Gianluca Giacchi

Sara Avesani, Gianluca Giacchi, Michael Multerer

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.

Gianluca Giacchi, Michael Multerer, Jacopo Quizi

Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension and such that they exhibit vanishing moments with respect to any prescribed set of primitives. We consider the samplet construction in a probabilistic framework and show that, when choosing polynomials as primitives, the resulting samplet basis converges in the infinite data limit to signed measures with broken polynomial densities. These densities amount to multiwavelets with respect to a hierarchical partition of the region containing the data sites. As a byproduct, we therefore obtain a construction of general multiwavelets that allows for a flexible prescription of vanishing moments going beyond tensor product constructions. For congruent partitions we particularly recover classical multiwavelets with scale- and partition- independent filter coefficients. The theoretical findings are complemented by numerical experiments that illustrate the convergence results in case of random as well as low-discrepancy data sites.

Giacomo Elefante, Gianluca Giacchi, Michael Multerer et al.

We present a novel framework for discrete multiresolution analysis of graph signals. The main analytical tool is the samplet transform, originally defined in the Euclidean framework as a discrete wavelet-like construction, tailored to the analysis of scattered data. The first contribution of this work is defining samplets on graphs. To this end, we subdivide the graph into a fixed number of patches, embed each patch into a Euclidean space, where we construct samplets, and eventually pull the construction back to the graph. This ensures orthogonality, locality, and the vanishing moments property with respect to properly defined polynomial spaces on graphs. Compared to classical Haar wavelets, this framework broadens the class of graph signals that can efficiently be compressed and analyzed. Along this line, we provide a definition of a class of signals that can be compressed using our construction. We support our findings with different examples of signals defined on graphs whose vertices lie on smooth manifolds. For efficient numerical implementation, we combine heavy edge clustering, to partition the graph into meaningful patches, with landmark \texttt{Isomap}, which provides low-dimensional embeddings for each patch. Our results demonstrate the method's robustness, scalability, and ability to yield sparse representations with controllable approximation error, significantly outperforming traditional Haar wavelet approaches in terms of compression efficiency and multiresolution fidelity.

Gianluca Giacchi, Isidoros Iakovidis, Bastien Milani et al.

Magnetic Resonance Imaging (MRI) is a powerful technique employed for non-invasive in vivo visualization of internal structures. Sparsity is often deployed to accelerate the signal acquisition or overcome the presence of motion artifacts, improving the quality of image reconstruction. Image reconstruction algorithms use TV-regularized LASSO (Total Variation-regularized LASSO) to retrieve the missing information of undersampled signals, by cleaning the data of noise and while optimizing sparsity. A tuning parameter moderates the balance between these two aspects; its choice affecting the quality of the reconstructions. Currently, there is a lack of general deterministic techniques to choose these parameters, which are oftentimes manually selected and thus hinder the reliability of the reconstructions. Here, we present ALMA (Algorithm for Lagrange Multipliers Approximation), an iterative mathematics-inspired technique that computes tuning parameters for generalized LASSO problems during MRI reconstruction. We analyze quantitatively the performance of these parameters for imaging reconstructions via TV-LASSO in an MRI context on phantoms. Although our study concentrates on TV-LASSO, the techniques developed here hold significant promise for a wide array of applications. ALMA is not only adaptable to more generalized LASSO problems but is also robust to accommodate other forms of regularization beyond total variation. Moreover, it extends effectively to handle non-Cartesian sampling trajectories, broadening its utility in complex data reconstruction scenarios. More generally, ALMA provides a powerful tool for numerically solving constrained optimization problems across various disciplines, offering a versatile and impactful solution for advanced computational challenges.