In transmitted X-ray tomography imaging, the acquired projections may be corrupted for various reasons, such as defective detector cells and beam-stop array scatter correction problems. In this study, we derive a consistency condition for cone-beam projections and propose a method to restore lost data in corrupted projections. In particular, the relationship of the geometry parameters in circular trajectory cone-beam computed tomography (CBCT) is utilized to convert an ultra-hyperbolic partial differential equation (PDE) into a second-order PDE. The second-order PDE is then transformed into a first-order ordinary differential equation in the frequency domain. The left side of the equation for the newly derived consistency condition is the projection derivative of the current and adjacent views, whereas the right side is the projection derivative of the geometry parameters. A projection restoration method is established based on the newly derived equation to restore corrupted data in projections in circular trajectory CBCT. The proposed method is tested in beam-stop array scatter correction, metal artifact reduction, and abnormal pixel correction cases to evaluate the performance of the consistency condition and corrupted projection restoration method. Qualitative and quantitative results demonstrate that the present method has considerable potential in restoring lost data in corrupted projections.
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