Nonlinear multigrid optimization for image reconstruction
by
Rick Millane
Purdue University, USA
Coauthors: J.C. Ye (University of Illinois, USA), C.A. Bouman (Purdue University, USA), K.J. Webb (Purdue University, USA)

We describe a nonlinear multigrid scheme for reducing the computational burden of non-quadratic optimisation problems. This method is applied to maximum a posteriori image reconstruction in optical diffusion tomography. The multigrid method is applied directly to the optimisation problem. Given an approximate solution to the optimisation problem at one grid level, we derive an update scheme that involves minimising a modified cost function at the next coarser grid level, and then use this solution to improve the solution at the finer level. We show that for a convex cost function, the exact solution to the fine grid optimisation problem is a fixed point to this update procedure. The update procedure is applied using standard V-cycle and full multigrid recursions. Optical diffusion tomography is a technique for imaging highly scattering media using measurements of scattered and attenuated light, which can be modelled using the diffusion equation. This represents a difficult and computationally intensive inverse problem because of the nonlinear forward model. We use a forward model based on the diffusion equation that incorporates shot-noise statistics to represent measurement noise. The solution is regularised using a Bayesian formalism with a generalised Gaussian Markov random field prior model, and the resulting optimisation problem is solved for the maximum a posteriori (MAP) estimate of the image. Numerical simulations show that this method converges much faster, and tends to find a better local mimimum, than do fixed-grid methods.

Date received: October 3, 2000