自适应三角剖分算法及其关键技术研究
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摘要
三角剖分在科学计算可视化、逆向工程、三维有限元方法的预处理、医学成像、三维扫描系统及地球物理等领域有着广泛的应用，是计算机辅助几何设计、几何造型及计算机图形学中的重要研究内容之一。实际应用中，实体模型通常包含大量的几何特征。在构建网格模型的过程中，希望在几何特征附近进行高分辨率分割，保证网格离散的几何精度和单元质量；在其余部分生成大尺度单元，避免网格规模不必要扩大，以减少存储空间和处理速度。利用人工控制分辨率达到上述目的费时且易错，开展自适应三角网格剖分研究可有效缓解上述瓶颈问题。
本文研究、改进和实现了自适应三角网格剖分的自动生成和拓扑修补。本文研究了基于移动抛物线逼近（MPA）的自适应三角剖分算法，对其进行修改和优化，结合Shepard插值和改进的八叉树方法，提出并实现基于Shepard插值的自适应三角剖分算法和框架。利用改进的算法，很容易实现包括薄片类、稀疏\均匀类、大规模复杂点云在内的各种点云的三角网格剖分。另外，本文针对点云数据三角化网格生成过程中产生的拓扑缺陷，提出基于数学形态学和拓扑规则相结合的网格拓扑修补算法。
本文针对三角剖分的自适应生成算法进行研究，在理解和掌握自适应三角剖分相关概念的基础上，对采用基于移动抛物线逼近（MPA）的自适应三角剖分算法克服经典三角剖分算法的一些不足的思想和过程进行了较为详细的论述。在此基础上，提出了将Shepard曲面插值与多尺度分析方法相结合，同时引入改进的八叉树搜索思想，计算点云中每个测量点的曲率，生成带自适应分辨率的分层空间栅格，最终实现三角网格重构。既节省内存，又减少了计算量，提高了算法的整体性能，且形成的三角网格质量较高，能够较好地再现原三维物体的细节特征，适用广泛。
本文针对散乱点云数据三角剖分过程中产生的拓扑缺陷，提出一种基于数学形态学运算和拓扑规则的网格拓扑修补算法。通过自适应分层栅格的缺陷识别技术分析有拓扑缺陷的区域，从而确定待修复区域的边界，然后用数学形态学开启运算和闭合运算去除该修复区域的拓扑缺陷。实验结果验证了该方法的可行性与准确性。
大多数算法在采用形态算子修复存在拓扑缺陷的区域后，要将体素集转化为二维流形网格，即重新对点云进行三角剖分。本文利用基于柄体理论（Handlebody理论）与星形理论（Stellar理论）的拓扑运算法则对待修复区域进行局部拓扑修改。应用实例表明，由于不需要对整个点云数据重新进行三角剖分，该算法具有运算速度快、结果准确性好的优点，并能较好地消除网格中的拓扑缺陷，有效地提高三角网格的显示精度，最终得到具有几何一致性和网格单元拓扑一致性的三角网格模型。
本文提出一种基于欧拉示性数及形状尺度因子的三角网格拓扑完备性检测算法，通过计算欧拉示性数，避免了点云重新三角化、几何微分属性重新估计等复杂的计算，使问题简化，从而大大提高了网格模型拓扑特征的提取效率；形状尺度因子的引入可以使我们在一个较大的尺度上来观察网格曲面，根据曲面的形状对不同尺度的曲面几何特征采用不同的尺度，即大特征曲面段用大尺度检测，小特征曲面段用小尺度检测，有效提高建模效率与重建模型的精度。
Triangulation for3D surfaces is widely used in such areas as the ScientificVisualization, reverse engineering, three-dimensional pretreatment of finite elementmethod, medical visualization,3D scanning system, etc., which is one of the keyparts in the field of computer-aided design, geometric modeling and computergraphics. As an input of surface mesh generation, many geometrical features areusually contained in complex geometry models. To generate a good triangular meshfor numerical analyses, the high resolution should be small near the features toachieve high geometry accuracy and element quality, and large elsewhere to avoidincreasing the number of mesh elements unnecessarily. However, manual resolutioncontrol on complex models is time consuming and prone to errors to achieve suchgoals. Instead, the adaptive triangulation algorithm for three dimensionalunorganized point clouds is capable of overcoming this bottleneck problem,effectively.
An adaptive triangulation algorithm for three dimensional unorganized pointclouds is proposed in this paper, since the most existing algorithms are not veryadaptable and are difficult to express the detail characters of the real surface well. Inthe proposed method，we combine4D Shepard surface with multi-resolution analysisand implement the modified octree algorithm, which the curvature of each point inthe point cloud is calculated. Then a hierarchical grid with adaptive resolution isconstructed for generating a triangular mesh from point clouds. Experimental resultsshow that the original algorithm can preserve more characters, but inefficient and notflexible; the improved algorithm is greatly advanced and generally applicable, andforms high quality triangle grid surface and reproduces initial3D object’s detailcharacters, which is suited to popularize in CAGD and surface modeling.
Limitations of current3D acquisition technology often lead to triangle meshesexhibiting a number of topological defects. In this paper we present a new methodfor model repair which takes as input an arbitrary mesh and outputs a valid2-manifold triangle mesh. By means of a defect detecting technology based onadaptive hierarchical grid, our method allows users to conveniently identify areaswith potential topological errors. Then using morphological operators, the topologyof the model can be modified. The experiment validate the feasibility and accuracy of the repair strategies.
The conversion of the corrected model back into a2-manifold triangle mesh isable to be finished quickly through topological mesh operators based on Handlebodyand Stellar theory. This approach avoids the formation of handles and cavities andguarantees a topologically correct reconstruction of the object’s surface.Experimental results show that the proposed algorithm is greatly advanced andgenerally applicable, and forms high quality triangle grid surface, which is suited topopularize in CAGD and surface modeling.
In this paper we present a topology completeness detection algorithm fortriangular meshes based on the Euler characterlstic and the scale factor. Bycalculating Euler characterlstic, some complex calculations, such as triangularmeshes reconstruction, differential geometry property re-estimate, etc., are avoided,which simplified the problem so that the extraction efficiency for topological featureof mesh model is greatly enhanced. Through introducing the scale factor, meshsurfaces are able to observed on larger scale. According to surface shape, differentscales are adopted in geometric features on different scales, which large scalesurfaces are detected on large scale, and small scale surfaces are detected on smallscale. The precision and efficiency of reconstructed model are improved remarkably.
本文研究、改进和实现了自适应三角网格剖分的自动生成和拓扑修补。本文研究了基于移动抛物线逼近（MPA）的自适应三角剖分算法，对其进行修改和优化，结合Shepard插值和改进的八叉树方法，提出并实现基于Shepard插值的自适应三角剖分算法和框架。利用改进的算法，很容易实现包括薄片类、稀疏\均匀类、大规模复杂点云在内的各种点云的三角网格剖分。另外，本文针对点云数据三角化网格生成过程中产生的拓扑缺陷，提出基于数学形态学和拓扑规则相结合的网格拓扑修补算法。
本文针对三角剖分的自适应生成算法进行研究，在理解和掌握自适应三角剖分相关概念的基础上，对采用基于移动抛物线逼近（MPA）的自适应三角剖分算法克服经典三角剖分算法的一些不足的思想和过程进行了较为详细的论述。在此基础上，提出了将Shepard曲面插值与多尺度分析方法相结合，同时引入改进的八叉树搜索思想，计算点云中每个测量点的曲率，生成带自适应分辨率的分层空间栅格，最终实现三角网格重构。既节省内存，又减少了计算量，提高了算法的整体性能，且形成的三角网格质量较高，能够较好地再现原三维物体的细节特征，适用广泛。
本文针对散乱点云数据三角剖分过程中产生的拓扑缺陷，提出一种基于数学形态学运算和拓扑规则的网格拓扑修补算法。通过自适应分层栅格的缺陷识别技术分析有拓扑缺陷的区域，从而确定待修复区域的边界，然后用数学形态学开启运算和闭合运算去除该修复区域的拓扑缺陷。实验结果验证了该方法的可行性与准确性。
大多数算法在采用形态算子修复存在拓扑缺陷的区域后，要将体素集转化为二维流形网格，即重新对点云进行三角剖分。本文利用基于柄体理论（Handlebody理论）与星形理论（Stellar理论）的拓扑运算法则对待修复区域进行局部拓扑修改。应用实例表明，由于不需要对整个点云数据重新进行三角剖分，该算法具有运算速度快、结果准确性好的优点，并能较好地消除网格中的拓扑缺陷，有效地提高三角网格的显示精度，最终得到具有几何一致性和网格单元拓扑一致性的三角网格模型。
本文提出一种基于欧拉示性数及形状尺度因子的三角网格拓扑完备性检测算法，通过计算欧拉示性数，避免了点云重新三角化、几何微分属性重新估计等复杂的计算，使问题简化，从而大大提高了网格模型拓扑特征的提取效率；形状尺度因子的引入可以使我们在一个较大的尺度上来观察网格曲面，根据曲面的形状对不同尺度的曲面几何特征采用不同的尺度，即大特征曲面段用大尺度检测，小特征曲面段用小尺度检测，有效提高建模效率与重建模型的精度。
Triangulation for3D surfaces is widely used in such areas as the ScientificVisualization, reverse engineering, three-dimensional pretreatment of finite elementmethod, medical visualization,3D scanning system, etc., which is one of the keyparts in the field of computer-aided design, geometric modeling and computergraphics. As an input of surface mesh generation, many geometrical features areusually contained in complex geometry models. To generate a good triangular meshfor numerical analyses, the high resolution should be small near the features toachieve high geometry accuracy and element quality, and large elsewhere to avoidincreasing the number of mesh elements unnecessarily. However, manual resolutioncontrol on complex models is time consuming and prone to errors to achieve suchgoals. Instead, the adaptive triangulation algorithm for three dimensionalunorganized point clouds is capable of overcoming this bottleneck problem,effectively.
An adaptive triangulation algorithm for three dimensional unorganized pointclouds is proposed in this paper, since the most existing algorithms are not veryadaptable and are difficult to express the detail characters of the real surface well. Inthe proposed method，we combine4D Shepard surface with multi-resolution analysisand implement the modified octree algorithm, which the curvature of each point inthe point cloud is calculated. Then a hierarchical grid with adaptive resolution isconstructed for generating a triangular mesh from point clouds. Experimental resultsshow that the original algorithm can preserve more characters, but inefficient and notflexible; the improved algorithm is greatly advanced and generally applicable, andforms high quality triangle grid surface and reproduces initial3D object’s detailcharacters, which is suited to popularize in CAGD and surface modeling.
Limitations of current3D acquisition technology often lead to triangle meshesexhibiting a number of topological defects. In this paper we present a new methodfor model repair which takes as input an arbitrary mesh and outputs a valid2-manifold triangle mesh. By means of a defect detecting technology based onadaptive hierarchical grid, our method allows users to conveniently identify areaswith potential topological errors. Then using morphological operators, the topologyof the model can be modified. The experiment validate the feasibility and accuracy of the repair strategies.
The conversion of the corrected model back into a2-manifold triangle mesh isable to be finished quickly through topological mesh operators based on Handlebodyand Stellar theory. This approach avoids the formation of handles and cavities andguarantees a topologically correct reconstruction of the object’s surface.Experimental results show that the proposed algorithm is greatly advanced andgenerally applicable, and forms high quality triangle grid surface, which is suited topopularize in CAGD and surface modeling.
In this paper we present a topology completeness detection algorithm fortriangular meshes based on the Euler characterlstic and the scale factor. Bycalculating Euler characterlstic, some complex calculations, such as triangularmeshes reconstruction, differential geometry property re-estimate, etc., are avoided,which simplified the problem so that the extraction efficiency for topological featureof mesh model is greatly enhanced. Through introducing the scale factor, meshsurfaces are able to observed on larger scale. According to surface shape, differentscales are adopted in geometric features on different scales, which large scalesurfaces are detected on large scale, and small scale surfaces are detected on smallscale. The precision and efficiency of reconstructed model are improved remarkably.
引文
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