Wavelet-based Heat Kernel Derivatives: Towards Informative Localized Shape Analysis

In this paper, we propose a new construction for Mexican hat wavelets on shapes with applications to partial shape matching. Our approach takes its main inspiration in the well-established methodology of diffusion wavelets. Our novel construction enables us to rapidly compute a multiscale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this approach allows our functions to inherit many attractive properties of the heat kernel. Due to its natural ability to encode high-frequency details on a shape, our method allows us to reconstruct and transfer $\delta$-functions more accurately than the Laplace-Beltrami eigenfunction basis and other related bases. Finally, we apply our approach to the challenging problems of partial and large-scale shape matching. An extensive comparison to state-of-the-art approaches shows that our method, while comparable in performance, is both simpler and much faster than competing approaches.