Local Prior Knowledge in Tomography

Computed tomography (CT) is a technique that uses computation to form an image of the inside of an object or person, by combining projections of that object or person. The word tomography is derived from the Greek word tomos, meaning slice. The basis for computed tomography was laid in 1917 by Johann Radon, an Austrian mathematician. Computed tomography has a broad range of applications, the best known being medical imaging (the CT scanner where X-rays are used for making the projection images. The rst practical application of CT was, however, in astronomy, by Ronald Bracewell in 1956. He used CT to improve the resolution of radio-astronomical observations. The practical applications in this thesis are from electron tomography, where the images are made with an electron microscope, and from preclinical research, where the images are made with a CT scanner. There are two important techniques for the reconstruction of the images, namely analytical and algebraic methods. Medical scanners mostly employ ltered backprojection, an analytical method. This thesis builds on algebraic methods, where the problem is cast as a system of linear equations Wx = p. Here, x is the unknown image, p represents the projection data, and W the system matrix, which describes the relation between both. In practical CT, it is not always possible to record the projection images that are necessary for creating an accurate reconstruction (their number, from which angles, etc.). There are several reasons for this, an important one being limiting the radiation dose. A straightforward example of this are X-rays, which can be harmful for man, but lifeless objects can also be damaged by radiation. To avoid this, the radiation dose must often be limited, which decreases the signal-to-noise ratio and makes the reconstruction more dicult. The imaging modality itself can also have inherent limitations. In electron tomography, e.g., the projection images are often made manually, so that recording even a small number of projections can be quite labor intensive. Apart from the limits of the microscope itself, there is also the fact that the samples are mostly planar, so that their apparent thickness increases when they are imaged under a large angle. The result of this is that projections cannot be made over the full range of 180. The consequence of these kinds of limitations is that the reconstruction algorithms only have access to limited data, which can make it impossible to create accurate reconstructions. A solution for the limited data problem is adding prior knowledge to the reconstruction algorithm. In this thesis, two assumptions are employed. The rst one is that the scanned object consists of homogeneous regions. The second one is that the gray levels of these regions are known. Together, these assumptions lead to discrete tomography (DT). The rst assumption by itself leads to algorithms that employ total variation minimization (TVMin). The practical algorithms that are used in this thesis are DART for DT, and NESTA and FISTA for TVMin. Using these algorithms results in a large improvement of the quality of the reconstructions. A problem with applying prior knowledge is that there are cases where the prior knowledge is not valid for the entire object. The methods that were mentioned before can then fail. In this thesis, we investigate the possibility of only applying the prior knowledge to part of the object. There are two ways to do this, either developing a specic algorithm for each specic situation, or developing a general technique for creating local versions of existing algorithms. Both approaches are investigated.