Two-dimensional region-of-interest reconstruction: Analyzing the difference between virtual fanbeam and DBP-Hilbert reconstructions

This work addresses theoretical advances classical (2D) tomographic image reconstruction. During the past several years, inversion formulas have been established that allow ROI reconstruction from incomplete (yet sufficient) data. Such reconstructions have important consequences in certain practical situations, such as truncated projections. The precise relationship between the largest ROI that can be reconstructed and the incompleteness of the sinogram is a complex question which has still not been completely answered in the 2D case. These relationships are inherent to the system and have consequences for iterative/statistical reconstruction methods, because they describe which part of the reconstructed image is determined completely by the data; the other parts of the image will have been more heavily influenced by the regularization method or by the nature of the objective function. Our understanding of the nature of reconstruction from incomplete yet sufficient data relies mainly on formulas obtained from the virtual fanbeam (VFB) method and from the DBP-Hilbert method. The purpose of this work is to provide a structure in which to examine the inherent differences in these two approaches. Using a common reconstruction problem, we reformulate VFB and DBP-Hilbert reconstruction formulas into weight functions that are applied in the sense of an inner product to the sinogram. A common regularization is used for the Hilbert transform in both methods. Unlike the usual Fourier windows used in analytic methods, the regularization we used is applied locally to the singularity to avoid the regularization obscuring the nature of the reconstruction. The weight functions clearly show how truncated projections are being correctly handled. The dissimilarity in the weight functions of the two methods illustrates fundamental differences in managing incomplete data, and suggests that many other such methods exist.