arXiv subjects

Alessandro Lanza

Publications and source records attributed to Alessandro Lanza.

13 recordsLinked to original sources

arXiv2026

Guided Variational Network for Image Decomposition

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.

math.NA↗

arXiv2025

Additive decomposition of one-dimensional signals using Transformers

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.

cs.LG↗

arXiv2025

Whiteness-based bilevel estimation of weighted TV parameter maps for image denoising

We consider a bilevel optimisation strategy based on normalised residual whiteness loss for estimating the weighted total variation parameter maps for denoising images corrupted by additive white Gaussian noise. Compared to supervised and semi-supervised approaches relying on prior knowledge of (approximate) reference data and/or information on the noise magnitude, the proposal is fully unsupervised. To avoid noise overfitting an early stopping strategy is used, relying on simple statistics of optimal performances on a set of natural images. Numerical results comparing the supervised/unsupervised procedures for scalar/pixel-dependent \mbox{parameter maps are shown.

math.OC↗

arXiv2024

Whiteness-based bilevel learning of regularization parameters in imaging

We consider an unsupervised bilevel optimization strategy for learning regularization parameters in the context of imaging inverse problems in the presence of additive white Gaussian noise. Compared to supervised and semi-supervised metrics relying either on the prior knowledge of reference data and/or on some (partial) knowledge on the noise statistics, the proposed approach optimizes the whiteness of the residual between the observed data and the observation model with no need of ground-truth data.We validate the approach on standard Total Variation-regularized image deconvolution problems which show that the proposed quality metric provides estimates close to the mean-square error oracle and to discrepancy-based principles.

math.OC↗

arXiv2022

Whiteness-based parameter selection for Poisson data in variational image processing

We propose a novel automatic parameter selection strategy for variational imaging problems under Poisson noise corruption. The selection of a suitable regularization parameter, whose value is crucial in order to achieve high quality reconstructions, is known to be a particularly hard task in low photon-count regimes. In this work, we extend the so-called residual whiteness principle originally designed for additive white noise to Poisson data. The proposed strategy relies on the study of the whiteness property of a standardized Poisson noise process. After deriving the theoretical properties that motivate our proposal, we solve the target minimization problem with a linearized version of the alternating direction method of multipliers, which is particularly suitable in presence of a general linear forward operator. Our strategy is extensively tested on image restoration and computed tomography reconstruction problems, and compared to the well-known discrepancy principle for Poisson noise proposed by Zanella at al. and with a nearly exact version of it previously proposed by the authors.

math.NA↗

arXiv2021

ADMM-based residual whiteness principle for automatic parameter selection in super-resolution problems

We propose an automatic parameter selection strategy for the problem of image super-resolution for images corrupted by blur and additive white Gaussian noise with unknown standard deviation. The proposed approach exploits the structure of both the down-sampling and the blur operators in the frequency domain and computes the optimal regularisation parameter as the one optimising a suitable residual whiteness measure. Computationally, the proposed strategy relies on the fast solution of generalised Tikhonov $\ell_2$-$\ell_2$ problems as proposed in a work from Zhao et al. These problems naturally appear as substeps of the Alternating Direction Method of Multipliers (ADMM) optimisation approach used to solve super-resolution problems with non-quadratic and often non-smooth, sparsity-promoting regularisers both in convex and in non-convex regimes. After detailing the theoretical properties defined in the frequency domain which allow to express the whiteness functional in a compact way, we report an exhaustive list of numerical experiments proving the effectiveness of the proposed approach for different type of problems, in comparison with well-known parameter selection strategy such as, e.g., the discrepancy principle.

math.NA↗

arXiv2021

On and beyond Total Variation regularisation in imaging: the role of space variance

Over the last 30 years a plethora of variational regularisation models for image reconstruction has been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin-Osher-Fatemi (ROF) model \cite{ROF} which relies on the minimisation of the edge-preserving Total Variation (TV) semi-norm as regularisation term. Despite its (often limiting) simplicity, this model is still very much employed in many applications and used as a benchmark for assessing the performance of modern learning-based image reconstruction approaches, thanks to its thorough analytical and numerical understanding. Among the many extensions to TV proposed over the years, a large class is based on the concept of \emph{space variance}. Space-variant models can indeed overcome the intrinsic inability of TV to describe \emph{local} features (strength, sharpness, directionality) by means of an adaptive mathematical modelling which accommodates local regularisation weighting, variable smoothness and anisotropy. Those ideas can further be cast in the flexible Bayesian framework of generalised Gaussian distributions and combined with maximum likelihood and hierarchical optimisation approaches for efficient hyper-parameter estimation. In this work, we review and connect the major contributions in the field of space-variant TV-type image reconstruction models, focusing, in particular, on their Bayesian interpretation which paves the way to new exciting and unexplored research directions.

math.NA↗

arXiv2021

Residual whiteness principle for automatic parameter selection in $\ell_2$-$\ell_2$ image super-resolution problems

We propose an automatic parameter selection strategy for variational image super-resolution of blurred and down-sampled images corrupted by additive white Gaussian noise (AWGN) with unknown standard deviation. By exploiting particular properties of the operators describing the problem in the frequency domain, our strategy selects the optimal parameter as the one optimising a suitable residual whiteness measure. Numerical tests show the effectiveness of the proposed strategy for generalised $\ell_2$-$\ell_2$ Tikhonov problems.

math.NA↗

arXiv2019

Space-adaptive anisotropic bivariate Laplacian regularization for image restoration

In this paper we present a new regularization term for variational image restoration which can be regarded as a space-variant anisotropic extension of the classical isotropic Total Variation (TV) regularizer. The proposed regularizer comes from the statistical assumption that the gradients of the target image distribute locally according to space-variant bivariate Laplacian distributions. The highly flexible variational structure of the corresponding regularizer encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. A minimization algorithm based on the Alternating Direction Method of Multipliers (ADMM) is presented for the efficient numerical solution of the proposed variational model. Some experimental results are reported which demonstrate the high-quality of restorations achievable by the proposed model, in particular with respect to classical Total Variation regularization.

eess.IV↗

arXiv2019

Space-variant Generalized Gaussian Regularization for Image Restoration

We propose a new space-variant regularization term for variational image restoration based on the assumption that the gradient magnitudes of the target image distribute locally according to a half-Generalized Gaussian distribution. This leads to a highly flexible regularizer characterized by two per-pixel free parameters, which are automatically estimated from the observed image. The proposed regularizer is coupled with either the $L_2$ or the $L_1$ fidelity terms, in order to effectively deal with additive white Gaussian noise or impulsive noises such as, e.g, additive white Laplace and salt and pepper noise. The restored image is efficiently computed by means of an iterative numerical algorithm based on the alternating direction method of multipliers. Numerical examples indicate that the proposed regularizer holds the potential for achieving high quality restorations for a wide range of target images characterized by different gradient distributions and for the different types of noise considered.

eess.IV↗

arXiv2019

Space-variant TV regularization for image restoration

We propose two new variational models aimed to outperform the popular total variation (TV) model for image restoration with L$_2$ and L$_1$ fidelity terms. In particular, we introduce a space-variant generalization of the TV regularizer, referred to as TV$_p^{SV}$, where the so-called shape parameter $p\,$ is automatically and locally estimated by applying a statistical inference technique based on the generalized Gaussian distribution. The restored image is efficiently computed by using an alternating direction method of multipliers procedure. We validated our models on images corrupted by Gaussian blur and two important types of noise, namely the additive white Gaussian noise and the impulsive salt and pepper noise. Numerical examples show that the proposed approach is particularly effective and well suited for images characterized by a wide range of gradient distributions.

eess.IV↗

arXiv2019

Adaptive parameter selection for weighted-TV image reconstruction problems

We propose an efficient estimation technique for the automatic selection of locally-adaptive Total Variation regularisation parameters based on an hybrid strategy which combines a local maximum-likelihood approach estimating space-variant image scales with a global discrepancy principle related to noise statistics. We verify the effectiveness of the proposed approach solving some exemplar image reconstruction problems and show its outperformance in comparison to state-of-the-art parameter estimation strategies, the former weighting locally the fit with the data (Dong et al. '11), the latter relying on a bilevel learning paradigm (Hintermüller et al., '17)

math.OC↗

arXiv2019

A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection

We propose a new space-variant anisotropic regularisation term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalised Gaussian distribution. The highly flexible variational structure of the corresponding regulariser encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. For the numerical solution of the corresponding image restoration model, we use an iterative algorithm based on the Alternating Direction Method of Multipliers (ADMM). A suitable preliminary variable splitting together with a novel result in multivariate non-convex proximal calculus yield a very efficient minimisation algorithm. Several numerical results showing significant quality-improvement of the proposed model with respect to some related state-of-the-art competitors are reported, in particular in terms of texture and detail preservation.

math.NA↗