Mixed Total Variation and L

Optical tomography is an emerging and important molecular imaging modality. The aim of optical tomography is to reconstruct optical properties of human tissues. In this paper, we focus on reconstructing the absorption coefficient based on the radiative transfer equation (RTE). It is an ill-posed parameter identification problem. Regularization methods have been broadly applied to reconstruct the optical coefficients, such as the total variation (TV) regularization and the L1 regularization. In order to better reconstruct the piecewise constant and sparse coefficient distributions, TV and L1 norms are combined as the regularization. The forward problem is discretized with the discontinuous Galerkin method on the spatial space and the finite element method on the angular space. The minimization problem is solved by a Jacobian-based Levenberg-Marquardt type method which is equipped with a split Bregman algorithms for the L1 regularization. We use the adjoint method to compute the Jacobian matrix which dramatically improves the computation efficiency. By comparing with the other imaging reconstruction methods based on TV and L1 regularizations, the simulation results show the validity and efficiency of the proposed method

Inversion of the Star Transform

We define the star transform as a generalization of the broken ray transform for image reconstruction in single scattering tomography. Using the star transform provides advantages including possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously. We derive the star transform from physical principles, and derive several computationally efficient algorithms for its inversion. We discuss mathematical properties and analyze numerical stability of inversion, and obtain necessary conditions for stable reconstruction. An approach combining scattered rays and ballistic rays to improve reconstruction is provided, and total variation and L1 regularization are utilized to remove noise. Numerical experiments are carried out to test the algorithms of inversion, the possibility to recover the absorption and the scattering coefficients, and the effect of different regularizations

Instability of an inverse problem for the stationary radiative transport near the diffusion limit

In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean free path (the Knudsen number) 0 < \eps \ll 1. It is well-known that there is a transition of stability from H\"{o}lder type to logarithmic type with \eps\to 0, the theory of this transition of stability is still an open problem. In this study, we show the transition of stability by establishing the balance of two different regimes depending on the relative sizes of \eps and the perturbation in measurements. When \eps is sufficiently small, we obtain exponential instability, which stands for the diffusive regime, and otherwise we obtain H\"{o}lder instability instead, which stands for the transport regime.Comment: 20 page

Analytical and Iterative Regularization Methods for Nonlinear Ill-posed Inverse Problems: Applications to Diffuse Optical and Electrical Impedance Tomography

Electrical impedance tomography (EIT) and Diffuse Optical Tomography (DOT) are imaging methods that have been gaining more popularity due to their ease of use and non-ivasiveness. EIT and DOT can potentially be used as alternatives to traditional imaging techniques, such as computed tomography (CT) scans, to reduce the damaging effects of radiation on tissue. The process of imaging using either EIT or DOT involves measuring the ability for tissue to impede electrical flow or absorb light, respectively. For EIT, the inner distribution of resistivity, which corresponds to different resistivity properties of different tissues, is estimated from the voltage potentials measured on the boundary of the object being imaged. In DOT, the optical properties of the tissue, mainly scattering and absorption, are estimated by measuring the light on the boundary of the tissue illuminated by a near-infrared source at the tissue\u27s surface. In this dissertation, we investigate a direct method for solving the EIT inverse problem using mollifier regularization, which is then modified and extended to solve the inverse problem in DOT. First, the mollifier method is formulated and then its efficacy is verified by developing an appropriate algorithm. For EIT and DOT, a comprehensive numerical and computational comparison, using several types of regularization techniques ranging from analytical to iterative to statistical method, is performed. Based on the comparative results using the aforementioned regularization methods, a novel hybrid method combining the deterministic (mollifier and iterative) and statistical (iterative and statistical) is proposed. The efficacy of the proposed method is then further investigated via simulations and using experimental data for damage detection in concrete