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Seamless Parametrization with Arbitrary Cones for Arbitrary Genus

机译:具有任意锥度的任意属的无缝参数化

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Seamless global parametrization of surfaces is a key operation in geometry processing, e.g., for high-quality quad mesh generation. A common approach is to prescribe the parametric domain structure, in particular, the locations of parametrization singularities (cones), and solve a non-convex optimization problem minimizing a distortion measure, with local injec-tivity imposed through either constraints or barrier terms. In both cases, an initial valid parametrization is essential to serve as a feasible starting point for obtaining an optimized solution. While convexified versions of the constraints eliminate this initialization requirement, they narrow the range of solutions, causing some problem instances that actually do have a solution to become infeasible. We demonstrate that for arbitrary given sets of topologically admissible parametric cones with prescribed curvature, a global seamless parametrization always exists (with the exception of one well-known case). Importantly, our proof is constructive and directly leads to a general algorithm for computing such parametrizations. Most distinctively, this algorithm is bootstrapped with a convex optimization problem (solving for a confor-mal map), in tandem with a simple linear equation system (determining a seamless modification of this map). This initial map can then serve as a valid starting point and be optimized for low distortion using existing injectivity preserving methods.
机译:表面的无缝全局参数化是几何处理中的关键操作,例如,用于生成高质量四边形网格。一种常见的方法是规定参数域结构,特别是参数化奇异点(圆锥)的位置,并通过将约束或势垒项强加局部注射来解决最小化失真度量的非凸优化问题。在这两种情况下,初始有效参数化对于获得优化解决方案的可行起点都是必不可少的。尽管约束的凸版本消除了此初始化要求,但它们缩小了解决方案的范围,导致实际上确实具有解决方案的某些问题实例变得不可行。我们证明,对于具有给定曲率的任意给定集合的拓扑上可接受的参数锥,始终存在全局无缝参数化(一个众所周知的情况除外)。重要的是,我们的证明是建设性的,直接导致了用于计算此类参数化的通用算法。最有特色的是,此算法与一个简单的线性方程组(确定此图的​​无缝修改)并排,出现一个凸优化问题(求解一个保形图)。然后,该初始图可以用作有效的起点,并使用现有的注入保留方法对低失真进行优化。

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