Conjugate-gradient preconditioning methods for shift-variant PET image reconstruction

Gradient-based iterative methods often converge slowly for tomographic imag e reconstruction and image restoration problems, but can be accelerated by suitable preconditioners, Diagonal preconditioners offer some improvement i n convergence rate, but do not incorporate the structure of the Hessian mat rices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessia ns, However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited opti cal imaging, the Hessian of the weighted least-squares objective function i s quite shift-variant, and circulant preconditioners perform poorly, Additi onal shift-variance is caused by edge-preserving regularization methods bas ed on nonquadratic penalty functions. This paper describes new precondition ers that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the ne w preconditioners lead to significantly faster convergence rates for the un constrained conjugate-gradient (CG) iteration. We also propose a new effici ent method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.