Serena Morigi

We present a continuous-domain variational model for the morphological decomposition of 1D noisy signals into a sum of meaningful constituent components. The model relies on sparsifying fractional-order derivatives of the sought components to capture intricate signal structures. An in-depth analysis of the model leads to a representer theorem , establishing the equivalence between the infinite-dimensional problem and a finite-dimensional counterpart, which serves as its exact discretization. To efficiently solve the resulting discrete and convex optimization problem, an alternating direction method of multipliers-based algorithm is presented. Furthermore, we introduce a bilevel optimization framework for the automatic selection of all free model parameters, including the fractional derivative orders, based on the generalized whiteness principle. Numerical results validate the effectiveness of our approach, which can provide accurate decompositions even in demanding scenarios characterized by high noise levels and abrupt signal discontinuities.

Sparse signal recovery from underdetermined systems presents significant challenges when using conventional [Formula: see text] and [Formula: see text] penalties, primarily due to computational complexity and estimation bias. This paper introduces a truncated Huber penalty, a nonconvex metric that effectively bridges the gap between unbiased sparse recovery and differentiable optimization. The proposed penalty applies quadratic regularization to small entries while truncating large magnitudes, avoiding nondifferentiable points at optimal solutions. Theoretical analysis demonstrates that, for an appropriately chosen threshold, any [Formula: see text]-sparse solution recoverable via conventional penalties remains a local optimum under the truncated Huber function. This property allows the exact and robust recovery theories developed for other penalty regularization functions to be directly extended to the truncated Huber function. To solve the optimization problem, we develop a block coordinate descent (BCD) algorithm with finite-step convergence guarantees under spark conditions. Numerical experiments validate the effectiveness and robustness of the proposed method in both synthetic and real scenarios. Furthermore, we demonstrate the flexibility of the truncated Huber framework through two extensions: one to an adaptively weighted variant inspired by sorted penalties, and the other to the gradient domain for applications such as signal denoising and image smoothing.

Multi-frequency Electrical Impedance Tomography (mfEIT) represents a promising biomedical imaging modality that enables the estimation of tissue conductivities across a range of frequencies. Addressing this challenge, we present a novel variational network, a model-based learning paradigm that strategically merges the advantages and interpretability of classical iterative reconstruction with the power of deep learning. This approach integrates graph neural networks (GNNs) within the iterative Proximal Regularized Gauss Newton (PRGN) framework. By unfolding the PRGN algorithm, where each iteration corresponds to a network layer, we leverage the physical insights of nonlinear model fitting alongside the GNN’s capacity to capture inter-frequency correlations. Notably, the GNN architecture preserves the irregular triangular mesh structure used in the solution of the nonlinear forward model, enabling accurate reconstruction of overlapping tissue fraction concentrations.

Cartoon-texture image decomposition is a critical preprocessing problem bottlenecked by the numerical intractability of classical variational or optimization models and the tedious manual tuning of global regularization parameters.We propose a Guided Variational Decomposition (GVD) model which introduces spatially adaptive quadratic norms whose pixel-wise weights are learned either through local probabilistic statistics or via a lightweight neural network within a bilevel framework.This leads to a unified, interpretable, and computationally efficient model that bridges classical variational ideas with modern adaptive and data-driven methodologies. Numerical experiments on this framework, which inherently includes automatic parameter selection, delivers GVD as a robust, self-tuning, and superior solution for reliable image decomposition.

We introduce and study a variational model for signal and image denoising based on Riemann-Liouville fractional derivatives of every positive order higher than zero. Both the one-dimensional and two-dimensional cases are studied. The model exploits an L^1 L 1 fitting data term together with both right and left Riemann-Liouville fractional derivatives as regularizing terms, with the aim of achieving an orientation independent analysis. To provide evidence of effectiveness for the proposed model we introduce a discretisation based on a second-order consistent Grünwald Letnikov scheme and show some numerical simulations aiming to denoise images corrupted by impulsive noise, which can be well modeled by the Laplace distribution.

Eigenfunctions and the spectrum of the Laplace-Beltrami operator are fundamental tools in geometry processing, providing a robust framework for analyzing complex geometries. This paper introduces an incremental framework specifically designed for the computation of the leading eigenpairs of the Laplace-Beltrami operator of 2-manifolds represented by triangular surface meshes in R³. The proposed approach relies on a discrete physics-informed neural network (dPINN), which leverages Graph Neural Networks to effectively model the intrinsic geometry and topology of the manifold. dPINNs are ideally suited for learning on a triangular surface mesh in R³ because they leverage its discrete representation and numerical differentiation. Conversely, continuous PINNs (cPINNs) prove inadequate when the spatial domain (manifold) is not readily available for arbitrary sampling. Numerical experiments on several 2- manifolds in R²/R³ demonstrate the accuracy, robustness to geometric and topological variations, and generalization capabilities of the proposed neural network approach in case of 2-manifolds with parametric representations. The presented method aims to demonstrate how learning-based approaches can improve upon traditional numerical methods under adverse meshing conditions or domain discretization changes.

One-dimensional signal decomposition is a well-established and widely used technique across various scientific fields. It serves as a highly valuable pre-processing step for data analysis. While traditional decomposition techniques often rely on mathematical models, recent research suggests that applying the latest deep learning models to this problem presents an exciting, unexplored area with promising potential. This work presents a novel method for the additive decomposition of one-dimensional signals. We leverage the Transformer architecture to decompose signals into their constituent components: piece-wise constant, smooth (low-frequency oscillatory), textured (high-frequency oscillatory), and a noise component. Our model, trained on synthetic data, achieves excellent accuracy in modeling and decomposing input signals from the same distribution, as demonstrated by the experimental results.