Comparison of three undersampling approaches in computed tomography reconstruction

Background: Projection data undersampling is an effective approach to reduce X-ray radiation dose in computed tomography (CT). In modern CT technologies, undersampling is also a favorable method to reduce projection data size to facilitate rapid CT scan and imaging. It is an intriguing question that given an undersampling ratio, what is the optimal undersampling approach that enables the best CT image reconstruction. While this is in general a challenging mathematical question, it is the motivation of this paper to compare three types of undersampling operations, which we hope to shed some light to this question. Methods: We considered regular view undersampling that acquires X-ray projections at equiangular projection angles, regular ray undersampling that acquires projections at all angles but with X-ray lines blocked within each projection under a periodic pattern, and random ray undersampling that acquires each X-ray line with a certain probability. By representing the undersampling projection operators under the basis of singular vectors of full projection operator, we generated matrix representations of these undersampling operators and numerically perform singular value decomposition (SVD). Singular value spectra and singular vectors were compared. Results: For a given undersampling ratio, the random ray undersampling approach preserves the properties of the full projection operator better than the other two approaches. This translates to advantages of reconstructing a CT image at a lower error, which has also been demonstrated in the numerical experiments. Conclusions: We compared three undersampling strategies and found that random undersampling preserves the most information and outperforms the other two in terms of reconstruction quality.