The Study Of The BPF And Window Functions Based Hilbert Transform

The CT image reconstruction is a kind of technology which reconstruct the original image by the projection data. This technology is wildly applied in industry and medical. The most important part of CT technology is the reconstruction algorithm. Those algorithm can be divided into two categories which are analytical and discrete. Two mathematicians Zou and Pan obtained a new analytical reconstruction algorithm named Backprojection-Filter (BPF) in2004. The BPF can be separated in two steps. The first step is the relationship between the Differential Backprojection of the projection data and the Hilbert transform of the original image. The second step is inverse Hilbert transform line by line.This paper will detailed introduce the theory of BPF. As an advantage of inverse Hilbert transform line by line, the BPF can be applied in the Region of Interest (ROI). As a result, this paper will select three kinds of region to reconstruct for example. Otherwise, this paper will study some about reconstruction of truncated data.This paper will focus on the window functions of BPF. The BPF make use of window function twice in the process of image reconstruction with BPF. The window function will be first used in the filter of inverse Fourier transform and second used in the inverse Hilbert transform. This paper will study four kinds of window functions which are rectangular window, cosine window, sine window, generalized Hamming window. This paper will analyze advantages and disadvantages of these window functions. And this paper will complete numerical experiment of reconstruction with rectangular window, cosine window and sine window. As the best window with minimum error of reconstruction, the cosine window will be used to the reconstruction of ROI and truncated data.This paper will innovatively give two kinds of new window functions. One is the modified cosine window. The other is exponential window. This paper will study this two window functions and complete their numerical experiment. As comparison, this two window functions with less error are better than those four in this paper.