A preconditioned Krylov-subspace conjugate gradient solver for emission tomograph

The authors have implemented the mathematical tools and computational techniques that may render canonical formulation of the inverse problem solvable while compensating for noise, statistical fluctuation of data acquisition, physical effects and geometric distortions in emission tomography. The authors proposed and developed a new reconstruction approach that is based on a Krylov-subspace pixel basis decomposition of the tomographic space. Their strategy obviates the assumption of pixel geometries which may cause systematic and statistical errors. No special feature of the matrix A is assumed and so the authors´ solution is general and is independent of detectors´s geometry. Krylov-space algorithms are iterative numerical methods for sparse linear and eigenvalue problems. Given a linear system Ax=b, and a number of iterations T, the methods find the vector x^(T)∈K_T which is closest to z in some appropriate norm, where K_T={b, Ab, A²b, …, A^T-1}. The authors´ Krylov techniques are efficient and exploit the architecture of the computing machines to provide optimal performance guarantees. The authors describe a preconditioned Krylov-subspace conjugate gradient (CG) solver for PCR2, a cylindrical emission tomograph built at MGH. The out-of-core Krylov-subspace algorithm minimizes data flow between the computer´s primary and secondary memories and improves performance by one order of magnitude when compared to naive implementations relying on paging to perform I/O