Robust algorithms for discrete tomography

Tomography methods concentrate on reconstructing objects from multiple projections that are obtained by sending, for example, X-rays through the object. Applications of these methods are, among others, radiology (CT-, MRI- and PET scans), geophysics and material science. The tomographic problems can be formulated as a system of linear equations. Unfortunately, these systems are not symmetric nor positive (semi)definite, rank deficient and not square. In material science one is often presented with very small objects (like crystals or nano-structures) that consist of one or a small number of different materials, each with its own density. Scanning these small objects can cause damage to the structure and thus one can only take a very limited amount of projections. Fortunately, one can use the prior knowledge about the object to arrive at a reconstruction of the original object. How to arrive at this reconstruction is studied by the field of discrete tomography (DT). With every kind of tomography, and thus also with DT, one is faced with noisy data. Because of this noise the reconstruction process becomes more difficult since the system of linear equations becomes inconsistent. The DART (Discrete Algebraic Reconstruction Method) algorithm was developed to solve DT problems. This algorithm deals with noise in a very heuristic method. The goal of this project is to investigate how the problem can be regularized such that it deals with the noise in a more efficient and robust manner.