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3D point cloud (PC) -- a collection of discrete geometric samples of aphysical object's surface -- is typically large in size, which entailsexpensive subsequent operations like viewpoint image rendering and objectrecognition. Leveraging on recent advances in graph sampling, we propose a fastPC sub-sampling algorithm that reduces its size while preserving the overallobject shape. Specifically, to articulate a sampling objective, we first assumea super-resolution (SR) method based on feature graph Laplacian regularization(FGLR) that reconstructs the original high-resolution PC, given 3D pointschosen by a sampling matrix $\H$. We prove that minimizing a worst-case SRreconstruction error is equivalent to maximizing the smallest eigenvalue$\lambda_{\min}$ of a matrix $\H^{\top} \H + \mu \cL$, where $\cL$ is asymmetric, positive semi-definite matrix computed from the neighborhood graphconnecting the 3D points. Instead, for fast computation we maximize a lowerbound $\lambda^-_{\min}(\H^{\top} \H + \mu \cL)$ via selection of $\H$ in threesteps. Interpreting $\cL$ as a generalized graph Laplacian matrix correspondingto an unbalanced signed graph $\cG$, we first approximate $\cG$ with a balancedgraph $\cG_B$ with the corresponding generalized graph Laplacian matrix$\cL_B$. Second, leveraging on a recent theorem called Gershgorin disc perfectalignment (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 agraph sampling algorithm to maximize $\lambda^-_{\min}(\H^{\top} \H + \mu\cL_p)$ in roughly linear time. Experimental results show that 3D points chosenby our algorithm outperformed competing schemes both numerically and visuallyin SR reconstruction quality.

Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment