Spectral analysis of medial axis for shape description

Abstract

In this thesis, we make several significant achievements towards defining a medial axis based shape descriptor which is compact, yet discriminative.

First, we propose a novel medial axis spectral shape descriptor called the medial axis spectrum for a 2D shape, which applies spectral analysis directly to the medial axis of a 2D shape. We extend the Laplace-Beltrami operator onto the medial axis of a 2D shape, and take the solution to an extended Laplacian eigenvalue problem defined on this axis as the medial axis spectrum. The medial axis spectrum of a 2D shape is certainly more efficient to compute than spectral analysis of a 2D region, since the efficiency of solving the Laplace eigenvalue problem strongly depends on the domain dimension. We show that the medial axis spectrum is invariant under uniform scaling and isometry of the medial axis. It could also overcome the medial axis noise problem automatically, due to the incorporation of the hyperbolic distance metric. We also demonstrate that the medial axis spectrum inherits several advantages in terms of discriminating power over existing methods.

Second, we further generalize the medial axis spectrum to the description of medial axes of 3D shapes, which we call the medial axis spectrum for a 3D shape. We develop a newly defined Minkowski-Euclidean area ratio inspired by the Minkowski inner product to characterize the geometry of the medial axis surface of a 3D mesh. We then generalize the Laplace-Beltrami operator to the medial axis surface, and take the solution to an extended Laplacian eigenvalue problem defined on the surface as the medial axis spectrum. As the 2D case, the medial axis spectrum of a 3D shape is invariant under rigid transformation and isometry of the medial axis, and is robust to shape boundary noise as shown by our experiments. The medial axis spectrum is finally used for 3D shape retrieval, and its superiority over previous work is shown by extensive comparisons.