arXiv subjects

Stefano Aleotti

Publications and source records attributed to Stefano Aleotti.

5 recordsLinked to original sources

arXiv2026

Iterated graph Laplacian for image restoration problems

We study the graph Laplacian operator as a regularizer in a generalized Tikhonov framework for linear ill-posed problems. The Laplacian is updated iteratively from the current reconstruction, so that progressively sharper structural information about the solution is fed into the regularization term. We introduce three schemes: a standard one that rebuilds the Laplacian from each new iterate; an error-equation scheme that, following the error-based formulation of iterated Tikhonov regularization, builds the Laplacian from an estimate of the reconstruction error rather than of the image itself; and a mixed scheme combining the two. We establish convergence of all three schemes for noisy data under a priori parameter and stopping rules as the noise level tends to zero. Numerical experiments in two-dimensional computed tomography and image deblurring show consistent gains in reconstruction quality and sharper recovery of fine details.

math.NA↗

arXiv2026

Multilevel Preconditioning Strategies for Convex Optimization Methods in Image Deblurring

Proximal gradient methods are widely used in imaging, and their speed of convergence can be accelerated by incorporating variable metrics and/or extrapolation steps. Recent works have shown that preconditioning strategies can significantly enhance this acceleration, in particular, for image deblurring problems. In parallel, a multilevel framework has been introduced to speed up inertial and inexact forward-backward schemes for image restoration problems. In this paper, we combine preconditioning and multilevel strategies to design a robust and consistent acceleration framework for both standard and inexact forward-backward schemes applied to regularized convex optimization problems. Numerical experiments in image deblurring confirm that our approach yields a substantial improvement in convergence speed compared to standard methods.

math.NA↗

arXiv2025

Variable Projected Augmented Lagrangian Methods for Generalized Lasso Problems

We introduce variable projected augmented Lagrangian (VPAL) methods for solving generalized nonlinear Lasso problems with improved speed and accuracy. By eliminating the nonsmooth variable via soft-thresholding, VPAL transforms the problem into a smooth reduced formulation. For linear models, we develop a preconditioned variant that mimics Newton-type updates and yields significant acceleration. We prove convergence guarantees for both standard and preconditioned VPAL under mild assumptions and show that variable projection leads to sharper convergence and higher solution quality. The method seamlessly extends to nonlinear inverse problems, where it outperforms traditional approaches in applications such as phase retrieval and contrast enhanced MRI (LIP-CAR). Across tasks including deblurring, inpainting, and sparse-view tomography, VPAL consistently delivers state-of-the-art reconstructions, positioning variable projection as a powerful tool for modern large-scale inverse problems.

math.OC↗

arXiv2024

A data-dependent regularization method based on the graph Laplacian

We investigate a variational method for ill-posed problems, named $\texttt{graphLa+}Ψ$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method $Ψ$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\texttt{graphLa+}Ψ$ is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in 2D computerized tomography, wherein we integrate the $\texttt{graphLa+}Ψ$ method with various reconstruction techniques $Ψ$, including Filter Back Projection ($\texttt{graphLa+FBP}$), standard Tikhonov ($\texttt{graphLa+Tik}$), Total Variation ($\texttt{graphLa+TV}$), and a trained deep neural network ($\texttt{graphLa+Net}$). The $\texttt{graphLa+}Ψ$ approach significantly enhances the quality of the approximated solutions for each method $Ψ$. Notably, $\texttt{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.

math.NA↗

arXiv2024

A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical $\ell^2$--norm on the fidelity term.

math.NA↗