Multiresolution local smoothness detection in non-uniformly sampled multivariate signals

• URL: http://arxiv.org/abs/2507.13480v1
• Date: Thu, 17 Jul 2025 18:46:01 GMT
• Title: Multiresolution local smoothness detection in non-uniformly sampled multivariate signals
• Authors: Sara Avesani, Gianluca Giacchi, Michael Multerer,
• Abstract summary: We introduce a (near) linear-time algorithm for detecting the local regularity in non-uniformly sampled multivariate signals.<n>Our approach quantifies regularity within the framework of microlocal spaces introduced by Jaffard.<n>We derive decay estimates for functions belonging to classical H"older spaces and Sobolev-Slobodeckij spaces.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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