Minimal residual cone-beam reconstruction with attenuation correction in SPECT

This paper presents an iterative method based on the minimal residual algorithm for tomographic attenuation compensated reconstruction from attenuated cone-beam projections given the attenuation distribution. Unlike conjugate-gradient based reconstruction techniques, the proposed minimal residual based algorithm solves directly a quasisymmetric linear system, which is a preconditioned system. Thus it avoids the use of normal equations, which improves the convergence rate. Two main contributions are introduced. First, a regularization method is derived for quasisymmetric problems, based on a Tikhonov-Phillips regularization applied to the factorization of the symmetric part of the system matrix. This regularization is made spatially adaptive to avoid smoothing the region of interest. Second, our existing reconstruction algorithm for attenuation correction in parallel-beam geometry is extended to cone-beam geometry. A circular orbit is considered. Two preconditioning operators are proposed: the first one is Grangeat's inversion formula and the second one is Feldkamp's inversion formula. Experimental results obtained on simulated data are presented and the shadow zone effect on attenuated data is illustrated.