Fabiana Zama

The conjugate gradient regularization method in Computed Tomography problems

In this work we solve inverse problems coming from the area of Computed Tomography by means of regularization methods based on conjugate gradient iterations. We develop a stopping criterion which is efficient for the computation of a regularized solution for the least-squares normal equations. The stopping rule can be suitably applied also to the Tikhonov regularization method. We report computational experiments based on different physical models and with different degrees of noise. We compare the results obtained with those computed by other currently used methods such as Algebraic Reconstruction Techniques (ART) and Backprojection.

The active-set method for nonnegative regularization of linear ill-posed problems

Abstract In this work, we analyze the behavior of the active-set method for the nonnegative regularization of discrete ill-posed problems. In many applications, the solution of a linear ill-posed problem is known to be nonnegative. Standard Tikhonov regularization often provides an approximated solution with negative entries. We apply the active-set method to find a nonnegative approximate solution of the linear system starting from the Tikhonov regularized one. Our numerical experiments show that the active-set method is effective in reducing the oscillations in the Tikhonov regularized solution and in providing a nonnegative regularized solution of the original linear system.

Uniform Penalty inversion of two-dimensional NMR relaxation data

The inversion of two-dimensional NMR data is an ill-posed problem related to the numerical computation of the inverse Laplace transform. In this paper we present the 2DUPEN algorithm that extends the Uniform Penalty (UPEN) algorithm [Borgia, Brown, Fantazzini, {\em Journal of Magnetic Resonance}, 1998] to two-dimensional data. The UPEN algorithm, defined for the inversion of one-dimensional NMR relaxation data, uses Tikhonov-like regularization and optionally non-negativity constraints in order to implement locally adapted regularization. In this paper, we analyze the regularization properties of this approach. Moreover, we extend the one-dimensional UPEN algorithm to the two-dimensional case and present an efficient implementation based on the Newton Projection method. Without any a-priori information on the noise norm, 2DUPEN automatically computes the locally adapted regularization parameters and the distribution of the unknown NMR parameters by using variable smoothing. Results of numerical experiments on simulated and real data are presented in order to illustrate the potential of the proposed method in reconstructing peaks and flat regions with the same accuracy.

A descent method for regularization of ill-posed problems

In this paper, we describe an iterative algorithm, called descent-TCG, based on truncated conjugate gradients iterations to compute Tikhonov regularized solutions of linear ill-posed problems. The sequence of approximate solutions and regularization parameters, computed by the algorithm, is shown to decrease the value of the Tikhonov functional. Suitable termination criteria are built-up to define an inner–outer iteration scheme that computes a regularized solution. Numerical experiments are performed to compare this algorithm with other well established regularization methods. We observe that the best descent-TCG results occur for highly noised data and we always get fairly reliable solutions, thus it prevents the dangerous error growth often appearing in other well established regularization methods. Finally, the descent-TCG method is computationally advantageous especially for large size problems.

A domain decomposition technique for spline image restoration on distributed memory systems

The problem of image restoration is considered when the point spread function is Space Variant Non Separable. The algorithm determines a continuous approximation of the original object, following the continuous object-discrete image approach. The image spatial domain is decomposed into subdomains and the local approximants are computed on a distributed memory environment. The continuity of the solution across the image subdomains is obtained by adding a suitable overlapping area to the sides of the subdomains. Numerical experiments have been carried out on a Hypercube Intel iPSC/860 and the most interesting results are reported.

AN EXPERIMENT IN IMAGE RESTORATION USING TRANSPUTER NETWORKS

The problem of image restoration, with a blurring function linear, space-variant and nonseparable, has been solved on a transputer network, using primitives of Parasoft express environment and A.CS.Arnia Library. A domain decomposition strategy has been introduced to split the problem among the processors. Some interesting computational results are reported.

Iterative Constrained Minimization for Vectorial TV Image Deblurring

In this paper, we consider the problem of restoring blurred noisy vectorial images where the blurring model involves contributions from the different image channels (cross-channel blur). The proposed method restores the images by solving a sequence of quadratic constrained minimization problems where the constraint is automatically adapted to improve the quality of the restored images. In the present case, the constraint is the Total Variation extended to vectorial images, and the objective function is the

A total-variation-based regularization strategy in magnetic resonance imaging

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