In this dissertation, we focus on analytic and numerical inversion of an integral transform (cone or Compton transform) that maps a function to its integrals over conical surfaces with a weight equal to some power of the distance from the cone's vertex. It arises in various imaging techniques, most prominently, in modeling of the data provided by the so-called Compton camera, which has novel applications in various fields, including biomedical and industrial imaging, homeland security, and gamma ray astronomy. In the case of pure surface measure on the cone, an integral identity relating cone, Radon and cosine transforms is presented, which enables us to derive an inversion formula for the cone transform in any dimension. The image reconstruction algorithms, based on the inversion formulas, and their numerical implementation results in dimensions two and three are provided. In 3D, the implementation of the inversion algorithms is challenging due to the high dimensionality of the forward data, and the fact that the application of a fourth order differential operator on the unit sphere to a singular integral is required. We thus develop and apply three different inversion algorithms and study their feasibility. The weighted divergent beam transform, which integrates a function over rays with a weight equal to some power of the distance to the starting point (source) of the ray, is closely intertwined with the weighted cone transform. We study it in some details, which leads eventually to other weighted cone transform inversions. The image reconstruction algorithm, based on one of the inversion formulas, and its numerical implementation results for various weight factors in dimensions two and three are also provided. All inversion formulas presented in this dissertation are applicable for a wide variety of detector geometries in any dimension.
In this dissertation, we focus on analytic and numerical inversion of an integral transform (cone or Compton transform) that maps a function to its integrals over conical surfaces with a weight equal to some power of the distance from the cone's vertex. It arises in various imaging techniques, most prominently, in modeling of the data provided by the so-called Compton camera, which has novel applications in various fields, including biomedical and industrial imaging, homeland security, and gamma ray astronomy.
In the case of pure surface measure on the cone, an integral identity relating cone, Radon and cosine transforms is presented, which enables us to derive an inversion formula for the cone transform in any dimension. The image reconstruction algorithms, based on the inversion formulas, and their numerical implementation results in dimensions two and three are provided. In 3D, the implementation of the inversion algorithms is challenging due to the high dimensionality of the forward data, and the fact that the application of a fourth order differential operator on the unit sphere to a singular integral is required. We thus develop and apply three different inversion algorithms and study their feasibility.
The weighted divergent beam transform, which integrates a function over rays with a weight equal to some power of the distance to the starting point (source) of the ray, is closely intertwined with the weighted cone transform. We study it in some details, which leads eventually to other weighted cone transform inversions. The image reconstruction algorithm, based on one of the inversion formulas, and its numerical implementation results for various weight factors in dimensions two and three are also provided.
All inversion formulas presented in this dissertation are applicable for a wide variety of detector geometries in any dimension.