Isometrically Invariant Description of Deformable Objects Based on the Fractional Heat Equation

Abstract

Recently, a number of researchers have turned their attention to the creation of isometrically invariant shape descriptors based on the heat equation. The reason for this surge in interest is that the Laplace-Beltrami operator, associated with the heat equation, is highly dependent on the topology of the underlying manifold, which may lead to the creation of highly accurate descriptors. In this paper, we propose a generalisation based on the fractional heat equation. While the heat equation enables one to explore the shape with a Markovian Gaussian random walk, the fractional heat equation explores the manifold with a non-Markovian Lévy random walk. This generalisation provides two advantages. These are, first, that the process has a memory of the previously explored geometry and, second, that it is possible to correlate points or vertices which are not part of the same neighbourhood. Consequently, a highly accurate, contextual shape descriptor may be obtained.