Backprojection with fourier series expansion and FFT

The filtered backprojection (FBP) method has recently been used for almost all the practical CT scanners because of its simple principle and high accuracy in reconstruction of images. However, it is difficult to use this method for some applications such as a real time display in which images are adjusted during displaying images, since a backprojection of the FBP method requires a very large amount of computation. This paper introduces a method of backprojection in which the projection data compensated by a filter are expanded using Fourier series, and the sum of the series is evaluated by a high-speed synthesis algorithm of complex exponential using Gaussian functions and the fast Fourier transform (FFT) method. The theory is explained based on a conventional FBP method; then the usefulness of the proposed method is demonstrated. The results show that the proposed method can realize interpolations with various frequency characteristics requiring about one-sixth the amount of computation compared with a conventional backprojection using a linear interpolation.