LBO-Shape Densities: Efficient 3D Shape Retrieval Using Wavelet Density Estimation

Abstract

Driven by desirable attributes such as topological characterization and invariance to isometric transformations, the use of the Laplace-Beltrami operator (LBO) and its associated spectrum have been widely adopted among the shape analysis community. Here we demonstrate a novel use of the LBO for shape matching and retrieval by estimating probability densities on its Eigen space, and subsequently using the intrinsic geometry of the density manifold to categorize similar shapes. In our framework, each 3D shape's rich geometric structure, as captured by the low order eigenvectors of its LBO, is robustly characterized via a nonparametric density estimated directly on these eigenvectors. By utilizing a probabilistic model where the square root of the density is expanded in a wavelet basis, the space of LBO-shape densities is identifiable with the unit hyper sphere. We leverage this simple geometry for retrieval by computing an intrinsic Karcher mean (on the hyper sphere of LBO-shape densities) for each shape category, and use the closed-form distance between a query shape and the means to classify shapes. Our method alleviates the need for superfluous feature extraction schemes-required for popular bag-of-features approaches-and experiments demonstrate it to be robust and competitive with the state-of-the-art in 3D shape retrieval algorithms.