Tracking Evolving Geometric Data by Local Graph Laplacian Operators

Abstract

Geometric data are samples on geometric surfaces of physical or abstract objects. When the underlying objects evolve over time, their geometric data change as well. Tracking their evolution is critical to many emerging areas, such as autonomous driving, edge computing networks, and drug interactions. Lacking well-defined grids in geometric data prohibits easy temporal matching for tracking evolution of geometric data. This paper considers the framework of graph signal processing to exploit spectral analysis on geometric surfaces to achieve the tracking task.This paper begins with modeling an underlying geometric surface by a graph, followed by using the spectra of local graph Laplacians to detect graph patches corresponding to regions of high curvatures on the geometric surfaces. The Laplacian spectra of feature graph patches and the structural information of graphs are analyzed together to perform temporal matching across times. Sparse temporal transforms are reconstructed based on the matched feature graph patches, and extrapolating the transforms to the full-scale graph derives the estimation of geometric evolution.