Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

Point cloud (PC)& mdash;a collection of discrete geometric samples of a 3D object's surface & mdash;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 Aminof matrix HTH + mG, where G 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 Amin, we maximize a lower bound A-min(HTH + mG) via selection of H & mdash;this translates to a graph sampling problem for a signed graph g with self-loops specified by graph Laplacian G. We tackle this general graph sampling problem in three steps. First, we approximate g with a balanced graph gB specified by Laplacian GB. Second, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform Gp = SGBS-1, so that all Gershgorin disc left-ends of Gp are aligned exactly at Amin(GB). Finally, we choose samples on gB using a previous graph sampling algorithm to maximize A-min(HTH +mGp) in linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in reconstruction quality.