Succinct Representation of Well-Spaced Point Clouds

TL;DR

This paper introduces a lossily compressed point cloud representation that maintains approximate inter-point distances and supports standard spatial operations efficiently, enabling space-efficient surface reconstruction and mesh refinement.

Contribution

It presents a novel O(n)-bit lossy compression method for point clouds that preserves approximate distances and supports key spatial operations efficiently.

Findings

Achieves 3x space savings over traditional representations.

Maintains inter-point distances within 10% error.

Supports quadtree operations and Voronoi cell computation in near-logarithmic time.

Abstract

A set of n points in low dimensions takes Theta(n w) bits to store on a w-bit machine. Surface reconstruction and mesh refinement impose a requirement on the distribution of the points they process. I show how to use this assumption to lossily compress a set of n input points into a representation that takes only O(n) bits, independent of the word size. The loss can keep inter-point distances to within 10% relative error while still achieving a factor of three space savings. The representation allows standard quadtree operations, along with computing the restricted Voronoi cell of a point, in time O(w^2 + log n), which can be improved to time O(log n) if w is in Theta(log n). Thus one can use this compressed representation to perform mesh refinement or surface reconstruction in O(n) bits with only a logarithmic slowdown.