Compression of Currents and Varifolds

TL;DR

This paper introduces a fast, theoretically-guaranteed compression algorithm for shape representations using RLS sampling and Nystrom approximation, improving efficiency in shape analysis tasks.

Contribution

It presents a novel compression method for currents and varifolds with theoretical error decay guarantees, enhancing speed and scalability in shape processing.

Findings

Faster compression than existing methods

Theoretical guarantees on error decay

Effective in large-scale shape analysis

Abstract

We derive an algorithm for compression of the currents and varifolds representations of shapes, using ridge leverage score (RLS) sampling, and the theory of Nystrom approximation in Reproducing Kernel Hilbert Spaces. Our method is faster than existing compression techniques and comes with theoretical guarantees on the rate of decay of the compression error as a function of the smoothness of the associated shape representation. The obtained compressions are shown to be useful for accelerating downstream tasks such as nonlinear shape registration in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework without loss of quality, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets, and is shown to be fast and scalable with rapid error decay.