# Point Cloud Sampling via Graph Balancing and Gershgorin Disc Alignment

IEEE Transactions on Pattern Analysis and Machine Intelligence（2023）

Abstract

Point cloud (PC)—a collection of discrete geometric samples of a 3D object’s surface—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
${\mathbf H}$
. We prove that minimizing a worst-case SR reconstruction error is equivalent to maximizing the smallest eigenvalue
$\lambda _{\min }$
of matrix
${\mathbf H}^{\top } {\mathbf H}+ \mu {\boldsymbol{\mathcal{L}}}$
, where
${\boldsymbol{\mathcal{L}}}$
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
$\lambda _{\min }$
, we maximize a lower bound
$\lambda ^-_{\min }({\mathbf H}^{\top } {\mathbf H}+ \mu {\boldsymbol{\mathcal{L}}})$
via selection of
${\mathbf H}$
—this translates to a graph sampling problem for a signed graph
${\mathcal G}$
with self-loops specified by graph Laplacian
${\boldsymbol{\mathcal{L}}}$
. We tackle this general graph sampling problem in three steps. First, we approximate
${\mathcal G}$
with a balanced graph
${\mathcal G}_B$
specified by Laplacian
${\boldsymbol{\mathcal{L}}}_B$
. Second, leveraging a recent linear algebraic theorem called Gershgorin disc perfect alignment (GDPA), we perform a similarity transform
${\boldsymbol{\mathcal{L}}}_p~=~{\mathbf S}{\boldsymbol{\mathcal{L}}}_B {\mathbf S}^{-1}$
, so that all Gershgorin disc left-ends of
${\boldsymbol{\mathcal{L}}}_p$
are aligned exactly at
$\lambda _{\min }({\boldsymbol{\mathcal{L}}}_B)$
. Finally, we choose samples on
${\mathcal G}_B$
using a previous graph sampling algorithm to maximize
$\lambda ^-_{\min }({\mathbf H}^{\top } {\mathbf H}+ \mu {\boldsymbol{\mathcal{L}}}_p)$
in linear time. Experimental results show that 3D points chosen by our algorithm outperformed competing schemes both numerically and visually in reconstruction quality.

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Key words

Point cloud processing,graph signal processing,graph sampling,Gershgorin circle theorem,graph Laplacian regularizer

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