Two Inverse Problems Arising in Medical Imaging

Abstract

In this thesis, we discuss two inverse problems arising in medical imaging. The first problem is about a hybrid imaging method using coupled boundary measurements, which combines electrical impedance tomography (EIT) with heat conduction. We prove uniqueness for this method in the two dimensional case, i.e. it uniquely determines the anisotropic electrical conductivity, the anisotropic thermal conductivity, and the product of heat capacity and heat density within a bounded domain in the plane up to a boundary- fixing diffeomorphism. The second problem arises in positron emission tomography (PET), a medical imaging tech- nique that aims to determine the distribution of radioactive tracers in a patient’s body. The PET data needs some attenuation correction before it can be used to reconstruct the dis- tribution, and the correction is usually achieved using attenuation coefficients reconstructed from another computed tomography (CT) scan. We show that the PET data is enough to recover certain types of singularities of the distribution (like jump discontinuities), which is usually enough for medical purpose, and thus that the extra CT scan is unnecessary.