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

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 multi‐scale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this leads to a family of functions that inherit many attractive properties of the heat kernel (e.g. 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 δ‐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.

Joint Symmetry Detection and Shape Matching for Non-Rigid Point Cloud

“…In the previous section, we outlined a basic mechanism to compute a functional map given a set of basis functions. Due to the instability of Laplace-Beltrami operator, LBO, on partial 3D shapes [15] and noise [21], our main goal is to avoid using its eigenfunctions and instead aim to learn an embedding that can replace the spectral embedding given by the LBO. This section details how to learn such an embedding whilst working in the symmetric space.…”

Multiscale Anisotropic Harmonic Filters on non Euclidean domains

“…We firstly introduce MAHFs on a continuous shape M with LBO ∆. As concern the smoothing kernel, we refer to the heat kernel [1], [18], the fundamental solution of the heat equation:…”

“…Additionally from ( 2)-( 3), we can deduce the Fourier representation Kt (s) = e −tλs , namely a Gaussian function in the frequency domain expressed wrt the frequency-like variable √ λ s , supplying the isomorphic property between the two domains. These important properties can be exploited to define diffusion wavelets [20] on manifolds and graphs, based on the concept "to scale is equivalent to diffuse"; moreover, the kernels' derivatives with respect to the variable t are exploited in Computer Graphics communities to define the Mexican Hat Wavelets (MHWs) [18], [19], inherently isotropic filters mainly used in shape matching and geometry processing. MHWs 2D-filters are widely applied in image processing for edge detection and thus will be taken as comparison in Sec.…”

“…It is worth to note that, referred to the multi-scale approach, the heat kernel for different t values can be conveniently evaluated in a recursive manner K t1+t2 = K t1 * K t2 [16]. Additionally, [18] relates the parameter t to the global area of the discrete manifold M in multi-scale analysis.…”

“…Herein, we presents three MAHFs applications aiming to demonstrate: i) signal variations detection and multi-scale behavior; ii) Normal field's local variation estimate, providing a comparisons with MHWs [18]; iii) edge extraction for an image textured on a face shape.…”