Abstract
This article presents a method for solving large-scale linear inverse imaging problems regularized with a nonlinear, edge-preserving penalty term such as total variation or the Perona–Malik technique. Our method is aimed at problems defined on unstructured meshes, where such regularizers naturally arise in unfactorized form as a stiffness matrix of an anisotropic diffusion operator and factorization is prohibitively expensive. In the proposed scheme, the nonlinearity is handled with lagged diffusivity fixed point iteration, which involves solving a large-scale linear least squares problem in each iteration. Because the convergence of Krylov methods for problems with discontinuities is notoriously slow, we propose to accelerate it by means of priorconditioning (Bayesian preconditioning). priorconditioning is a technique that, through transformation to the standard form, embeds the information contained in the prior (Bayesian interpretation of a regularizer) directly into the forward operator and thence into the solution space. We derive a factorization-free preconditioned LSQR algorithm (MLSQR), allowing implicit application of the preconditioner through efficient schemes such as multigrid. The resulting method is also matrix-free i.e. the forward map can be defined through its action on a vector. We illustrate the performance of the method on two numerical examples. Simple 1D-deblurring problem serves to visualize the discussion throughout the paper. The effectiveness of the proposed numerical scheme is demonstrated on a three-dimensional problem in fluorescence diffuse optical tomography with total variation regularization derived algebraic multigrid preconditioner, which is the type of large scale, unstructured mesh problem, requiring matrix-free and factorization-free approaches that motivated the work here.
Highlights
Inverse problems arise in almost all fields of science when details of a model have to be determined from a set of observed data
Using general singular value decomposition (GSVD) decomposition, we show that the Krylov subspace corresponding to the priorconditioned problem amplifies the directions spanning the covariance of the prior, in contrast to the Krylov subspace for the original regularized problem
The GSVD form of (17) reveals that each Krylov iteration boosts the directions in the range of the regularizer M spanned by X−T corresponding to the large singular values while damping those corresponding to the small ones with respect to the last Krylov vector; there is no direct relation to the initial vector as in case of (18)
Summary
Inverse problems arise in almost all fields of science when details of a model have to be determined from a set of observed data. We are interested in linear problems after appropriate discretization, where X = nX, nX ∈ is the space of images (large-scale for two or three spatial dimensions) and Y = nY , nY ∈ is the data space Broad class of such problems includes image deblurring, denoising and inpainting, tomography based on the radon transform (x-ray CT) or attenuated radon transform (PET, SPECT), or fluorescence diffuse optical tomography (fDOT). For many of those problems, the matrix representation of A becomes too large to form explicitly, even for moderately sized images. Examples are projection operators in x-ray CT, PET and SPECT, and the solution of direct and adjoint partial differential equations in fDOT
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