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.

中文翻译：

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基于小波的热核导数：面向信息局部形状分析

在本文中，我们提出了一种新的墨西哥帽小波结构，适用于部分形状匹配。我们的方法的主要灵感来自于成熟的扩散小波方法。这种新颖的构造使我们能够通过近似热核的导数来快速计算墨西哥帽子小波函数的多尺度族。我们证明它导致一系列函数继承了热核的许多有吸引力的特性（例如，局部支持、从单点恢复等距的能力、高效计算）。由于其对形状上的高频细节进行编码的天然能力，所提出的方法比拉普拉斯-贝尔特拉米特征函数基和其他相关基更准确地重建和传递 $\delta$-函数。最后，我们将我们的方法应用于部分和大规模形状匹配的挑战性问题。与最先进技术的广泛比较表明，它在性能上具有可比性，同时比竞争方法更简单且速度更快。