Computing the alpha complex using dual active set methods

TL;DR

This paper introduces a novel dual active set method for computing the alpha complex in computational geometry, avoiding full Delaunay triangulation and improving efficiency in higher dimensions.

Contribution

It presents a dual quadratic programming approach that efficiently computes the alpha complex by ruling out simplices, bypassing the need for full Delaunay triangulation.

Findings

Reduces computational complexity for high-dimensional alpha complexes

Avoids full Delaunay triangulation by ruling out non-simplices

Applicable to weighted and unweighted alpha complexes

Abstract

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" $\sigma = \{x_0, ..., x_k \} \subset S$, 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.