Optimal Cone Singularities for Conformal Flattening
Optimal Cone Singularities for Conformal Flattening
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DOI：
10.1145/3197517.3201367
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发表时间：
2018-08-01
影响因子：
6.2
通讯作者：
Crane, Keenan
中科院分区：
文献类型：
--
作者：
Soliman, Yousuf;Slepcev, Dejan;Crane, Keenan
保角或保角曲面参数化已被证明是从几何处理到数字制造再到机器学习的各种应用中的一种强大工具，但保角映射仍然会遭受严重的面积失真。锥奇异性提供了一种减轻这种失真的方法，但是找到锥的最佳配置是非常困难的。本文开发了一种策略，是全局最优的意义上说，它最大限度地减少总面积失真之间的所有可能的锥配置（数量，位置和大小），有不超过固定的总锥角。一个关键的见解是，为了优化的目的，不应该直接与曲率测量（自然代表锥配置），而是可以应用Fenchel-Rockafellar对偶来获得只涉及普通函数的公式。结果是一个凸优化问题，可以通过一系列稀疏线性系统来解决，这些系统很容易从通常的余切拉普拉斯算子中构建出来。该方法支持用户定义的重要性概念、对锥角的约束（例如，正的或在给定范围内），以及复杂的边界条件（例如，凸形或多边形）。我们将我们的方法与以前的技术在各种具有挑战性的模型上进行比较，通常可以实现显着降低的失真，并证明全局最优性在存在噪声或离散化差的情况下具有极强的鲁棒性。
Angle-preserving or conformal surface parameterization has proven to be a powerful tool across applications ranging from geometry processing, to digital manufacturing, to machine learning, yet conformal maps can still suffer from severe area distortion. Cone singularities provide a way to mitigate this distortion, but finding the best configuration of cones is notoriously difficult. This paper develops a strategy that is globally optimal in the sense that it minimizes total area distortion among all possible cone configurations (number, placement, and size) that have no more than fixed total cone angle. A key insight is that, for the purpose of optimization, one should not work directly with curvature measures (which naturally represent cone configurations), but can instead apply Fenchel-Rockafellar duality to obtain a formulation involving only ordinary functions. The result is a convex optimization problem, which can be solved via a sequence of sparse linear systems easily built from the usual cotangent Laplacian. The method supports user-defined notions of importance, constraints on cone angles (e.g., positive, or within a given range), and sophisticated boundary conditions (e.g., convex, or polygonal). We compare our approach to previous techniques on a variety of challenging models, often achieving dramatically lower distortion, and demonstrating that global optimality leads to extreme robustness in the presence of noise or poor discretization.