Iterative Reconstruction Methods for Inverse Problems in Tomography with Hybrid Data

Abstract

This thesis provides a theoretical and numerical investigation of two important problems in the field of tomography based on hybrid data from coupled physics phenomena. The first problem is related to Acousto-Electrical Tomography, while the second problem deals with Quantitative Elastography. The goal of these modalities is to quantify physical parameters of materials or tissues inside an object from given interior data, which is measured everywhere inside the object. The advantage of these modalities is that large variations in physical parameters can be resolved and therefore, they have important applications in both medical and industrial imaging. Mathematically, we face a nonlinear Inverse Problem of parameter identification type in both modalities. The applied physical phenomena are typically chosen in such a way that they interact with and complement each other, and most often they are described by models based on coupled partial dierential equations. In contrast to common methods, e.g., Electrical Impedance Tomography, where the reconstruction is solely based on boundary measurements, methods based on coupled phenomena lead to internal measurements. Availability of this so-called hybrid data is precisely the reason why reconstructions with a high contrast and a high resolution can be expected. The main contributions of this thesis consist in formulating the underlying mathematical problems with interior data as nonlinear operator equations, theoretically analysing them within the framework of nonlinear Inverse Problems and designing computational methods for identifying the unknown parameters. Furthermore, the theoretical investigations are supported by a number of numerical examples from both simulated and experimental data. Iterative regularization methods based on Landweber iteration and the Levenberg-Marquardt method are employed for solving the problems.The first problem considered in this thesis is a problem of conductivity estimation from interior measurements of the power density, known as Acousto-Electrical Tomography. A special case of limited angle tomography is studied for this problem, where only a part of the boundary is accessible to electrostatic measurements. Numerical examples support the intuition that stably reconstructing the conductivity becomes difficult far away from the accessible part of the measurement boundary. This is also supported by a quantitative numerical study of the ill-posedness of the problem in dependence on the completeness of the data. The second problem deals with Quantitative Elastography, where Lamé parameters are estimated from full internal static displacement field measurements obtained using both Photo-Acoustic Tomography and Optical Coherence Tomography. The developed computational method is successfully applied to both numerically simulated and experimental data