Laura Girometti

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.

In this work, we propose a parameter-free and efficient method to tackle the structure-texture image decomposition problem. In particular, we present a neural network LPR-NET based on the unrolling of the Low Patch Rank model. On the one hand, this allows us to automatically learn parameters from data, and on the other hand to be computationally faster while obtaining qualitatively similar results compared to traditional iterative model-based methods. Moreover, despite being trained on synthetic images, numerical experiments show the ability of our network to generalize well when applied to natural images.

Many tasks in real life scenarios can be naturally formulated as nonconvex optimization problems. Unfortunately, to date, the iterative numerical methods to find even only the local minima of these nonconvex cost functions are extremely slow and strongly affected by the initialization chosen. We devise a predictor–corrector strategy that efficiently computes locally optimal solutions to these problems. An initialization-free convex minimization allows to predict a global good preliminary candidate, which is then corrected by solving a parameter-free nonconvex minimization. A simple algorithm, such as alternating direction method of multipliers works surprisingly well in producing good solutions. This strategy is applied to the challenging problem of decomposing a 1D signal into semantically distinct components mathematically identified by smooth, piecewise-constant, oscillatory structured and unstructured (noise) parts.