网格曲面的分割算法及其应用
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摘要
网格曲面分割是数字几何处理中的重要组成部分,在包括网格参数化、网格简化、模型匹配、网格压缩、网格编辑等众多数字几何处理领域中都有着重要的应用。由于网格曲面分割算法应用范围广,实用性强,已经成为当前数字几何领域中的研究热点之一。本文围绕网格曲面的分割算法,及其在网格简化中的应用,展开了深入的研究,主要获得以下成果：
1.为了根据网格模型上的尖锐几何特征对三角网格曲面进行合理分片,提出一种新的基于张量投票(tensor voting)理论的三角网格曲面分割算法。该算法将给定的网格模型上所有的三角面片聚类成若干个集合,使得集合内部三角面片上点的尖锐几何特征尽可能接近。根据网格模型顶点上基于法向的张量投票矩阵的特征值分布与顶点尖锐几何特征的对应关系,算法将网格分割问题转化为目标能量函数最小化问题,并适当简化能量函数的形式,用快速聚类算法求解。通过引入启发式约束,算法较好地防止了分割区域的分离。实验表明,与已有算法相比,该算法具有较快的速度,同时能够较好地分割网格曲面上的尖锐几何特征区域。
2.以分割后得到的子网格曲面片的尽量接近平面为目标,提出一种新的基于平坦性的三角网格曲面分割算法。该算法通过建立刻划子网格曲面片平坦性的能量函数,并用聚类方法将该能量函数极小化,最终使得分割得到的子网格曲面片尽量平坦。与经典的网格分割算法相比,该算法具有实现简单,运行效率高,保证收敛,以及分割效果好等特点。对中等规模或以下的模型,算法能够在普通PC机上达到实时的效果。
3.提出一种适用于CAD模型的快速三角网格分割算法。给定一个原始网格曲面及目标分割数,算法通过构造反映子网格内部曲率相似度的能量函数,将网格分割问题转化为能量函数最大化问题,并通过最大化该能量函数对原始网格中三角面片进行聚类,从而将网格分割成用户指定数量的子网格片。实验表明算法是快速和有效的,对于中小网格模型可以达到实时或接近实时的速度。
4.给出一种新的自适应三角网格简化算法。算法主要包含两个步骤,首先通过红绿细分算子将输入的网格模型按照网格曲面上的曲率分布情况进行细分。然后将细分后的网格模型通过基于重心Voronoi剖分(centroidal Voronoi tessellations)的聚类算法进行简化。简化后的模型与原网格模型相比具有几何误差小,三角面片质量高等特点。实验结果表明该算法是鲁棒和有效的。
以上算法以一个统一的网格曲面分割算法框架为基础,采用不同的几何特征作为指标进行聚类求解,取得了较好的效果。同时,该算法框架还具有运行效率较高,适合于大型网格曲面的特点。如何在本文给出的算法框架基础上,进一步结合网格曲面的语义特征进行网格曲面分割,是未来要研究的问题之一
Mesh segmentation has become a key ingredient of digital geometry processing. It assists mesh parameterization, mesh simplification, mesh compression, mesh editing and more. Due to the widely usage of mesh segmentation, it has become a very hot research area in digital geometry processing. In this thesis, we made a profoundly research on mesh segmentation and its application in mesh simplification. The main contributions are listed as follows:
1. To segment the meshes according to the sharp features on the mesh models, we present a new mesh segmentation algorithm based on the tensor voting theory. All the triangles on the mesh cluster to the user-defined number of regions, such that the sharp features of triangles in the same region are as similar as possible. According to the correspondence between the distribution of eigenvalues from the tensor voting matrices based on normals and the sharp features on the mesh model, we converts the mesh segmentation problem to an energy term minimization problem. We simplified the energy term and solve the minimization problem by clustering. By introducing an heuristic constraint, we successfully prevent the separation of regions. The test results show that our algorithm can obtain much higher performance compared with existing algorithm. At the same time, it can detect the sharp area better.
2. We present a novel mesh segmentation algorithm based on the planarity, whose target is to make the result regions as planar as possible. The algorithm maximizes regions' planarity by driving an energy term which reflects the planarity of regions down on the mesh directly. It is easy to implement and extremely efficient. The convergence is guaranteed and the results are better compared with existing method. For models of moderate size, interactive performance is achieved with commodity PCs.
3. We present a fast method for segmentation, which is suitable for commonly used CAD models. Given a mesh surface, all faces of it cluster to a user-specified number of patches according to similarity of curvatures. Experimental results show that our algorithm is efficient and robust.
4. We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on Centroidal Voronoi Tessellations (CVTs). The accuracy and good quality of the output triangular mesh is achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.
The algorithms described above are built upon a unified mesh segmentation framework by adopting different geometrical features. And they can not only obtain good segmentation results, but also achieve high performance. How to combine our algorithm framework and semantic feature on the mesh surface is one of our future work.
1.为了根据网格模型上的尖锐几何特征对三角网格曲面进行合理分片,提出一种新的基于张量投票(tensor voting)理论的三角网格曲面分割算法。该算法将给定的网格模型上所有的三角面片聚类成若干个集合,使得集合内部三角面片上点的尖锐几何特征尽可能接近。根据网格模型顶点上基于法向的张量投票矩阵的特征值分布与顶点尖锐几何特征的对应关系,算法将网格分割问题转化为目标能量函数最小化问题,并适当简化能量函数的形式,用快速聚类算法求解。通过引入启发式约束,算法较好地防止了分割区域的分离。实验表明,与已有算法相比,该算法具有较快的速度,同时能够较好地分割网格曲面上的尖锐几何特征区域。
2.以分割后得到的子网格曲面片的尽量接近平面为目标,提出一种新的基于平坦性的三角网格曲面分割算法。该算法通过建立刻划子网格曲面片平坦性的能量函数,并用聚类方法将该能量函数极小化,最终使得分割得到的子网格曲面片尽量平坦。与经典的网格分割算法相比,该算法具有实现简单,运行效率高,保证收敛,以及分割效果好等特点。对中等规模或以下的模型,算法能够在普通PC机上达到实时的效果。
3.提出一种适用于CAD模型的快速三角网格分割算法。给定一个原始网格曲面及目标分割数,算法通过构造反映子网格内部曲率相似度的能量函数,将网格分割问题转化为能量函数最大化问题,并通过最大化该能量函数对原始网格中三角面片进行聚类,从而将网格分割成用户指定数量的子网格片。实验表明算法是快速和有效的,对于中小网格模型可以达到实时或接近实时的速度。
4.给出一种新的自适应三角网格简化算法。算法主要包含两个步骤,首先通过红绿细分算子将输入的网格模型按照网格曲面上的曲率分布情况进行细分。然后将细分后的网格模型通过基于重心Voronoi剖分(centroidal Voronoi tessellations)的聚类算法进行简化。简化后的模型与原网格模型相比具有几何误差小,三角面片质量高等特点。实验结果表明该算法是鲁棒和有效的。
以上算法以一个统一的网格曲面分割算法框架为基础,采用不同的几何特征作为指标进行聚类求解,取得了较好的效果。同时,该算法框架还具有运行效率较高,适合于大型网格曲面的特点。如何在本文给出的算法框架基础上,进一步结合网格曲面的语义特征进行网格曲面分割,是未来要研究的问题之一
Mesh segmentation has become a key ingredient of digital geometry processing. It assists mesh parameterization, mesh simplification, mesh compression, mesh editing and more. Due to the widely usage of mesh segmentation, it has become a very hot research area in digital geometry processing. In this thesis, we made a profoundly research on mesh segmentation and its application in mesh simplification. The main contributions are listed as follows:
1. To segment the meshes according to the sharp features on the mesh models, we present a new mesh segmentation algorithm based on the tensor voting theory. All the triangles on the mesh cluster to the user-defined number of regions, such that the sharp features of triangles in the same region are as similar as possible. According to the correspondence between the distribution of eigenvalues from the tensor voting matrices based on normals and the sharp features on the mesh model, we converts the mesh segmentation problem to an energy term minimization problem. We simplified the energy term and solve the minimization problem by clustering. By introducing an heuristic constraint, we successfully prevent the separation of regions. The test results show that our algorithm can obtain much higher performance compared with existing algorithm. At the same time, it can detect the sharp area better.
2. We present a novel mesh segmentation algorithm based on the planarity, whose target is to make the result regions as planar as possible. The algorithm maximizes regions' planarity by driving an energy term which reflects the planarity of regions down on the mesh directly. It is easy to implement and extremely efficient. The convergence is guaranteed and the results are better compared with existing method. For models of moderate size, interactive performance is achieved with commodity PCs.
3. We present a fast method for segmentation, which is suitable for commonly used CAD models. Given a mesh surface, all faces of it cluster to a user-specified number of patches according to similarity of curvatures. Experimental results show that our algorithm is efficient and robust.
4. We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on Centroidal Voronoi Tessellations (CVTs). The accuracy and good quality of the output triangular mesh is achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.
The algorithms described above are built upon a unified mesh segmentation framework by adopting different geometrical features. And they can not only obtain good segmentation results, but also achieve high performance. How to combine our algorithm framework and semantic feature on the mesh surface is one of our future work.
引文
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