Interior tomography with radial Hilbert filtering and a priori information in a small circular area

Interior tomography problem can be solved using the so-called differentiated backprojection-projection onto convex sets (DBP-POCS) method, which requires a priori information within a small area interior to the region of interest (ROI) to be imaged. In theory, the small area wherein the a priori information is required can be in any shape, but most of the existing implementations carry out the Hilbert filtering either horizontally or vertically, leading to a vertical or horizontal strip that may be across a large area in the object. In this work, we specifically re-derive the reconstruction formula in the DBP-POCS fashion with radial Hilbert filtering (namely radial DBP-POCS method henceforth). We implement the radial DBP-POCS method, and thus the small area with the a priori information can be roughly circular (e.g., a sinus or ventricles among other anatomic cavities in human or animal body). We also conduct an experimental evaluation to verify the performance of this practical implementation. The performance of the radial DBP-POCS method with the a priori information in a small circular area is evaluated with projection data of the standard Shepp-Logan phantom simulated by computer. The preliminary performance study shows that, if the a priori information in a small circular area is available, the radial DBP-POCS method can solve the interior tomography problem in a much more practical way at high accuracy. In comparison to the implementations of DBP-POCS method demanding the a priori information in horizontal or vertical strip, the radial DBP-POCS method requires the a priori information within a small circular area only. Such a relaxed requirement on the availability of a priori information can be readily met in practice, because a variety of small circular areas (e.g., air-filled sinuses or fluid-filled ventricles among other anatomic cavities) exist in human or animal body. Therefore, the radial DBP-POCS method with a priori information in a small circular area is feasible in clinical and preclinical practice.