Abstract: The inverse scattering problem refers to the mathematical imaging problem of recovering information about an unknown object from measurements of waves scattered by this object. 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 itself 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.
If the scattering induced by the object is sufficiently weak, the well-known Born approximation linearizes the inverse problem. In this setting, diffraction tomography enables a computationally efficient reconstruction method based on the Fourier diffraction theorem, which relates the Fourier transform of the measured data to that of the scattering potential.
Classical diffraction tomography relies on the central assumption that the incident wave is a plane wave. This, however, constitutes a significant limitation in many practical imaging systems, as they often employ focused or other non-planar illuminations. A prominent example, and the main motivation for this thesis, is medical ultrasound imaging, where focused beams are routinely used to achieve high spatial resolution in depth. As a consequence, the classical results and reconstruction method of diffraction tomography cannot be applied.
The goal of this cumulative thesis is therefore to extend diffraction tomography to generalized incident fields. To this end, we use the concept of incident Herglotz waves, which allows us to model general incident waves including focused beams. Based on this framework, we derive a new Fourier diffraction theorem that relates the measured data to the Fourier transform of the scattering potential under Herglotz wave illumination. In the first part of this thesis, this relation is studied in a setting in which the incident Herglotz wave can be rotated around the object, leading to an explicit reconstruction formula that is evaluated by numerical experiments.
In the second part, we consider a raster scan experiment motivated by ultrasound imaging, in which the incident beam is translated across the object rather than rotated. For this setting, we further extend the Fourier diffraction relation to describe the entire scan in the Fourier domain. Unlike the classical case, this relation does not directly yield an explicit reconstruction formula but instead leads to a linear system for the Fourier coefficients of the scattering potential. We therefore analyze which of these coefficients are uniquely determined by the measurements, thereby identifying the Fourier data that can be reliably used in the reconstruction.