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

A new algorithm for the reconstruction of two-dimensional (2D) images from projections is described. The algorithm is based on the decomposition of th e 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 b ackprojection (FBP) reconstruction. In contrast to FBP, the filter is appli ed to the Chebyshev coefficients and not to the Fourier coefficients of the projections. Then the reconstructed image is simply obtained by means of b ackprojection. Consequently, the method can be considered as a Chebyshev-do main filtered backprojection (CD-FBP). The total calculation time is domina ted 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 f inite degree, and if the sampling is adequate. Otherwise a polynomial appro ximation results. (b) The algorithm is inherently discrete. (c) It is parti cularly well suited for reconstructions from projections with non-equidista nt samples that occur for instance in 2D PET (positron emission tomography) imaging and in a special form of fan beam scanning. Examples of applicatio ns comprise reconstructions of the Shepp and Logan head phantom in various sampling geometries, and a real PET test object. In the PET example an incr eased resolution is observed in comparison with standard FBP.