Development and analysis of image reconstruction algorithms in diffraction tomography

Diffraction tomography (DT) is a wavefield inversion technique that reconstructs the spatially variant refractive index distribution of a scattering object. Recently, there has been a strong interest in developing ultrasound-, microwave- and optical-based DT systems as biomedical imaging modalities. The potential usefulness of these systems stems from their unique sources of image contrast and, in certain cases, their potential for physiological imaging of biological tissues. Unlike the X-rays used in computed tomography (CT) that travel along straight lines, the radiation employed in DT has to be treated in terms of wavefronts and fields scattered by inhomogeneities in the object. Therefore, a wide variety of techniques suitable for reconstruction of CT images cannot be used directly for reconstruction of diffraction tomographic images. The development of quantitatively accurate, statistically optimal, and computationally tractable reconstruction algorithms for DT remains a formidable task that has yet to be satisfactorily accomplished.; In this work, the reconstruction problem in DT is addressed. A major contribution of this work is the development of a new class of reconstruction algorithms for 2D and 3D DT referred to as estimate-combination (E-C) reconstruction algorithms. The E-C algorithms effectively operate by transforming the DT reconstruction problem into an X-ray CT reconstruction problem, which can subsequently be inverted using conventional CT reconstruction algorithms. The practical and theoretical advantages of this approach are exploited and discussed for both the linear and nonlinear DT reconstruction problems. We also develop and investigate minimal-scan reconstruction algorithms for 2D DT that do not require measurements to be taken over the full angular range [0, 2π) in order to perform an exact reconstruction. A limited-view reconstruction problem for DT is examined, in which measurements are available only for view angles in [0, Φc], where π ≤ Φ c < 2π. In this situation, we reveal a counter-intuitive relationship between the maximum scanning angle and resolution in the reconstructed image. A mathematical analysis of the DT reconstruction problem is presented, in which we provide range characterizations and a singular value decomposition for the DT imaging operator. Throughout the dissertation, extensive numerical results are presented to corroborate our theoretical developments.