LBO-Shape densities: A unified framework for 2D and 3D shape classification on the hypersphere of wavelet densities

Recent years have seen a sharp rise in shape classification applications, and following suit, several frameworks have been proposed for efficient indexing of shape models. Here we propose a state-of-the-art shape matching framework which concomitantly provides transformation invariance and computationally efficient querying. Shapes are represented as probability density functions estimated on the eigenspace of the shape's Laplace-Beltrami operator (LBO) and the ensuing manifold geometry is leveraged to classify query shapes. Specifically, we estimate a nonparametric, square-root wavelet density on the low-order eigenvectors of the LBO, capturing the rich geometric structure of 2D and 3D shapes with very minimal pre-processing requirements. By estimating 3D, square-root wavelet densities on each shape's eigenspace (LBO-Shape densities), both 2D and 3D shapes become identifiable with the unit hypersphere. Leveraging the hypersphere's simple geometry, our avant-garde model-to-mean indexing scheme computes the intrinsic Karcher mean for each shape category, and then uses the closed-form distance between a query shape and the means to assign labels. In 2D, the need for burdensome preprocessing like extracting closed curves along with topological and equal point set cardinality requirements are eliminated. Similarly, in 3D we gain isometry invariance and rid ourselves of the need for superfluous feature extraction schemes. The extensive experimental results demonstrate that our approach is competitive with the state-of-the-art in 2D/3D shape matching algorithms.