Uniform multi-penalty regularization for linear ill-posed inverse problems

This study examines, in the framework of variational regularization methods, a multi-penalty regularization approach which builds upon the Uniform PENalty (UPEN) method, previously proposed by the authors for Nuclear Magnetic Resonance (NMR) data processing. The paper introduces two iterative methods, UpenMM and GUpenMM, formulated within the Majorization-Minimization (MM) framework. These methods are designed to identify appropriate regularization parameters and solutions for linear inverse problems utilizing multi-penalty regularization. The paper demonstrates the convergence of these methods and illustrates their potential through numerical examples in one and two-dimensional scenarios, showing the practical utility of point-wise regularization terms in solving various inverse problems

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements

Denoising and Segmentation of MR Images by Coupled Diffusive Filters

The image denoising and segmentation is a fundamental task in many medical applications based on magnetic resonance image processing. This problem can be solved by means of nonlinear diffusive filters requiring the solution of evolutive partial differential equations. In this work a coupled system of linear and nonlinear diffusion-reaction equations is proposed and tested for denoising and segmentation of magnetic resonance images. The discretization of the coupled system by means of the Finite Element method is reported. The effectiveness of the model has been tested on MR images affected by gaussian, impulsive noise and also in the case of dynamic magnetic resonance images where the data are affected by noise in the frequency domain

Computation of Regularization Parameters using the Fourier Coefficients

In the solution of ill-posed problems by means of regularization methods, a crucial issue is the computation of the regularization parameter. In this work we focus on the Truncated Singular Value Decomposition (TSVD) and Tikhonov method and we define a method for computing the regularization parameter based on the behavior of Fourier coefficients. We compute a safe index for truncating the TSVD and consequently a value for the regularization parameter of the Tikhonov method. An extensive numerical experimentation is carried out on the Hansen's Regtool test problems and the results confirm the effectiveness and robustness of the method proposed

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