N discrete points The outer boundary segment is represented by a chord length parameterization method to determine the relevant parameters. Then, a p-order Bezier curve C(u) is fitted using control points c. This process is achieved by minimizing...

[0021]

[0022] We obtain . Here, λ is the positive weight, and the regularization term is...

[0023]

[0024] Used to measure the smoothness of the resulting curve. This is verified by the mean square error defined by the following formula.

[0025]

[0026] Is it greater than 4×10 -8 The specified tolerance is used to assess the fitting accuracy. If the error value exceeds the specified tolerance, the corresponding patch needs to be further decomposed. Patches with larger boundary fitting errors are usually quadrilaterals; therefore, the decomposition process is as follows: For the longer of the two boundary segments, select a point k1, whose geodesic distance to the two endpoints is approximately the same, and find its closest point k2 on the opposite boundary. By connecting k1 and k2, this patch is decomposed into two parts. This decomposition and fitting process will be iteratively performed until the boundary fitting error of all patches is reduced to below the specified tolerance.

[0027] Then, the internal boundary segments are processed. For internal boundary segments (straight lines) with endpoints p1 and p4, interpolation is performed at p1 and p4 to construct cubic Bezier curves to replace the original line segments. Then, two more control points p2 and p3 are constructed based on the tangent directions of p1 and p4. Specifically, let... It is the tangent vector of two adjacent Bezier curves at p1, and It is the tangent vector of two adjacent Bezier curves at p4, and the control points p2 and p3 are constructed as follows:

[0028] ∠p2p1p4≤σ,

[0029] and

[0030] ∠p3p4p1≤σ,

[0031] Where ρ and These are the degrees of p1 and p4, respectively, and θ is determined by... and The resulting included angle, It is by and The resulting included angle, σ, is a threshold. When the above two equations contradict each other, the following condition applies.

[0032] ∠p2p1p4=σ,

[0033] ∠p3p4p1=σ,

[0034] 2) Construct a global G 1 Smoothed parametric model:

[0035] 2.1) The initial multi-piece parameterization is calculated using the discrete Coons interpolation method; the multi-piece parameterization process is as follows:

[0036] For the first... represented by the rectangular Bezier surface A quadrilateral facet, Its boundary control points are Initial control point The structure is

[0037]

[0038] For the A triangular Bezier surface represented by a triangular patch. The formula for calculating internal control points that satisfy i+j+k=p is:

[0039]

[0040] 2.2) Apply G to the facet interface 1 The continuity condition is calculated to satisfy the global G. 1 Smooth control points; the application of G at the surface interface1 The process for establishing continuity conditions is as follows:

[0041] Let a p-order Bezier curve

[0042]

[0043] The control points are the common boundary of two adjacent rectangular Bezier faces. Then these two patches are G 1 If scalar weight functions λ(u) and μ(u) exist continuously, then the following equations are given.

[0044] (1-λ(u))s(u)+λ(u)t(u)=(1-μ(u))q l (u)+μ(u)q r (u)

[0045] It holds for all u, where

[0046]

[0047] and These are two rows of control points adjacent to the common boundary.

[0048] Here, the functions λ(u) and μ(u) are set as linear and quadratic polynomials, respectively, as shown below:

[0049]

[0050] Substituting λ(u) and μ(u) into the previous equation, we obtain the following G: 1 Continuity constraints

[0051]

[0052] They constitute a linear system.

[0053] When dealing with combinations of adjacent triangles and rectangles, first increase the order of the triangle from p to p+1. Then, the previously described method can be used to construct the aforementioned G. 1 Constraint system.

[0054] In this linear system, the boundary control points are known and fixed. s0,s p ,t0,t p Then, the remaining control points need to be calculated. The coefficients λ0, λ1 of λ(t) and the coefficients μ0, μ2 of μ(t) are determined by the first and last equations of the system, respectively. The coefficient μ1 of μ(t) is set to (μ0 + μ2) / 2. G 1 The continuous system is divided into two groups, namely, the vertex G.1 Continuous system (corresponding to k = 0, 1, p, p+1) and edge G 1 Continuous system (corresponding to k = 2, ..., p-1).

[0055] For each vertex (singular or regular corner) in the constructed polyhedral structure, i.e., the intersection of ρ Bezier faces, control the vertices o1, o2, ..., o ρ The twist vector at the vertex should satisfy the G corresponding to k=1 or p in the system. 1 Continuity constraints. All these constraints constitute the linear system Vx. v =y v , where vector x v Representative control point The set is obtained by solving the following problem.

[0056]

[0057] stVx v =y v ,

[0058] x v initial value It is constructed using the discrete Coons interpolation method.

[0059] For each common edge (boundary) shared by two Bezier faces, the two rows of control points adjacent to that boundary... and The G condition corresponding to k = 2, ..., p-1 should be satisfied. 1 Continuity constraints. These constraints constitute a linear system Ex. e =y e , where vector x e Representative control point and The set is obtained by solving the following problem.

[0060]

[0061] stEx e =y e

[0062] x e initial value It is also constructed using the discrete Coons interpolation method.

[0063] 3) QC mapping technology is used to obtain high-quality parameterization for each facet; the process of calculating the high-quality parameterization of the facet is as follows:

[0064] First, the first The parameterization of a quadrilateral patch (i.e., a rectangular Bezier surface) is rewritten as a complex function.

[0065]

[0066] in and here and Control point The components are as follows. In this parameterization, the boundary control points and control points adjacent to the common boundary are known, and the remaining internal control points are determined by solving the following optimization problem.

[0067]

[0068] in yes The Beltrami coefficient, where the parameter γ is the positive weight. and They represent and The Hessian matrix.

[0069] In solving this nonlinear and nonconvex problem, an auxiliary variable v is introduced and used to replace... This leads to an interleaved problem.

[0070]

[0071] ||v|| ∞ <1.

[0072] This is further relaxed to such a model

[0073]

[0074] st||v|| ∞ <1.

[0075] The weight η is relatively large. An energy descent method is used here to alternately solve and optimize the two subproblems. The specific iterative process is as follows: starting from constructing the initial G... 1 Parameterization Initially, solve v alternately. k and

[0076]

[0077] Among them, v k By fixed The model was then solved. By fixed v kThe model is then solved. This process continues until, for a pre-given threshold ε, the solution of the model satisfies the condition ||v. k+1 -v k ||<ε until.

[0078] The beneficial effects of this invention are as follows:

[0079] 1. This invention uses a combination of triangular and rectangular Bezier surfaces to represent complex domains in a hybrid manner, which is more flexible than traditional representation methods, can adapt to complex domains with arbitrary topologies, and has a wider range of applications.

[0080] 2. This invention employs a method combining skeleton-guided coarse decomposition with a more refined decomposition procedure, resulting in a superior patch layout. Unlike existing QMGLGO and PeDP methods, one advantage of the skeleton-guided domain decomposition process proposed in this invention is the absence of internal singularities in the generated polypatch structure. This helps generate better parameterization for complex domains, thereby reducing distortion. Furthermore, by implementing a series of operations to optimize the polypatch structure layout, such as automatically merging or further subdividing adjacent triangular patches during the coarse decomposition process, the number of patches is reduced while maintaining geometric accuracy, which is particularly desirable in the case of IgA.

[0081] 3. This invention uses QC mapping technology to ensure the injectivity and low distortion of the final parameterization. QC mapping is locally injective, and its angular distortion can be measured by the Beltrami coefficient. Using this as a framework to construct constraints for parameterization ensures its injectivity. Furthermore, this invention employs an energy descent method to alternately solve the difficult optimization problem during parameterization, thus generating higher quality patches than other techniques while producing the same number of patches. Attached Figure Description

[0082] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0083] Figure 1 This is a schematic diagram of the overall process of the present invention;

[0084] Figure 2 This is a diagram illustrating the initial decomposition process based on the skeleton in this invention;

[0085] Figure 3 This refers to the classification process of adjacent triangular facets during the initial decomposition of this invention.

[0086] Figure 4 This is a diagram illustrating the detailed decomposition process in this invention;

[0087] Figure 5 This is a schematic diagram of control points during the initial parameterization process of this invention;

[0088] Figure 6 This is a schematic diagram illustrating the initial parameterization and the use of QC mapping technology to obtain high-quality patches in this invention. Detailed Implementation

[0089] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0090] The process of this invention is as follows Figure 1 As shown, for complex domains with arbitrary topology, the region is first automatically refined stepwise based on Bezier patches, and then a global G is constructed. 1 Smoothing parameterization is performed, and finally, QC mapping technology is used to achieve high-quality parameterization for each facet. This can be divided into three steps:

[0091] 1) Automatic region decomposition with progressive refinement

[0092] The skeleton of an object depicts its complete shape and reflects its structure. Therefore, this invention performs domain decomposition based on the skeleton information of the input shape.

[0093] First, this invention extracts the skeleton from the boundary representation of the physical domain, which is a dense set of points. To this end, a sparse optimization-based method is employed to extract the skeleton, computing a compact median transformation for any two-dimensional object in a point cloud representation containing noisy or missing data. The key innovation distinguishing this invention from existing technologies lies in using spline curves instead of traditional polygonal lines to represent skeleton branches. This representation is more compact because it only needs to store the control points of the spline curves, thus avoiding the need to store a large number of discrete points and optimizing data compactness and efficiency. Then, the branch points, branches, the maximum inscribed circle centered at each branch point, and the corresponding tangent points on the skeleton are determined, such as... Figure 2 The left part is shown.

[0094] It is worth noting that some small terminal branches (such as those located in the lower right of the skeleton) may provide very little structural information about the domain. Therefore, they may not be necessary elements in subsequent decomposition processes. Assume c t (u) is the terminal branch spline curve connected to the branch point b, r bLet be the radius of the largest inscribed circle centered at b. To simplify the original skeleton of the region, delete the terminal branches that satisfy the following conditions.

[0095]

[0096] The left-hand portion represents the arc length of the terminal branch, r. max It is the maximum value of the radius of the circle centered at each branch point, where α and β are positive parameters. In the example, α is set to 1.0 and β is set to 0.35.

[0097] Use such as Figure 2 The second part is a simplified skeleton. Then, the corresponding tangent points on the domain boundary are connected to obtain preliminary decomposition results, automatically generating triangular and rectangular patches to construct the basic geometric structure, such as... Figure 2 The second part from the right is shown.

[0098] For a general domain, the degree of a branch point is typically 3, as is the case in this example, resulting in the generation of several triangular patches in this step. To optimize the layout of the generated multi-patch structure and reduce the number of patches, this invention employs a strategy of pairing adjacent triangular patches and merging them into quadrilateral patches, such as... Figure 2 As shown in the right part. However, there are two cases that require special handling: in Figure 2 In the right part, there is only one pair of adjacent triangular facets. In the coarse decomposition structure, a specific domain may be as follows: Figure 3 As shown on the left, there are two or more consecutively adjacent triangular faces. To address this, these faces should first be grouped, and then adjacent faces should be merged to form quadrilateral faces, as shown below. Figure 3 As shown in the middle image. Alternatively, you can choose to merge adjacent middle faces to maintain symmetry, such as... Figure 3 As shown on the right, this is more conducive to subsequent parameterization processing.

[0099] For a rotationally symmetric domain, some branch points may have a degree exceeding 3. Connecting the corresponding tangent points of each branch point sequentially along the region boundary in a clockwise direction may result in generating patches with more than four edges. If the number of edges of a patch is odd, it should be further subdivided into multiple quadrilateral patches and one triangular patch. Conversely, if the number of edges is even, it should be subdivided into several quadrilateral patches.

[0100] Next, based on the initial decomposition, the shapes of the remaining facets are further identified, and appropriate corner points are selected for each facet.

[0101] During the coarse decomposition process, quadrilateral patches (represented by "Q") and triangular patches (represented by "T") are automatically generated from the input shape, such as... Figure 4 As shown in the first row. Some of the remaining facets are triangular, such as... Figure 4The area defined by the red curve is shown. This invention identifies these patches by analyzing the shape differences between them and triangles. For a patch... for One of the vertices, Boundary (edge) )for An edge that lies inside Ω is also called an internal boundary. Represents a piece of dough The perimeter, L p Let δ represent the perimeter of Δpp′p″, and let δ>0 as a threshold. If the facet satisfy

[0102]

[0103] Then it is believed It is a triangle. At this point, point p and the two endpoints p′ and p″ of the corresponding internal boundary are selected as the three corner points required for subsequent parameterization.

[0104] After identifying and distinguishing triangular facets, the remaining facets are classified as quadrilateral facets according to default rules. For these quadrilateral facets, the prerequisite for subsequent parameterization is the selection of four corner points on the domain boundary. Two of these tangent points have already been determined and can be used as corner points. For the remaining two corner points, this invention employs an automated method based on the optimal transmission principle for selection. Figure 4 As shown in the first row from the right, the corner points of all the facets have been determined. Specifically, each triangular facet has three boundary segments, while each quadrilateral facet has four boundary segments.

[0105] The region is further refined, and a mathematical model of the resulting patch boundary segments is constructed using Bezier curves.

[0106] For a given set of N discrete points The outer boundary segment is represented by a chord length parameterization method to determine the relevant parameters. Then, a p-order Bezier curve C(u) is fitted using control points c. This process is achieved by minimizing...

[0107]

[0108] We obtain . Here, λ is the positive weight, and the regularization term is...

[0109]

[0110] Used to measure the smoothness of the resulting curve. This is verified by the mean square error defined by the following formula.

[0111]

[0112] Is it greater than 4×10 -8 The specified tolerance is used to assess the fitting accuracy. If the error value exceeds the specified tolerance, the corresponding patches need to be further decomposed. Figure 4 The leftmost figure in the second row shows the fitting results for the outer boundary segments, indicating that the four patches marked with "×" need to be decomposed. As can be seen from the test examples, the patches with larger boundary fitting errors are usually quadrilaterals. This is because the boundary segments of triangular patches have relatively simple shapes and can be accurately represented by Bezier curves.

[0113] Taking one of the patches with a relatively large fitting error as an example, the process of further decomposition is explained. Figure 4 In the middle of the second line, for the longer of the two boundary segments, a point p1 is selected, whose geodesic distance to the two endpoints is approximately the same. The nearest point p2 on the opposite boundary is then found. By connecting p1 and p2, this patch is decomposed into two parts. This decomposition and fitting process is iterated until the boundary fitting error of all patches is reduced to below the specified tolerance. Figure 4 The rightmost image in the second row shows the detailed decomposition results of the giraffe domain.

[0114] The shape of the internal boundary significantly influences the subsequent parameterization results. For an internal boundary segment (straight line segment) with endpoints p1 and p4, interpolation is performed at p1 and p4 to construct a cubic Bezier curve to replace the original line segment. Then, two additional control points p2 and p3 are constructed based on the tangent directions of p1 and p4. For example... Figure 4 As shown in the third line, let... It is the tangent vector of two adjacent Bezier curves at p1, and It is the tangent vector of two adjacent Bezier curves at p4, and the control points p2 and p3 are constructed as follows:

[0115] ∠p2p1p4≤σ,

[0116] and

[0117] ∠p3p4p1≤σ,

[0118] Where ρ and These are the degrees of p1 and p4, respectively, and θ is determined by... and The resulting included angle, It is by and The resulting containment angle, σ, is a threshold. The second inequality is to prevent the constructed cubic Bezier curve from intersecting with the original interior boundary. Deviations exceeding the predetermined range are problematic because excessive deviations could potentially disrupt the overall polyhedral structure generated in the previous stage. The final steps ensure that p2 and p3 are within the predetermined range. The projection onto this line lies at one-third of the distance along this line. However, in certain specific cases, the above two equations contradict each other, in which case the following condition applies.

[0119] ∠p2p1p4=σ,

[0120] ∠p3p4p1=σ,

[0121] Once the internal boundary, represented by a cubic Bezier curve, is constructed, its degree can be further increased to p without changing its basic shape. Figure 4 The last image shows the final decomposition structure of the giraffe domain.

[0122] 2) Construct a global G 1 Smooth parameterized model

[0123] First, the discrete Coons interpolation method is used to calculate the initial multi-piece parameterization.

[0124] For the first... represented by the rectangular Bezier surface A quadrilateral facet, Its boundary control points are Initial control point The structure is

[0125]

[0126] For the A triangular Bezier surface represented by a triangular patch. The formula for calculating internal control points that satisfy i+j+k=p is:

[0127]

[0128] Then apply G to the facet interface 1 The continuity condition is calculated to satisfy the global G. 1 Smooth control points.

[0129] Let a p-order Bezier curve

[0130]

[0131] The control points are the common boundary of two adjacent rectangular Bezier faces. Then these two patches are G 1If scalar weight functions λ(u) and μ(u) exist continuously, then the following equations are given.

[0132] (1-λ(u))s(u)+λ(u)t(u)=(1-μ(u))q l (u)+μ(u)q r (u)

[0133] It holds for all u, where

[0134]

[0135]

[0136] and These are two rows of control points adjacent to the common boundary, such as Figure 5 As shown.

[0137] The functions λ(u) and μ(u) can be expressed in various mathematical forms. Here, they are set as linear and quadratic polynomials, respectively, as follows:

[0138]

[0139] Substituting λ(u) and μ(u) into the previous equation, we obtain the following G: 1 Continuity constraints

[0140]

[0141] They constitute a linear system.

[0142] Clearly, for some higher-order rational functions λ and μ, almost all pairs of patches with shared boundaries satisfy the fundamental G... 1 Continuity condition. This invention uses only linear and quadratic polynomials for parameterization. Only when a pair of patches satisfies the conditions described here are they considered to conform to G. 1 Standards of continuity.

[0143] When dealing with combinations of adjacent triangles and rectangles, first increase the order of the triangle from p to p+1. Then, the previously described method can be used to construct the aforementioned G. 1 Constraint system. There are no adjacent triangular patches here, because they have been merged into a single quadrilateral patch.

[0144] In this linear system, the boundary control points are known and fixed. s0,s p ,t0,t pThen, the remaining control points need to be calculated. The coefficients λ0, λ1 of λ(t) and the coefficients μ0, μ2 of μ(t) are determined by the first and last equations of the system, respectively. The coefficient μ1 of μ(t) is set to (μ0 + μ2) / 2. G 1 The continuous system is divided into two groups, namely, the vertex G. 1 Continuous system (corresponding to k = 0, 1, p, p+1) and edge G 1 Continuous system (corresponding to k = 2, ..., p-1).

[0145] For each vertex (singular or regular corner) in the constructed polyhedral structure, i.e., the intersection of ρ Bezier faces, control the vertices o1, o2, ..., o r The twist vector at the vertex should satisfy the G corresponding to k=1 or p in the system. 1 Continuity constraints. All these constraints constitute the linear system Vx. v =y v , where vector x v Representative control point The set can be obtained by solving the following problem.

[0146]

[0147] stVx v =y v ,

[0148] x v initial value It is constructed using the discrete Coons interpolation method mentioned earlier.

[0149] For each common edge (boundary) shared by two Bezier faces, the two rows of control points adjacent to that boundary... and The G condition corresponding to k = 2, ..., p-1 should be satisfied. 1 Continuity constraints. These constraints constitute a linear system Ex. e =y e , where vector x e Representative control point and The set can be obtained by solving the following problem.

[0150]

[0151] stEx e =y e

[0152] x e initial value It is also constructed using the discrete Coons interpolation method. The constraints of the two problems mentioned above always have solutions, therefore both problems have a global optimal solution.

[0153] Except for the control points adjacent to the common boundary, the remaining control points of each patch are retained as control points obtained by Coons interpolation or by degree elevation. Figure 6 The two figures on the left show the calculated G of the giraffe domain. 1 Smoothing parameterization reveals that the resulting mapping is not globally injective (||μ) g || ∞ =1.48>1).

[0154] 3) High-quality parameterization of each facet is obtained using QC mapping technology.

[0155] The QC mapping is locally injective, and its angular distortion can be measured using the Beltrami coefficient. This invention uses this framework to generate high-quality parameterizations for each facet. Here, a quadrilateral facet is used as an example; similar processing should be applied to the triangular case.

[0156] Since QC mappings are defined in complex form, first let the first... The parameterization of a quadrilateral patch (i.e., a rectangular Bezier surface) is rewritten as a complex function.

[0157]

[0158] in and here and Control point The components are as follows. In this parameterization, the boundary control points and control points adjacent to the common boundary are known, and the remaining internal control points are determined by solving the following optimization problem.

[0159]

[0160] in yes The Beltrami coefficient, where the parameter γ is the positive weight. and They represent and The Hessian matrix. In the above model, the first term of the objective function measures the conformal distortion of the mapping, while the second term relates to the uniformity of the parameterization. The constraints ensure the injectivity of the parameterization.

[0161] because and The components are coupled together, and solving this nonlinear and nonconvex problem is not an easy task. Therefore, an auxiliary variable v is introduced and used to replace the nonlinear and nonconvex components. This leads to an interleaved problem.

[0162]

[0163] ||n|| ∞ <1.

[0164] This is further relaxed to such a model

[0165]

[0166] st||n|| ∞ <1.

[0167] The weight η is relatively large. To efficiently process the model, an energy descent method is used to alternately solve and optimize the two subproblems. The specific iterative process is as follows: starting from constructing the initial G... 1 Parameterization Initially, solve v alternately. k and

[0168]

[0169] Among them, v k By fixed The model was then solved. By fixed v k The model is then solved. This process continues until, for a pre-given threshold ε, the solution of the model satisfies the condition ||v. k+1 -v k ||<ε until.

[0170] In each iteration of the alternating solution process, there are several types of integrations, such as...

[0171]

[0172]

[0173] It needs to be calculated, where f1 is a rational function, given by... The partial derivatives of a bivariate function are expressed as follows: The value is

[0174]

[0175] Where f2 is v k The real part of f3 is v kThe imaginary part of the integral. In each iteration, the Bernstein basis functions appearing in the integral remain unchanged. Inspired by the validity evaluation of the integral generated in isogeometric discretization, the values ​​and derivatives of these basis functions are pre-computed. Therefore, a lookup strategy is used to compute the integral. Figure 6 The two figures on the right show the parameterization results of the giraffe domain based on the QC mapping, which are significantly better than... Figure 6 The initial results are depicted in the two figures on the left.

[0176] The various embodiments in this specification are described in a progressive manner. Similar or identical parts between embodiments can be referred to interchangeably. Each embodiment focuses on its differences from other embodiments. In particular, for the device embodiments, the above descriptions are merely preferred embodiments of the present invention. Since they are fundamentally similar to the method embodiments, the descriptions are relatively simple, and relevant parts can be referred to the descriptions of the method embodiments. The above descriptions are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention, without departing from the principle of the present invention, should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A planar parameterization method applicable to complex geometric regions and possessing G1 smoothness characteristics, characterized by the following steps: 1) Automatic region decomposition with progressive refinement, using triangular and rectangular Bezier patches to represent complex domains, specifically including: 1.1) Extract the skeleton based on the region boundary using a sparse optimization method; the skeleton of the object depicts its complete shape and reflects its structure, and performs domain decomposition based on the skeleton information of the input shape; the skeleton is extracted using a sparse optimization method, and a compact median transformation is calculated for any two-dimensional object represented by a point cloud containing noise or missing data; then the branch points, branches, the maximum inscribed circle centered at each branch point, and the corresponding tangent points on the skeleton are determined. 1.2) The skeleton guides the initial decomposition process, connects the corresponding tangent points on the domain boundary, and automatically generates triangles and rectangular patches to construct the basic geometric structure; 1.3) Based on the initial decomposition, further identify the shapes of the remaining facets and select appropriate corner points for each facet; 1.4) Further refine the decomposition of the region, and construct the mathematical model of the resulting patch boundary segments using Bezier curves, as follows: 1.4.1) For a given set of N discrete points The outer boundary segment is represented by a chord length parameterization method to determine the relevant parameters. Then, a p-order Bezier curve C(u) is fitted using control points c. This process is achieved by minimizing... Obtain; among them, λ is the positive weight, the regularization term. Used to measure the smoothness of the resulting curve; 1.4.2) Verify the mean square error as defined by the following formula: Is it greater than 4×10 -8 The specified tolerance is used to evaluate the fitting accuracy; if the error value exceeds the specified tolerance, the corresponding patches need to be further decomposed. 1.4.3) Processing internal boundary segments: For internal boundary line segments with two endpoints p1 and p4, interpolate at p1 and p4 to construct cubic Bezier curves to replace the original line segments; then construct two other control points p2 and p3 based on the tangent directions of p1 and p4. 2) Construct a global G 1 Smoothed parametric model: 2.1) The initial multi-piece parameterization is calculated using the discrete Coons interpolation method; 2.2) Apply G to the facet interface 1 The continuity condition is calculated to satisfy the global G. 1 Smooth control points; 3) Employ QC mapping technology to obtain high-quality parameterization for each facet, as detailed below: 3.1) The first The parameterization of a quadrilateral patch, i.e., a rectangular Bezier surface, can be rewritten as a complex function. in and here and Control point The components; in this parameterization, the boundary control points and control points adjacent to the common boundary are known, and the remaining internal control points are determined by solving the following optimization problem: in yes The Beltrami coefficient, where the parameter γ is the positive weight. and They represent and The Hessian matrix; 3.2) Introduce an auxiliary variable v and replace it with... This leads to an interleaved problem. ‖n‖ ∞ <1. This is further relaxed to the following model: st‖n‖ ∞ <1. The weight η is relatively large; 3.3) An energy descent method is used to alternately solve and optimize the two subproblems; 3.4) This process continues until, for a pre-given threshold ∈, the solution of the model satisfies the condition ‖v k+1 -v k ‖<∈ until.

2. The planar parameterization method applicable to complex geometric regions and possessing G1 smoothness characteristics according to claim 1, characterized in that: The process of extracting the skeleton described in step 1.1) is as follows: Assume c t (u) is the terminal branch spline curve connected to the branch point b, r b Given the radius of the largest incircle centered at b, delete the terminal branches that satisfy the following conditions. The left-hand portion represents the arc length of the terminal branch, r. max It is the maximum value of the radius of the circle centered at each branch point, where α and β are positive parameters.

3. The planar parameterization method according to claim 1, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: The process of obtaining corner points described in step 1.3) is as follows: 1.3.1) After completing the initial decomposition of region Ω, further analyze and identify the remaining facets to determine whether there are facets that can be approximated as triangles; 1.3.2) After identifying and distinguishing the triangular facets, the remaining facets are classified as quadrilateral facets according to the default rules. For each quadrilateral facet, four corner points need to be selected on the domain boundary. Two of the corner points have been determined and can be used as corner points. For the remaining two corner points, an automated method based on the optimal transmission principle is used for selection.

4. The planar parameterization method according to claim 3, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: Step 1.31) The specific process of determining whether there is a facet that can be approximated as a triangle is as follows: For a facet... p for One of the vertices, Boundary (edge) )for An edge that lies inside Ω is also called an internal boundary; represented by L P Represents a piece of dough The perimeter, L p Let δ represent the perimeter of Δppp, and let δ>0 as a threshold; if the facet satisfy: Then it is believed Since it is a triangle, point p and the two endpoints p′ and p of the corresponding internal boundary are selected as the three corner points required for subsequent parameterization.

5. The planar parameterization method according to claim 1, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: The specific process of decomposing the corresponding patch in step 1.42) is as follows: For the longer segment of the two boundaries, select a point k1, which is approximately the same geodesic distance from the two endpoints, and find its nearest point k2 on the opposite boundary; by connecting k1 and k2, decompose the patch into two parts; this decomposition and fitting process will be iteratively performed until the boundary fitting error of all patches is reduced to below the specified tolerance.

6. The planar parameterization method for complex geometric regions with G1 smoothness characteristics according to claim 1 or 5, characterized in that: Step 1.43) The specific process is as follows: set up It is the tangent vector of two adjacent Bezier curves at p1, and It is the tangent vector of two adjacent Bezier curves at p4, and the control points p2 and p3 are constructed as follows: and Where ρ and These are the degrees of p1 and p4, respectively, and θ is determined by... and The resulting included angle, It is by and The resulting included angle, where σ is the threshold; When the two equations above contradict each other, the following condition applies.

7. The planar parameterization method according to claim 1, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: The multi-chip parameterization process described in step 2.1) is as follows: For the first... represented by the rectangular Bezier surface A quadrilateral facet, Its boundary control points are Initial control point The structure is as follows: For the A triangular Bezier surface represented by a triangular patch. The formula for calculating internal control points that satisfy i+j+k=p is:

8. The planar parameterization method according to claim 1, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: Step 2.2) describes applying G to the facet interface. 1 The process for establishing continuity conditions is as follows: Let a p-order Bezier curve The control points are the common boundary of two adjacent rectangular Bezier faces. Then these two patches are G 1 Continuous; if there exist scalar weight functions λ(u) and μ(u) such that the following equations hold: (1-λ(u))s(u)+λ(u)t(u)=(1-μ(u))q l (u)+μ(u)q r (u) It holds for all u, where and These are two rows of control points adjacent to the common boundary; Here, the functions λ(u) and μ(u) are set as linear and quadratic polynomials, respectively, as shown below: Substituting λ(u) and μ(u) into the previous equation, we obtain the following G: 1 Continuity constraints: They constitute a linear system; When processing combinations of adjacent triangles and rectangles, the order of the triangle is first increased from p to p+1, and then the aforementioned G is constructed. 1 Constraint system.

9. The planar parameterization method according to claim 8, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: In the above G 1 In the constraint system, the boundary control points are known and fixed. s0,s p ,t0,t p Then the remaining control points need to be calculated; The coefficients λ0, λ1 of λ(t) and the coefficients μ0, μ2 of μ(t) are determined by the first and last equations of the system, respectively; the coefficient μ1 of μ(t) is set to (μ0 + μ2) / 2; G 1 The continuous system is divided into two groups, namely, the vertex G. 1 Continuous system (corresponding to k = 0, 1, p, p+1) and edge G 1 A continuous system (corresponding to k = 2, ..., p-1); for each vertex, singular corner, or regular corner in the constructed polyhedral structure, i.e., the intersection of ρ Bezier faces, the controlling vertices o1, o2, ..., o ρ The twist vector at the vertex should satisfy the G corresponding to k=1 or p in the system. 1 Continuity constraints; all these constraints constitute the linear system Vx v =y v , where vector x v Representative control point The set is obtained by solving the following problem. s.t.Vx v =y v , x v initial value It is constructed using the discrete Coons interpolation method; For each common edge shared by two Bezier faces, the two rows of control points adjacent to that boundary... and The G condition corresponding to k = 2, ..., p-1 should be satisfied. 1 Continuity constraints; these constraints constitute a linear system Ex e =y e , where vector x e Representative control point and The set is obtained by solving the following problem. s.t.Ex e =y e x e initial value Constructed using discrete Coons interpolation.

10. The planar parameterization method according to claim 1, applicable to complex geometric regions and possessing G1 smoothness characteristics, is characterized in that: The specific iterative process of the energy descent method described in step 3.3) is as follows: starting from constructing the initial G... 1 Parameterization Initially, solve v alternately. k and Among them, v k By fixed The model was then solved. By fixed v k The model was then solved.

Images