Fully 3d Pet Image Reconstruction Using a Fourier Preconditioned Conjugate-gradient Algorithm

Since the data sizes in fully 3D PET imaging are very large, iterative image reconstruction algorithms must converge in very few iterations to be useful. One can improve the convergence rate of the conjugate-gradient (CG) algorithm by incorporating precondi-tioning operators that approximate the inverse of the Hessian of the objective function. If the 3D cylindrical PET geometry were not truncated at the ends, then the Hessian of the penalized least-squares objective function would be approximately shift-invariant, i.e. G 0 G would be nearly block-circulant, where G is the system matrix. We propose a Fourier precondi-tioner based on this shift-invariant approximation to the Hessian. Results show that this preconditioner signiicantly accelerates the convergence of the CG algorithm with only a small increase in computation.