Succinct Representation of Well-Spaced Point Clouds

09/17/2009

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by   Benoît Hudson, et al.

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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 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.