Variational Methods for Discrete Tomography

Abstract

Image reconstruction from tomographic sampled data has contoured as a stand alone research area with application in many practical situations, in domains such as medical imaging, seismology, astronomy, flow analysis, industrial inspection and many more. Already existing algorithms on the market (continuous) fail in being able to model the analysed object. In this thesis, we study discrete tomographic approaches that enable the addition of constraints in order to better fit the description of the analysed object and improve the end result. A particular focus is set on assumptions regarding the signals' sampling methodology, point at which we look towards the recently introduced Compressive Sensing (CS) approach, that has shown to return remarkable results based on how sparse a given signal is. However, research done in the CS field does not accurately relate to real world applications, as objects usually surrounding us are considered to be piece-wise constant (not sparse on their own) and the properties of the sensing matrices from the viewpoint of CS do not re ect real acquisition processes. Motivated by these shortcomings, we study signals that are sparse in a given representation, e.g. the forward-difference operator (total variation) and develop reconstruction diagrams (phase transitions) with the help of linear programming, convex analysis and duality that enable the user to pin-point the type of objects (with regard to their sparsity) which can be reconstructed, given an ensemble of acquisition directions. Moreover, a closer look is given to handling large data volumes, by adding different perturbations (entropic, quadratic) to the already constrained linear program. In empirical assessments, perturbation has lead to an increased reconstruction rate. Needless to say, the topic of this thesis is motivated by industrial applications where the acquisition process is restricted to a maximum of nine cameras, thus returning a severely undersampled inverse problem.