Progressive Convex Hull Simplification

TL;DR

This paper introduces an efficient greedy algorithm for simplifying convex hulls by reducing their half-spaces while minimizing volume or surface area increase, improving efficiency and tightness.

Contribution

It presents a novel $O(n \,\log n)$ greedy optimization method for convex hull simplification in dual space, outperforming existing approaches.

Findings

The method achieves efficient convex hull simplification with minimal volume increase.

It outperforms existing methods in efficiency, tightness, and safety.

Demonstrated success on various shapes and applications.

Abstract

Convex hulls are useful as tight bounding proxies for a variety of tasks including collision detection, ray intersection, and distance computation. Unfortunately, the complexity of polyhedral convex hulls grows linearly with their input. We consider the problem of conservatively simplifying a convex hull to a specified number of half-spaces while minimizing added volume or surface area. By working in the dual representation, we propose an efficient $O(n \log n)$ greedy optimization. In comparisons, we show that existing methods either exhibit poor efficiency, tightness or safety. We demonstrate the success of our method on a variety of input shapes and downstream application domains.