Statistical projection completion in X-ray CT using consistency conditions

Projection data incompleteness arises in many situations relevant to X-ray computed tomography (CT) imaging. We propose a penalized maximum likelihood statistical sinogram restoration approach that incorporates the Helgason-Ludwig (HL) consistency conditions to accommodate projection data incompleteness. Image reconstruction is performed by the filtered-backprojection (FBP) in a second step. In our problem formulation, the objective function consists of the log-likelihood of the X-ray CT data and a penalty term; the HL condition poses a linear constraint on the restored sinogram and can be implemented efficiently via fast Fourier transform (FFT) and inverse FFT. We derive an iterative algorithm that increases the objective function monotonically. The proposed algorithm is applied to both computer simulated data and real patient data. We study different factors in the problem formulation that affect the properties of the final FBP reconstructed images, including the data truncation level, the amount of prior knowledge on the object support, as well as different approximations of the statistical distribution of the available projection data. We also compare its performance with an analytical truncation artifacts reduction method. The proposed method greatly improves both the accuracy and the precision of the reconstructed images within the scan field-of-view, and to a certain extent recovers the truncated peripheral region of the object. The proposed method may also be applied in areas such as limited angle tomography, metal artifacts reduction, and sparse sampling imaging.