提出快速检测非均匀采样多维信号局部平滑性的算法
Multiresolution local smoothness detection in non-uniformly sampled multivariate signals
- 用样本变换分析散乱数据的局部光滑性
- 在1-3维信号中准确识别不连续点与边缘
- 适合处理高维、非规则数据的科研人员
受基于小波系数衰减行为的边缘检测启发，本文提出一种（近）线性时间算法，用于检测非均匀采样多维信号的局部正则性。方法基于Jaffard提出的微局部空间框架，核心工具是专为散乱数据设计的快速样本变换（samplet transform），一种分布型小波变换。我们建立了样本变换系数衰减与多维信号逐点正则性之间的联系。作为副产品，推导出属于经典Hölder空间和Sobolev-Slobodeckij空间函数的衰减估计。传统小波在低维规则数据上表现良好，而样本变换在高维及散乱数据上仍具鲁棒性。通过大量数值实验，验证了该方法在非均匀采样时间序列、图像分割及点云边缘检测中的有效性。
原文摘要 · Abstract (English)
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.
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