使用对偶有效集方法计算 alpha 复形
Computing the alpha complex using dual active set methods
AI总结：
针对高维下先求完整 Delaunay 复形导致 alpha 复形计算不可行的问题，本文利用 Lagrange 对偶提出基于对偶二次规划、以排除单纯形为目标的直接计算方法。
摘要：
alpha 复形是计算几何中的一种基础数据结构，它编码了点集 S 中各球 B(x; r) ⊂ ℝ^m 之并的拓扑类型，包括允许半径变化的加权版本。它由“单纯形” σ = {x_0, ..., x_k} ⊂ S 的集合构成，这些单纯形对应于半径受限版本 Voronoi 图中各单元的非空 (k+1) 重交集。现有的 alpha 复形计算算法要求点位于低维空间中，因为它们首先计算整个 Delaunay 复形；即使 alpha 复形本身规模合理，这一步也会迅速变得难以处理。本文提出一种无需计算完整 Delaunay 三角剖分即可计算 alpha 复形的方法，该方法运用 Lagrange 对偶，具体而言是一种基于对偶二次规划的算法，其目标是排除单纯形，而非将单纯形纳入。
英文摘要：
The alpha complex is a fundamental data structure from computational geometry, which encodes the topological type of a union of balls $B(x; r) \subset \mathbb{R}^m$ for $x\in S$, including a weighted version that allows for varying radii. It consists of the collection of "simplices" $σ= \{x_0, ..., x_k \} \subset S$, which correspond to nomempty $(k + 1)$-fold intersections of cells in a radius-restricted version of the Voronoi diagram. Existing algorithms for computing the alpha complex require that the points reside in low dimension because they begin by computing the entire Delaunay complex, which rapidly becomes intractable, even when the alpha complex is of a reasonable size. This paper presents a method for computing the alpha complex without computing the full Delaunay triangulation by applying Lagrangian duality, specifically an algorithm based on dual quadratic programming that seeks to rule simplices out rather than ruling them in.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
