Numerical methods for dynamic Magnetic Resonance Imaging

Nel rapporto vengono presentati alcuni metodi nuemrici per la ricostruzione di immagini dinamiche di risonanza magnetica utilizzando anche informazioni a priori, attraverso una formula unificata che li descrive tutti. In particolare, si studia l'uso delle basi B-spline e della regolarizzazione nella ricostruzione di immagini dinamiche di risonanza magnetica

To be or not to be stable, that is the question: understanding neural networks for inverse problems

The solution of linear inverse problems arising, for example, in signal and image processing is a challenging problem, since the ill-conditioning amplifies the noise on the data. Recently introduced deep-learning based algorithms overwhelm the more traditional model-based approaches but they typically suffer from instability with respect to data perturbation. In this paper, we theoretically analyse the trade-off between neural networks stability and accuracy in the solution of linear inverse problems. Moreover, we propose different supervised and unsupervised solutions, to increase network stability by maintaining good accuracy, by inheriting, in the network training, regularization from a model-based iterative scheme. Extensive numerical experiments on image deblurring confirm the theoretical results and the effectiveness of the proposed networks in solving inverse problems with stability with respect to noise.Comment: 26 pages, 9 figures, divided in 4 blocks of figures in the LaTeX code. Paper will be sent for publication on a journal soon. This is a preliminary version, updated versions will be uploaded on ArXi

Recurrent Neural Networks Applied to GNSS Time Series for Denoising and Prediction

Global Navigation Satellite Systems (GNSS) are systems that continuously acquire data and provide position time series. Many monitoring applications are based on GNSS data and their efficiency depends on the capability in the time series analysis to characterize the signal content and/or to predict incoming coordinates. In this work we propose a suitable Network Architecture, based on Long Short Term Memory Recurrent Neural Networks, to solve two main tasks in GNSS time series analysis: denoising and prediction. We carry out an analysis on a synthetic time series, then we inspect two real different case studies and evaluate the results. We develop a non-deep network that removes almost the 50% of scattering from real GNSS time series and achieves a coordinate prediction with 1.1 millimeters of Mean Squared Error

DeepCEL0 for 2D Single Molecule Localization in Fluorescence Microscopy

In fluorescence microscopy, Single Molecule Localization Microscopy (SMLM) techniques aim at localizing with high precision high density fluorescent molecules by stochastically activating and imaging small subsets of blinking emitters. Super Resolution (SR) plays an important role in this field since it allows to go beyond the intrinsic light diffraction limit. In this work, we propose a deep learning-based algorithm for precise molecule localization of high density frames acquired by SMLM techniques whose $\ell_{2}$-based loss function is regularized by positivity and $\ell_{0}$-based constraints. The $\ell_{0}$ is relaxed through its Continuous Exact $\ell_{0}$ (CEL0) counterpart. The arising approach, named DeepCEL0, is parameter-free, more flexible, faster and provides more precise molecule localization maps if compared to the other state-of-the-art methods. We validate our approach on both simulated and real fluorescence microscopy data

An Iterative Method for the Solution of Nonlinear Regularization Problems with Regularization Parameter Estimation

Ill posed problems constitute the mathematical model of a large variety of applications. Aim of this paper is to define an iterative algorithm finding the solution of a regularization problem. The method minimizes a function constituted by a least squares term and a generally nonlinear regularization term, weighted by a regularization parameter. The proposed method computes a sequence of iterates approximating the regularization parameter and a sequence of iterates approximating the solution. The numerical experiments performed on 1D test problems show that the algorithm gives good results with different regularization functions both in terms of precision and computational efficiency. Moreover, it could be easily applied to large size regularization problems

An Iterative Tikhonov Method for Large Scale Computations

In this paper we present an iterative method for the minimization of the Tikhonov regularization functional in the absence of information about noise. Each algorithm iteration updates both the estimate of the regularization parameter and the Tikhonov solution. In order to reduce the number of iterations, an inexact version of the algorithm is also proposed. In this case the inner Conjugate Gradient (CG) iterations are truncated before convergence. In the numerical experiments the methods are tested on inverse ill posed problems arising both in signal and image processing