Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

Abstract

3D point cloud (PC)-a collection of discrete geometric samples of a physical object's surface-is typically large in size, which entails expensive subsequent operations like viewpoint image rendering and object recognition. Leveraging on recent advances in graph sampling, we propose a fast PC sub-sampling algorithm that reduces its size while preserving the overall object shape. 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-resolution PC, given 3D points chosen by a sampling matrix H. We prove that minimizing a worst-case SR reconstruction error is equivalent to maximizing the smallest eigenvalue λ min of a matrix H H+µL, where L is a symmetric, positive semi-definite matrix computed from the neighborhood graph connecting the 3D points. Instead, for fast computation we maximize a lower bound λ − min (H H + µL) via selection of H in three steps. Interpreting L as a generalized graph Laplacian matrix corresponding to an unbalanced signed graph G, we first approximate G with a balanced graph G B with the corresponding generalized graph Laplacian matrix L B . Second, leveraging on a recent theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform Lp = SL B S −1 so that Gershgorin disc left-ends of Lp are all aligned at the same value λ min (L B ). Finally, we perform PC sub-sampling on G B using a graph sampling algorithm to maximize λ − min (H H+µLp) in roughly linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in SR reconstruction quality.