Kaczmarz extended algorithm for tomographic image reconstruction from limited-data

The algebraic reconstruction technique (ART), based on the well known algorithm proposed by Kaczmarz in 1937, is one of the most important class of solution methods for image reconstruction problems. But unfortunately, almost all the methods from the ART class give satisfactory results only in the case of consistent problems. In the inconsistent case (and unfortunately this is what happens in real applications, because of measurement errors) they give only more or less "controllable" versions of the exact solutions. This is exactly the case that we analyze in the present paper. We start with a theoretical analysis of the classical Kaczmarz's projection method in the case of an inconsistent linear least-squares problem and we prove that the approximations so obtained are at a certain distance from the set of exact least-squares solutions. This distance is controlled by the component of the right hand side of the problem lying in the orthogonal complement of the range of problem's matrix, i.e. exactly the component that makes the problem inconsistent. For overcoming this difficulty we consider an extended version of Kaczmarz's algorithm, previously analyzed by one of the authors. In the numerical experiments described in the last part of the paper we compare the above mentioned extension with two well known (ART) type algorithms for image reconstruction in two electromagnetic geotomography problems. The results indicate that the extended Kaczmarz algorithm gives much better results than the other two.