Smooth Distance Approximation

Ahmed Abdelkader, D. Mount
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Abstract

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in $\mathbb{R}^d$ from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute $\varepsilon$-approximation to this query in time $O(\log (1/\varepsilon))$ using $O(1/\varepsilon^{d/2})$ storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian.
平滑距离近似
计算几何中的传统问题涉及离散和连续两方面。其中一个例子是最近邻搜索,其中输入是离散的,但结果取决于连续变化的距离。在几何数据结构的许多实际应用中,假设查询结果是连续的,没有跳转不连续。这与计算几何中的许多现代数据结构不一致,它们使用近似值来实现效率,但这些近似值经常受到不连续的影响。在本文中,我们提出了一种将近似但不连续的数据结构转化为产生光滑近似的数据结构的一般方法,同时匹配原始数据结构的渐近空间效率。我们通过采用一种称为统一分割法的方法来实现这一点,该方法将多个局部近似平滑地混合到一个光滑的全局近似中。我们在一个具体的应用中说明了这种技术的使用,即从其内部的任何一点近似到$\mathbb{R}^d$中的凸多面体边界的距离。我们首先开发一种新的数据结构,它使用$O(1/\varepsilon^{d/2})$存储空间在$O(\log (1/\varepsilon))$时间上有效地计算这个查询的绝对$\varepsilon$ -近似值。然后,我们将提出的统一分割混合方法应用于保证近似距离场的光滑性,建立了其梯度和Hessian范数的最优渐近界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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