Basis functions provide a simple and effective way to interpolate signals on a surface with an arbitrary dimension, topology, and discretisation. However, the definition of local and shape-aware basis functions is still an open problem, which has been addressed through optimisation methods that are time-consuming, over-constrained, or admit more than one solution. In this context, we review the definition, properties, computation, and applications of the class of Laplacian spectral basis functions, which are defined as solutions to the harmonic equation, the diffusion equation, and PDEs involving the Laplace–Beltrami operator. The resulting functions are efficiently computed through the solution of sparse linear systems and satisfy different properties, such as smoothness, locality, and multi-scale shape encoding. Finally, the discussion of the properties of the Laplacian spectral basis functions is integrated with an analysis of their behaviour with respect to different measures (i.e., conformal and area-preserving metrics, transformation distance) and of their applications to spectral geometry processing and shape analysis.

基函数提供了一种简单有效的方法，可以以任意尺寸，拓扑和离散化对表面上的信号进行插值。但是，局部和形状感知基础函数的定义仍然是一个悬而未决的问题，已通过耗时，过度约束或允许多个解决方案的优化方法解决了该问题。在这种情况下，我们回顾了拉普拉斯谱基函数类的定义，性质，计算和应用，它们被定义为谐波方程，扩散方程和涉及Laplace-Beltrami算子的PDE的解。通过稀疏线性系统的解可以有效地计算所得函数，并满足不同的属性，例如平滑度，局部性和多尺度形状编码。最后，