Application of Krylov subspaces to SPECT imaging

Abstract

ABSTRACT:The application of the conjugate gradient (CG) algorithm to the problem of data reconstruction in SPECT imaging indicates that most of the useful information is already contained in Krylov subspaces of small dimension, ranging from 9 (two-dimensional case) to 15 (three-dimensional case). On this basis, a new, proposed approach can be basically summarized as follows: construction of a basis spanning a Krylov subspace of suitable dimension and projection of the projector-backprojector matrix (a 10 6 ϫ 10 6 matrix in the three-dimensional case) onto such a subspace. In this way, one is led to a problem of low dimensionality, for which regularized solutions can be easily and quickly obtained. The required SPECT activity map is expanded as a linear combination of the basis elements spanning the Krylov subspace and the regularization acts by modifying the coefficients of such an expansion. By means of a suitable graphical interface, the tuning of the regularization parameter(s) can be performed interactively on the basis of the visual inspection of one or some slices cut from a reconstruction.

Image deblurring with Poisson data: from cells to galaxies

“…and, therefore, all methods developed in the framework of the least-square approach can be used. In fact, this approximation, combined with a preconditioned conjugate gradient (CG), is used since a long time in medical imaging (see, for instance, [37,55,74]). It has recently been re-proposed in the context of image deblurring [13,14].…”