## EECS CSPL SEMINAR SERIES

WINTER TERM 1996

Jeffrey Fessler
# Jeffrey Fessler

### Department of Electrical Engineering and Computer
Science, University of Michigan

### fessler@eecs.umich.edu

Tuesday, March 12, 1996

### Preconditioned Conjugate Gradient Methods for
Statistical Image Reconstruction

Abstract -

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)

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