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.

Abstract. Electrical impedance tomography is an imaging modality for extracting information on the conductivity distribution inside a physical body from boundary measurements of current and voltage. In many practical applications, it is a priori known that the conductivity consists of embedded inhomogeneities in an approximately constant background. This work introduces an iterative reconstruction algorithm that aims at finding the maximum a posteriori estimate for the conductivity assuming an edge-preferring prior. The method is based on applying (a single step of) priorconditioned lagged diffusivity iteration to sequential linearizations of the forward model. The algorithm is capable of producing reconstructions on dense unstructured three-dimensional finite element meshes and with a high number of measurement electrodes. The functionality of the proposed technique is demonstrated with both simulated and experimental data in the framework of the complete electrode model, which is the most accurate model for practical impedance tomography.Key words. Electrical impedance tomography, priorconditioning, edge-preferring regularization, LSQR, complete electrode model AMS subject classifications. 65N21, 35R301. Introduction. The aim of electrical impedance tomography is to reconstruct the internal conductivity distribution of a physical body based on boundary measurements of current and voltage. This constitutes a nonlinear and severely illposed inverse problem. EIT can be used in medical imaging, process tomography and nondestructive testing of materials. Consult the review articles [2, 10, 33] for more information on EIT, the associated mathematical theory and the related reconstruction algorithms. In this work, we consider EIT under the prior assumption that the tobe-reconstructed conductivity consists of well defined inclusions in an approximately homogeneous background, which is a setting encountered in many practical applications: Consider, e.g., the localization of air bubbles or manufacturing defects in a piece of building material. We work exclusively with the complete electrode model (CEM), which is the most accurate model for real-world EIT [11,32]; in particular, the introduced algorithm also estimates the contact resistances that are an unavoidable nuisance of practical EIT.We tackle the reconstruction problem of EIT within the Bayesian paradigm and incorporate the prior information on the structure of the imaged object by introducing an edge-preferring prior density for the (discretized) conductivity. The prior for the contact resistances is chosen to be uninformative since their estimation from EIT measurements is not an illposed problem. Assuming an additive Gaussian measurement noise model, the computation of the maximum a posteriori (MAP) estimate, i.e., the maximizer of the posterior density, corresponds to finding a minimizer for a Tikhonovtype functional that exhibits nonquadratic behavior in both the discrepancy and the penalty term. In the terminology of regularization theory, this corresponds ...

This work considers electrical impedance tomography in the special case that the boundary measurements of current and voltage are carried out with two (infinitely) small electrodes. One of the electrodes lies at a fixed position while the other is moved along the object boundary in a sweeping motion, with the corresponding measurement being the (relative) potential difference required for maintaining a unit current between the two electrodes. Assuming that the two-dimensional object of interest has constant background conductivity but is contaminated by compactly supported inhomogeneities, it is shown that such sweep data represent the boundary value of a holomorphic function defined in the exterior of the embedded inclusions. This observation makes it possible to use the sweep data as the input for the convex source support method in order to localize conductivity inhomogeneities. The functionality of the resulting algorithm is demonstrated by numerical experiments both with idealized point electrode data and with simulated complete electrode model measurements.

This is the documentation of the tomographic X-ray data of a walnut made available at http://www.fips.fi/dataset.php. The data can be freely used for scientific purposes with appropriate references to the data and to this document in arXiv. The data set consists of (1) the X-ray sinogram of a single 2D slice of the walnut with three different resolutions and (2) the corresponding measurement matrices modeling the linear operation of the X-ray transform. Each of these sinograms was obtained from a measured 120-projection fanbeam sinogram by down-sampling and taking logarithms. The original (measured) sinogram is also provided in its original form and resolution. In addition, a larger set of 1200 projections of the same walnut was measured and a high-resolution filtered back-projection reconstruction was computed from this data; both the sinogram and the FBP reconstruction are included in the data set, the latter serving as a ground truth reconstruction.

In optical tomography a physical body is illuminated with near-infrared light and the resulting outward photon flux is measured at the object boundary. The goal is to reconstruct internal optical properties of the body, such as absorption and diffusivity. In this work, it is assumed that the imaged object is composed of an approximately homogeneous background with clearly distinguishable embedded inhomogeneities. An algorithm for finding the maximum a posteriori estimate for the absorption and diffusion coefficients is introduced assuming an edge-preferring prior and an additive Gaussian measurement noise model. The method is based on iteratively combining a lagged diffusivity step and a linearization of the measurement model of diffuse optical tomography with priorconditioned LSQR. The performance of the reconstruction technique is tested via threedimensional numerical experiments with simulated measurement data.

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