The problem of obtaining an artifact-free computed tomographic (CT) reconstruction from projection data that are limited in number and possibly angular coverage is a difficult one to solve in general. Similarly, restorations of blurred images are almost always marred by artifacts that appear to be related to the blurring function. These difficulties arise from a fundamental limitation inherent in incomplete data sets. This limitation may be viewed as resulting from an essential lack of information in the measurements about the unknown source function, which is codified in the concept of the null space of functions associated with the measurement geometry. The Bayesian approach allows one to incorporate a priori information about the source function based on the properties of the ensemble of source functions realizable in the specified imaging situation. If the prior knowledge is restrictive enough, reasonable estimates of the null-space components of the source function can be obtained, thereby reducing the artifacts in the reconstruction. The use of prior knowledge about the shape or structure of the source function will be emphasized here. The results of the maximum a posteriori (MAP) method will be compared with the fit and iterative reconstruction (FAIR) technique, in which the previously known shape of the object may be explicitly incorporated by means of a model of the object.
Keywords: Bayesian methods, use of a priori information, null space, limited-angle tomographic reconstruction, Fit and Iterative Reconstruction, FAIR
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