On Implementation and Stability Analysis of Two Types of Finite Hilbert Transforms Used in Backprojection Filtration Algorithms

Backprojection filtration(BPF) algorithms are important analytical reconstruction algorithms in Computed Tomography(CT), Magnetic Resonance Imaging(MRI) and Electron Paramagnetic Resonance Imaging(EPRI). An important step in the BPF algorithm is the finite Hilbert transform(FHT), which allows recovery of a compactly supported function from its inverse Hilbert transform using only a finite interval. Two simple and practical implementation methods were proposed to serve for CT and EPRI, respectively. Numerical experiments were made on three classical functions to verify the two implementation methods. The boundary stability analysis was given by numerical experiments on a parabola function. The results show that the two methods can get enough exact results using an appropriate interval bigger than the supporting interval of the compactly supported function and that the method not including definite integration is not stable and is sensitive to the change of the interval. However the method including definite integration is very stable and can get an exact result just using a small interval that is a little bigger than the supporting interval. The method including definite integration is fit for CT and the method not including definite integration is fit for MRI and EPRI.