Sparse Randomized Approximation of Normal Cycles
This work addresses computational efficiency in geometry processing for researchers and practitioners, but it is incremental as it extends prior methods for currents and varifolds to normal cycles.
The authors tackled the problem of compressing shape representations using normal cycles by developing a compression algorithm based on Nyström approximation and ridge leverage score sampling, achieving theoretical error guarantees and demonstrating effectiveness in accelerating nonlinear shape registration tasks with high compression ratios.
We extend our work for compression of currents and varifolds to a compression algorithm for the embedded normal cycles representation of shape, restricted to the constant normal kernel case, using the Nystrom approximation in Reproducing Kernel Hilbert Spaces (RKHS) and ridge leverage score (RLS) sampling. Our method comes with theoretical guarantees on the compression error decay, and the approximations are shown to be effective for downstream tasks such as nonlinear shape registration in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework, 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 accelerate downstream registration tasks significantly.