Abstract
To solve the interior problem in computed tomography, a priori knowledge of the sub-region located inside the region of interest (ROI) is usually required. However, this is somewhat paradoxical because the a priori knowledge of the sub-region is located inside the ROI that we want to obtain through reconstruction. Here we show that the interior problem has a unique solution even if the a priori knowledge of the sub-region is located outside the ROI when lines called exact lines exist and satisfy the following two conditions. The first condition is that the exact line region should belong to the a priori knowledge region or ROI, and the second condition is that overlap between the exact line region and ROI always exists. By using a property of the Hilbert transform, the part of the object function ƒ(x) located inside the ROI can be obtained from a priori knowl-edge located outside the ROI. As a result, we can obtain new a priori knowledge located inside the ROI, and thus existing interior problem methods can be used to generate the reconstructed image. This paper is an extension of the work reported in "Tiny a priori knowledge solves the interior problem in computed tomography", which was published in Physics in Medicine and Biology in 2008. The new results presented here indicate that under relaxed constraints, the a priori knowledge of a sub-region outside the target ROI can provide exact and stable reconstruction from interior truncated projections. The results of experiments performed using simulated data have shown the validity of the proposed method.
