Browsing by Author "Piila, Erna"

The word tomography refers to a variety of imaging methods where a penetrating wave is used to collect data about an unknown object of interest. The waves usually need to be sent through the object from a large number of different angles in order to have enough data for a successful reconstruction. The problems can be expressed in a form where the measured data is known to be equal to the unknown object (expressed as a function) multiplied by a known operator. Reconstructing either a two-, three-, or in case of dynamic tomography, four-dimensional image based on data is not a simple matter of inverting said operator. The measurement noise, which is always a factor in imaging situations, can be amplified greatly in the reconstruction, making the inverse problem in question ill-posed. To avoid this, some regularization method in which a stable, unique problem close to the original, ill-posed one, needs to be applied. A method called Tikhonov regularization is one of the most commonly used ones. Discrete tomography differs from general tomography by limiting the objects or images being reconstructed to ones consisting of only a small set of different densities or colours. This a priori knowledge of the object makes it possible to make successful reconstructions based on a much smaller amount of data. Traditionally discrete tomography has only focused on making reconstructions of binary images but more recently algorithms have been developed that allow the number of different colours or densities to be as large as five. There are some very promising new algorithms in the field of discrete tomography but due to the requirements set by new applications, an ever-increasing number of researchers are working on new ones. In this thesis a small, simulated example of tomographic reconstruction is made using both Tikhonov regularization and DART (discrete algebraic reconstruction technique), which is an algorithm of discrete tomography. Both methods give reasonably good results in all of the situations that were studied. It is found, however, that for an image fulfilling the requirements for using DART (small enough number of different colours), DART performs significantly better when the number of projection angles is decreased.