Discrete Image Reconstruction from Uncertain and Insufficient Data

The goal of the project is to develop efficient image acquisition methods to gain visual information from incorrect, noisy, and uncertain projections. Such projections typically arise in sensor-network applications. Projection signals can be detected in various forms (X-ray, gamma-ray, radar, ultrasound, electric or electromagnetic sensors, etc.). Due to the large variety in the number, modality and position of the detectors, there is no general method to obtain an accurate reconstruction in each application. In the past years it turned out, that discrete tomography can guarantee reconstructions of good quality even when the classical reconstruction methods are hardly applicable due to the small number of possibly noisy projections. Our aim is to investigate the theoretical background of reconstruction problems where projections are noisy, and/or their direction is not known precisely. We study uniqueness and stability questions. We also investigate how structural or geometrical prior information can facilitate an improve the reconstruction if the number of projections is insufficent to get a unique and exact reconstruction. We design and implement discrete tomographic reconstruction methods for different applications and analyse the performance of those algorithms from the viewpoint of speed, accuracy, and noise-sensitivity, both theoretically and numerically.