Nonlinear multigrid inversion algorithms with applications to statistical image reconstruction

Many tasks in image processing applications, such as reconstruction, deblurring, and registration, depend on the solution to inverse problems. In this thesis, we present nonlinear multigrid inversion methods for solving computationally expensive inverse problems. The multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems. The method works by dynamically adjusting the cost functionals at different scales so that they are consistent with, and ultimately reduce, the finest scale cost functional. In this way, the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem. While multigrid inversion is a general framework applicable to a wide variety of inverse problems, it is particulary well-suited for the inversion of nonlinear forward problems such as those modeled by the solution to partial differential equations since the new algorithm can greatly reduce computation by more coarsely descretizing both the forward and inverse problems at lower resolutions. An application of our method to optical diffusion tomography shows the potential for very large computational savings, better reconstruction quality, and robust convergence with a range of initialization conditions for this non-convex optimization problem. The method is extended to further reduce computations by reducing the resolutions of the data space as well as the parameter space at coarse scales. Applications of the approach to Bayesian reconstruction algorithms in transmission and emission tomography are presented, both with a Poisson noise model and with a quadratic data term. Simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixed-grid iterative coordinate descent (ICD) method and a multigrid method with fixed data resolution.