When the available CT projection data are incomplete, there exists a null space in the space of possible reconstructions about which the data provide no information. Deterministic CT reconstructions are impotent in regard to this null space. Furthermore, it is shown that consistency conditions based on projection moments do not provide the missing projections. When the projection data consist of a set of parallel projections that do not encompass a complete 180 degree rotation, the null space corresponds to a missing sector in the Fourier transform of the original 2-D function. The long-range streak artifacts created by the missing sector can be reduced by attenuating the Fourier transform of the reconstruction smoothly to zero at the sector boundary. It is shown that the Fourier transform of a reconstruction obtained under a maximum entropy constraint is nearly zero in the missing sector. Hence, maximum entropy does not overcome the basic lack of information. It is suggested that some portion of the null space might be filled in by use of a priori knowledge of the type of image expected.
Keywords: incomplete data, missing data, null space, streak artifacts, algebraic reconstruction technique (ART), maximum entropy reconstruction, nonnegativity constraint, prior knowledge, consistency conditions
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