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The reconstruction of a function on the rotation group from mean values along all geodesics is an overdetermined problem, i.e. it is sufficient to know the mean values for a three-dimensional subset of all geodesics on the rotation group. In this paper we give a Fourier slice theorem for the restricted problem. Based on the Fourier slice theorem and fast Fourier transforms on the rotation group and the sphere we introduce a fast algorithm for the forward transform. Analyzing the inverse problem we come up with an exact inversion formula for bandlimited functions on the rotation group. Unfortunately this inversion formula turns out to be extremely ill conditioned. Therefore we introduce an iterative approach which makes use of regularization and the fast algorithm for the forward transform. Numerical experiments indicate the applicability of our algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a new joint inversion approach to shape-based inverse problems. Given two sets of data from distinct physical models, the main objective is to obtain a unified characterization of inclusions within the spatial domain of the physical properties to be reconstructed. Although our proposed method generally applies to many types of inverse problems, the main motivation here is to characterize subsurface contaminant source zones by processing down-gradient hydrological data and cross-gradient electrical resistance tomography observations. Inspired by Newton's method for multi-objective optimization, we present an iterative inversion scheme in which descent steps are chosen to simultaneously reduce both data-model misfit terms. Such an approach, however, requires solving a non-smooth convex problem at every iteration, which is computationally expensive for a pixel-based inversion over the whole domain. Instead, we employ a parametric level set technique that substantially reduces the number of underlying parameters, making the inversion computationally tractable. The performance of the technique is examined and discussed through the reconstruction of source zone architectures that are representative of dense non-aqueous phase liquid (DNAPL) contaminant release in a statistically homogenous sandy aquifer. In these examples, the geometric configuration of the DNAPL mass is considered along with additional information about its spatial variability within the contaminated zone, such as the identification of low and high saturation regions. Comparison of the reconstructions with the true DNAPL architectures highlights the superior performance of the model-based technique and joint inversion scheme. [ABSTRACT FROM AUTHOR]

Consider a two-dimensional backward problem for a time-fractional diffusion process, which can be considered as image de-blurring where the blurring process is assumed to be slow diffusion. In order to avoid the over-smoothing effect for object image with edges and to construct a fast reconstruction scheme, the total variation regularizing term and the data residual error in the frequency domain are coupled to construct the cost functional. The well posedness of this optimization problem is studied. The minimizer is sought approximately using the iteration process for a series of optimization problems with Bregman distance as a penalty term. This iteration reconstruction scheme is essentially a new regularizing scheme with coupling parameter in the cost functional and the iteration stopping times as two regularizing parameters. We give the choice strategy for the regularizing parameters in terms of the noise level of measurement data, which yields the optimal error estimate on the iterative solution. The series optimization problems are solved by alternative iteration with explicit exact solution and therefore the amount of computation is much weakened. Numerical implementations are given to support our theoretical analysis on the convergence rate and to show the significant reconstruction improvements. [ABSTRACT FROM AUTHOR]