Tuesday, March 12, 1996
Iterative methods for inverse problems often converge slowly. Preconditioning methods can often accelerate gradient-based iterations. Previous preconditioning methods for tomographic reconstruction have used either diagonal or Fourier-based preconditioners. Fourier-based preconditioners are particularly well suited to problems with approximately circulant (e.g. Toeplitz-block-Toeplitz) Hessian matrices. However, in applications with nonuniform noise variance, such as arises from Poisson statistics, the circulant approximation to the Hessian is suboptimal. We present a new combined diagonal/Fourier preconditioner that more accurately approximates certain Hessian matrices ("modulated" Toeplitz-block-Toeplitz) and thus leads to significantly faster convergence rates than when either diagonal or Fourier preconditioning used alone. An application to position emission tomography illustrates the method.
(Joint work with Scott Booth)