Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

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 $\lambda_{\min}$ of a matrix \H^{\top} \H + \mu \cL, where \cL 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 \lambda^-_{\min}(\H^{\top} \H + \mu \cL) via selection of \H in three steps. Interpreting \cL as a generalized graph Laplacian matrix corresponding to an unbalanced signed graph \cG, we first approximate \cG with a balanced graph \cG_B with the corresponding generalized graph Laplacian matrix \cL_B. Second, leveraging on a recent theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform \cL_p = \S \cL_B \S^{-1} so that Gershgorin disc left-ends of \cL_p are all aligned at the same value \lambda_{\min}(\cL_B). Finally, we perform PC sub-sampling on \cG_B using a graph sampling algorithm to maximize \lambda^-_{\min}(\H^{\top} \H + \mu \cL_p) 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