Abstract
Inspired by edge detection based on the decay behavior of wavelet coefficients, we introduce a (near) linear-time algorithm for detecting the local regularity in non-uniformly sampled multivariate signals. Our approach quantifies regularity within the framework of microlocal spaces introduced by Jaffard. The central tool in our analysis is the fast samplet transform, a distributional wavelet transform tailored to scattered data. We establish a connection between the decay of samplet coefficients and the pointwise regularity of multivariate signals. As a by product, we derive decay estimates for functions belonging to classical Hölder spaces and Sobolev-Slobodeckij spaces. While traditional wavelets are effective for regularity detection in low-dimensional structured data, samplets demonstrate robust performance even for higher dimensional and scattered data. To illustrate our theoretical findings, we present extensive numerical studies detecting local regularity of one-, two- and three-dimensional signals, ranging from non-uniformly sampled time series over image segmentation to edge detection in point clouds.
Abstract (translated)
受基于小波系数衰减行为的边缘检测算法启发，我们提出了一种（近似）线性时间算法来探测非均匀采样多元信号中的局部正则性。我们的方法在Jaffard引入的小局域空间框架内量化了这种正则性。分析中主要使用的工具是快速样本变换，这是一种专为散乱数据设计的分布小波变换。我们建立了样本系数衰减与多变量信号点态正则性之间的联系。作为副产品，我们还推导出经典Hölder空间和Sobolev-Slobodeckij空间中函数的衰减估计值。 传统的波let在检测低维结构化数据中的正则性方面非常有效，但样本（samplets）即使面对高维度和散乱的数据也能展现出强大的性能。为了说明我们的理论发现，我们展示了广泛的数值研究案例，用于探测一维、二维及三维信号的局部正则性，这些应用范围从非均匀采样的时间序列到图像分割再到点云中的边缘检测。
URL
https://arxiv.org/abs/2507.13480
