Fast and exact 2D image reconstruction by means of Chebyshev decomposition and backprojection

A new algorithm for the reconstruction of two-dimensional (2D) images from projections is described. The algorithm is based on the decomposition of the projections into Chebyshev polynomials of the second kind, which are the ideal basis functions for this application. The Chebyshev decomposition is done via the fast discrete sine transform. A discrete reconstruction filter is applied that corresponds to the ramp filter used in standard filtered backprojection (FBP) reconstruction. In contrast to FBP, the filter is applied to the Chebyshev coefficients and not to the Fourier coefficients of the projections. Then the reconstructed image is simply obtained by means of backprojection. Consequently, the method can be considered as a Chebyshev-domain filtered backprojection (CD-FBP). The total calculation time is dominated by the backprojection step only and is comparable to FBP. The merits of CD-FBP as compared with standard FBP are that: (a) The result is exact if the 2D function to be reconstructed can be decomposed into polynomials of finite degree, and if the sampling is adequate. Otherwise a polynomial approximation results. (b) The algorithm is inherently discrete. (c) It is particularly well suited for reconstructions from projections with non-equidistant samples that occur for instance in 2D PET (positron emission tomography) imaging and in a special form of fan beam scanning. Examples of applications comprise reconstructions of the Shepp and Logan head phantom in various sampling geometries, and a real PET test object. In the PET example an increased resolution is observed in comparison with standard FBP.