Combined diagonal/Fourier preconditioning methods for image reconstruction in emission tomography

Iterative methods for tomographic image reconstruction often converge slowly. Preconditioning methods can often accelerate gradient-based iterations. Previous preconditioning methods for PET reconstruction have used either diagonal or Fourier-based preconditioners. Fourier-based preconditioners are well suited to problems with near-circulant Hessian matrices. However, due to the nonuniform Poisson noise variance in PET, the circulant approximation to the Hessian is suboptimal. This paper shows that a particular combined diagonal/Fourier preconditioner yields a more accurate approximation to the Hessian and gives significantly faster convergence rates than does either preconditioner used alone.