The inverse scattering problem is the mathematical imaging problem of recovering information about an unknown object from measurements of waves scattered by it. In the underlying experiment, the object is probed by an incident wave, and the resulting scattered waves are recorded. The interaction between the incident wave and the object is determined by the scattering potential, and the goal is to reconstruct the spatial distribution of this quantity from the measurements. While wave propagation is governed by a linear equation, the measured waves do not depend linearly on the scattering potential. Hence, the inverse problem is non-linear and requires computationally demanding reconstruction methods.
Under the assumption of weak scattering, the Born approximation provides a linearization of the inverse problem. In this setting, diffraction tomography yields computationally efficient reconstruction methods based on the Fourier diffraction theorem, which directly relates measured data to Fourier coefficients of the scattering potential.
Classical diffraction tomography, however, relies on the restrictive assumption of plane-wave illumination. This limitation is particularly relevant in medical ultrasound imaging, where focused beams are routinely used to achieve high spatial resolution. As a consequence, the classical reconstruction formulas are not directly applicable in realistic imaging setups.
The aim of this thesis is therefore to extend diffraction tomography to generalized incident fields. To this end, incident fields are modeled by Herglotz waves, allowing focused and other non-planar wave fields to be incorporated into the diffraction tomography framework. Based on this approach, a generalized Fourier diffraction theorem is derived and analyzed for different measurement configurations.
In the first part of the thesis, rotating Herglotz-wave illumination is studied, leading to explicit reconstruction formulas that are validated numerically. In the second part, motivated by ultrasound imaging, raster-scan measurement configurations are investigated in which the incident beam is translated across the object. In this setting, the generalized Fourier diffraction relation leads to coupled Fourier coefficients rather than direct reconstruction formulas. The thesis, therefore, further analyzes which Fourier coefficients are uniquely determined by the measurement data and can be reliably reconstructed.