Computing inversion-free mappings by simplex assembly
Computing inversion-free mappings by simplex assembly
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DOI：
10.1145/2980179.2980231
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发表时间：
2016-11
期刊：
影响因子：
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通讯作者：
Xiao-Ming Fu;Yang Liu
中科院分区：
文献类型：
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作者：
Xiao-Ming Fu;Yang Liu
我们提出了一种新的方法，称为单纯形集合，用于计算在单纯形网格上具有低或有界失真的无逆映射。我们的方法包括两个步骤：单纯形分解和单纯形组装。给定一个单纯形网格及其初始分段仿射映射，我们将与每个单纯形相关联的仿射变换投影到无逆且有界的变形空间。投影将输入网格分解为不相交的简单。然后，通过最小化映射失真和不相交顶点相对于分段仿射变换的差异来组装不相交的简化以恢复原始连通性，而分段仿射映射被限制在可行空间内。由于使用了仿射变换作为变量，我们的方法明确地保证了在优化过程中不会出现逆单纯形，并且映射失真低于界限。与已有方法相比，我们的方法对具有多个倒置元素和位置约束的初始化具有较强的鲁棒性。我们通过各种几何处理任务展示了我们方法的有效性和稳健性。
We present a novel method, called Simplex Assembly, to compute inversion-free mappings with low or bounded distortion on simplicial meshes. Our method involves two steps: simplex disassembly and simplex assembly. Given a simplicial mesh and its initial piecewise affine mapping, we project the affine transformation associated with each simplex into the inversion-free and distortion-bounded space. The projection disassembles the input mesh into disjoint simplices. The disjoint simplices are then assembled to recover the original connectivity by minimizing the mapping distortion and the difference of the disjoint vertices with respect to the piecewise affine transformations, while the piecewise affine mapping is restricted inside the feasible space. Due to the use of affine transformations as variables, our method explicitly guarantees that no inverted simplex occurs, and that the mapping distortion is below the bound during the optimization. Compared with existing methods, our method is robust to an initialization with many inverted elements and positional constraints. We demonstrate the efficiency and robustness of our method through a variety of geometric processing tasks.