Multiresolution local smoothness detection in non-uniformly sampled multivariate signals

TL;DR

This paper introduces a fast, near-linear algorithm using samplet transforms for detecting local regularity in non-uniformly sampled multivariate signals, extending wavelet-based edge detection to scattered data.

Contribution

It presents a novel, efficient method for local regularity detection in high-dimensional, non-uniform data using samplet transforms within the microlocal analysis framework.

Findings

Samplet coefficients decay characterizes pointwise regularity.

Method performs robustly on high-dimensional, scattered data.

Numerical experiments validate effectiveness across various data types.

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 H\"older 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.