Multiresolution local smoothness detection in non-uniformly sampled multivariate signals
This work addresses the challenge of local smoothness detection in scattered and high-dimensional data, which is incremental by building on wavelet-based edge detection and microlocal spaces.
The paper tackles the problem of detecting local regularity in non-uniformly sampled multivariate signals by introducing a near linear-time algorithm based on the fast samplet transform, achieving robust performance for higher-dimensional and scattered data as demonstrated in numerical studies across one-, two-, and three-dimensional signals.
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.