A mathematical framework for reconstructing shapes from tomographic projections at unknown angles

Tomographic reconstruction is a critical tool in many scientific and medical applications, allowing the visualization of internal structures non-destructively using only external measurements. Standard reconstruction methods typically rely on the knowledge of projection angles. However, this information may not always be available in many practical scenarios. This problem frequently arises in fields such as biomedical imaging and material science where constraints on equipment, experimental setups, or the nature of the subject being imaged make angle measurements infeasible. For instance, in single particle cryogenic electron microscopy (cryo-EM), determining the 3D structure of molecules involves analyzing tomographic projections of particles that are randomly oriented within a thin layer of vitreous ice. Although this problem has been studied for practical implementations, several fundamental questions remain unresolved. Chief among these are issues related to the uniqueness of the reconstruction and the existence of algorithms capable of accurately estimating the underlying structure. Moreover, many existing approaches do not take into account the conversion from the continuous nature of the underlying physical process to its discrete measurements, which is fundamentally governed by a sampling procedure. Typically, existing methods rely on a large number of projections to achieve reliable reconstruction. However, there is no theoretical framework that establishes the minimum number of projections required to guarantee perfect reconstruction of the structure. This limitation is further compounded in many practical scenarios where obtaining a large number of projections is infeasible. Therefore, in this thesis: 1) we develop a novel sampling and reconstruction framework for the problem of unknown angle tomography, 2) we present both theoretical and algorithmic solutions to the reconstruction problem, and 3) we investigate advanced signal models that overcome the limitations imposed by a restricted number of projections, enabling more robust and accurate reconstructions in practical scenarios, and for different applications.