Compression of Currents and Varifolds
This work addresses the computational bottleneck in shape analysis for geometry processing applications, offering a scalable compression method with practical speedups.
The authors developed a faster algorithm for compressing currents and varifolds representations of shapes using ridge leverage score sampling and Nyström approximation, achieving rapid error decay with theoretical guarantees. Their method accelerates downstream tasks like nonlinear shape registration in LDDMM without quality loss, even at high compression ratios, as demonstrated on large-scale geometry datasets.
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.