Sampling Of 3d Point Cloud Via Gershgorin Disc Alignment

Point cloud-a collection of geometric samples of a physical object in 3D space-can be very large in size, which entails a large computation cost for many imaging applications. In this paper, we reduce the size of a point cloud towards a more compact representation via optimal sub-sampling. Specifically, we first derive a sampling objective that maximizes the stability (maximizes the smallest eigenvalue λ
_min
(B) of a coefficient matrix B = H
^T
H + μL) of a linear system super-resolving a sub-sampled point cloud. To circumvent eigen-decomposition, we maximize instead a lower bound λ
_min
^-
(B) using a fast graph sampling scheme called Gershgorin disc alignment (GDA) based on the well-known Gershgorin circle theorem. However, GDA requires that the disc left-ends of real matrix L are initially aligned at the same value, which is not the case for point clouds. Orthogonally, we recently derived a matrix theorem proving that disc left-ends of a generalized graph Laplacian matrix for a balanced and irreducible signed graph can be perfectly aligned via a similarity transform using the matrix's first eigenvector. Leveraging this work, we first interpret L as a generalized graph Laplacian matrix and balance the underlying graph. We then align disc left-ends of the resulting generalized graph Laplacian of the balanced graph using its first eigenvector, in order to employ GDA for point cloud sampling. Experiments show that our sampling method outperforms competing methods in super-resolved point cloud quality.