The solution of
linear systems coming from the discretization of Fredholm integral equations of
the first kind is usually an ill-conditioned problem. Ordinary methods give
unacceptable solutions, so that regularization methods are necessary. Since the
systems arising from image reconstruction methods are usually large and sparse,
our research is mainly focused on iterative regularization methods, and in particular
conjugate gradients type regularization methods , that are
suitable for such systems.
Main results
The study of
regularization methods is mainly centred on the choice of the
regularization parameter. In conjugate gradient type methods, the
parameter is identified with the number of iterations of the method. Our
reserach is studying some heuristic criteria based on the behaviour of the
residuals and of the approximate solutions. These criteria do not need any
information about the noise and hence they can be applied on any linear system,
even on least squares problems. We tested these criteria on systems coming from
tomographic imaging reconstruction and from dynamic MR imaging reconstruction.
·
E.Loli
Piccolomini F. Zama G. Zanghirati, Regularization Methods in dynamic MRI
, Applied Mathematics and Computation, September 2002.
·
M.
Bertaja, S. Morigi, E. Loli Piccolomini, F. Sgallari, F. Zama, Regularization of Large Discrete ill posed
problems in image processing, Recent trends in Numerical analysis
(ed. Trigiante ), Advances in the Theory of Computational Mathematics, vol.
3, Nova Science, Books and Journals (2000) (ISBN 1-56072-885-X)
· E. Loli Piccolomini, F. Zama, Regularization Methods for the solution of Inverse Problems: Theory and Computational Aspects, Rendiconti del circolo Matematico di Palermo, ed. M. Maugeri, E. Galligani, serie II numero 58, 1999.
·
A. Baronio, E. Loli Piccolomini, F. Zama, A Method for solving the Indirect Approximation
Problem, Applied Mathematics and Computation, vol. 77, pag.97-107
(1996)
E. Loli Piccolomini, F. Zama, Regularization algorithms for image
reconstruction from projections, Atti dell'Accademia delle Scienze di
Bologna,1996.
·
E.
Loli Piccolomini, F. Zama, Regularization algorithms for image
reconstruction from projections, Series on Advances in Mathematics for
Applied Sciences, (edited by P. Ciarlini, M.G.Cox, F. Pavese, D. Richter),
World Scientific,vol. 45, 1997.
regularization methods, ill conditioned
systems, conjugate gradients, regularization parameter