Ultrafast Preconditioned Conjugate Gradient OSEM algorithm for fully 3D PET reconstruction

The Conjugate Gradient (CG) method is an optimization algorithm used to determine the numerical solution of particular systems of linear equations which may be expressed as a symmetric and positive definite matrix. The CG method is iterative, so it can be applied to systems which are too large to be handled by direct methods. The CG method can also be used to solve unconstrained optimization problems such as PET reconstruction. In the Bayesian PET reconstruction problem, Preconditioned Conjugate Gradient (PCG) algorithms were previously shown to have more favorable convergence rates than expectation maximization (EM) type algorithms. However, PCG fails to converge on partial datasets. Block iterative methods such as Ordered Subset Expectation Maximization (OSEM) have become the most commonly used methods in PET reconstruction, as they require less iteration than PCG. This work combines both algorithms, PCG-OSEM, to reduce the number of iterations and speed up the convergence of OSEM. The proposed search direction of the CG is orthogonal to previous search directions, and in the image space rather than projection domain. Therefore, single iteration can be performed to achieve an acceptable PET reconstructed image.