Inspired by recent interest in geometric deep learning, this work generalises the recently developed Slepian scale-discretised wavelets on the sphere to Riemannian manifolds. Through the sifting convolution, one may define translations and, thus, convolutions on manifolds - which are otherwise not well-defined in general. Slepian wavelets are constructed on a region of a manifold and are therefore suited to problems where data only exists in a particular region. The Slepian functions, on which Slepian wavelets are built, are the basis functions of the Slepian spatial-spectral concentration problem on the manifold. A tiling of the Slepian harmonic line with smoothly decreasing generating functions defines the scale-discretised wavelets; allowing one to probe spatially localised, scale-dependent features of a signal. By discretising manifolds as graphs, the Slepian functions and wavelets of a triangular mesh are presented. Through a wavelet transform, the wavelet coefficients of a field defined on the mesh are found and used in a straightforward thresholding denoising scheme.

翻译：受几何深度学习领域近期兴起的兴趣启发，本工作将球面上新近发展的Slepian尺度离散小波推广至黎曼流形。通过筛卷积分，可定义流形上的平移与卷积运算——而这些运算在一般情况下往往缺乏良好定义。Slepian小波构建于流形的局部区域，因而特别适用于数据仅存在于特定区域的问题。作为构建Slepian小波的基础，Slepian函数是流形上Slepian空间-谱集中问题的基函数。通过具有平滑递减特性的生成函数对Slepian谐波线进行剖分，可定义尺度离散小波，从而实现对信号中空间局域化、尺度相关特征的探测。通过将流形离散化为图结构，展示了三角网格上的Slepian函数与小波构造。利用小波变换获得网格定义场的小波系数，并将其应用于简单的阈值去噪方案。