The Laplace-Beltrami operator has established itself in the field of non-rigid shape analysis due to its many useful properties such as being invariant under isometric transformation, having a countable eigensystem forming an orthonormal basis, and fully characterizing geodesic distances of the manifold. However, this invariancy only applies under isometric deformations, which leads to a performance breakdown in many real-world applications. In recent years emphasis has been placed upon extracting optimal features using deep learning methods, however spectral signatures play a crucial role and still add value. In this paper we take a step back, revisiting the LBO and proposing a supervised way to learn several operators on a manifold. Depending on the task, by applying these functions, we can train the LBO eigenbasis to be more task-specific. The optimization of the LBO leads to enormous improvements to established descriptors such as the heat kernel signature in various tasks such as retrieval, classification, segmentation, and correspondence, proving the adaption of the LBO eigenbasis to both global and highly local learning settings.

翻译：拉普拉斯-贝尔特拉米算子因其诸多有用特性已在非刚性形状分析领域确立重要地位，例如在等距变换下具有不变性、拥有可数特征系统构成正交基，且能完全表征流形的测地距离。然而，这种不变性仅适用于等距变形场景，导致其在许多实际应用中出现性能退化。近年来研究重点转向通过深度学习方法提取最优特征，但谱特征仍发挥着关键作用并持续提供价值。本文回归基础，重新审视LBO并提出一种监督式学习流形上多种算子的方法。通过应用这些函数，我们可以根据具体任务训练LBO特征基使其更具任务针对性。LBO的优化显著提升了热核签名等经典描述符在检索、分类、分割与对应匹配等多种任务中的性能，证明了LBO特征基对全局与高度局部化学习场景的适应能力。