A general framework for nonlinear multigrid inversion.
A variety of new imaging modalities, such as optical diffusion tomography, require the inversion of a forward problem that is modeled by the solution to a three-dimensional partial differential equation. For these applications, image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate. In this paper, we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of 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. Importantly, the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions. An application of our method to Bayesian optical diffusion tomography with a generalized Gaussian Markov random-field image prior model shows the potential for very large computational savings. Numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem.
Oh
Seungseok
S
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-2035, USA. ohs@ecn.purdue.edu
Milstein
Adam B
AB
Bouman
Charles A
CA
Webb
Kevin J
KJ
D000465:Algorithms; D001185:Artificial Intelligence; D003196:Computer Graphics; D003198:Computer Simulation; D007089:Image Enhancement / Q000379:methods*; D007090:Image Interpretation, Computer-Assisted / Q000379:methods*; D021621:Imaging, Three-Dimensional / Q000379:methods*; D016247:Information Storage and Retrieval / Q000379:methods; D008954:Models, Biological; D015233:Models, Statistical; D017711:Nonlinear Dynamics; D009716:Numerical Analysis, Computer-Assisted; D010363:Pattern Recognition, Automated / Q000379:methods; D015203:Reproducibility of Results; D012680:Sensitivity and Specificity; D012815:Signal Processing, Computer-Assisted; D041622:Tomography, Optical / Q000379:methods*