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

TL;DR

This paper introduces a novel wavelet-based method for shape analysis that efficiently captures local details and improves partial shape matching by approximating heat kernel derivatives.

Contribution

It presents a new construction of Mexican hat wavelets on shapes inspired by diffusion wavelets, enabling rapid multiscale analysis and improved shape matching.

Findings

Accurately reconstructs and transfers delta functions on shapes.

Achieves comparable performance to state-of-the-art methods.

Significantly faster and simpler than existing approaches.

Abstract

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