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

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 3-D 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 optical diffusion tomography shows the potential for very large computational savings. Numerical data also indicates robust convergence with a range of initialization conditions for this non-convex optimization problem

Multigrid Tomographic Inversion with Variable Resolution Data and Image Spaces

A multigrid inversion approach that uses variable resolutions of both data space and image space is proposed. Since computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements, the proposed algorithm further reduces the computation relative to conventional multigrid approaches, which change only the image space resolution at coarse scales. The advantage is particularly important for data-rich applications, where data resolutions may differ for different scales. Applications of the approach to Bayesian reconstruction algorithms in transmission and emission tomography with a generalized Gaussian Markov random field image prior 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. Index Terms Multigrid algorithms, multiresolution, inverse problems, image reconstruction, computed tomography

Multigrid inversion algorithms with applications to optical diffusion tomography

In this paper, we propose a general framework for nonlinear multigrid inversion applicable to any inverse problem in which the forward model can be naturally represented at differing resolutions. In multigrid inversion, the problem is adjusted to be solved at each resolution by using the solutions at both finer and coarser resolutions. To do this, we formulate a consistent set of cost functionals across resolutions. At each resolution, both the forward and inverse problems are discretized at the lower resolution; thus reducing computation. Our simulation results for the problem of optical diffusion tomography indicate that multigrid inversion can dramatically reduce computation in this application.

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 3-D 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..

Fluorescence Optical Diffusion Tomography

Introduction Optical diffusion tomography #ODT# is emerging as a powerful tissue imaging modality. 1,2 In ODT, images are comprised of the spatially dependent absorption and scattering properties of the tissue. Boundary measurements from several sources and detectors are used to recover the unknown parameters from a scattering model described by a partial differential equation. Contrast between the properties of diseased and healthy tissue might then be used in clinical diagnosis. In principle, sinusoidally modulated, continuous-wave #cw#, or pulsed excitation light is launched into the biological tissue, where it undergoes multiple scattering and absorption before exiting. One can use the measured intensity and phase #or delay# information to reconstruct threedimensional #3-D# maps of the absorption and scattering properties by optimizing a fit to diffusion model computations. As a result of the nonlinear dependence of the diffusion equation photon flux on the unknown parameters