A Tikhonov Regularization of the Inverse Problem Solution in Gamma Ray Tomography.

Bruna G. M. Araújo1, Carlos C. Dantas2, Silvio B. Melo3, Renan F.Pires4,Valdemir A. dos Santos5

1,2Departamento de Energia Nuclear, Universidade Federal de Pernambuco -UFPE Av. Profesor Luiz Freire 1000, 50740540, Recife – PE. ccd@ufpe.br,

3,4,Centro de Informática da Universidade Federal de Pernambuco CIN / UFPE

sbm@cin.ufpe.br

5Departamento de Química, Universidade Católica de Pernambuco Rua do Príncipe 526, Boa Vista 50050410, Recife PE

vas@unicap.br

ABSTRACT

The inverse problem by means of Least Squares numerical solution was investigated, the theoretical methodology is considered in a broad sense as Tikhonov regularization and also including the Morozov concept for delta parameter control specifically. The quality of the imaging reconstruction shows effective improvement by means of Tikhonov regularization applied to simple gamma ray tomography algorithms. The effects on the results for solutions of linear system of equations are significant by the applied regularization techniques. The ART (algebraic reconstruction technique) type algorithm solved the linear system with Matlab building functions for ideal data to compare with regularization implemented with SVD (singular value decomposition) and also with SVD plus Toepelitz, tridiagonal and identity operators in hybrid algorithms. The quality of reconstruction is evaluated by RMSE. Comparison of the results shows for a high noise level = 10 and high delta parameter = 1000 the SVD plus tridiagonal operator are the best choice.

List keywords: regularization operators, SVD optimization, inverse problem.