Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

Point cloud (PC) —a collection of discrete geometric samples of a 3D object’s surface—is typically large, which entails expensive subsequent operations. Thus, PC sub-sampling is of practical importance. Previous model-based sub-sampling schemes are ad-hoc in design and do not preserve the overall shape sufficiently well, while previous data-driven schemes are trained for specific pre-determined input PC sizes and sub-sampling rates and thus do not generalize well. Leveraging advances in graph sampling, we propose a fast PC sub-sampling algorithm of linear time complexity that chooses a 3D point subset while minimizing a global reconstruction error. Specifically, to articulate a sampling objective, we first assume a super-resolution (SR) method based on feature graph Laplacian regularization (FGLR) that reconstructs the original high-res PC, given points chosen by a sampling matrix H. We prove that minimizing a worst-case SR reconstruction error is equivalent to maximizing the smallest eigenvalue of matrix H^ H+ {L}, where {L} is a symmetric, positive semi-definite matrix derived from a neighborhood graph connecting the 3D points. To arrive at a fast algorithm, instead of maximizing, we maximize a lower bound ^- (H^ H+ {L}) via selection of H —this translates to a graph sampling problem for a signed graph G with self-loops specified by graph Laplacian {L}. We tackle this general graph sampling problem in three steps. First, we approximate G with a balanced graph GB specified by Laplacian {L}B. Second, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform {L}ₚ~=~ S{L}B S^-1, so that all Gershgorin disc left-ends of {L}ₚ are aligned exactly at ({L}B). Finally, we choose samples on GB using a previous graph sampling algorithm to maximize ^- (H^ H+ {L}ₚ) in linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in reconstruction quality.