3D Fourier based discrete Radon transform. (Q1406573)

Following the guidelines of the technique employed in the papers by \textit{A. Averbuch} et al. [Fast slant stack: A notion of Radon transform for data in Cartesian grid which is rapidly computable, algebraically exact, geometrically faithful and invertible, SIAM J. Sci. Comput., submitted for publication] and \textit{Y. Shkolnisky} et al. [2D Fourier based discrete Radon transform, 2002, submitted for publication] for the definition of the 2D discrete Radon transform, the authors define a 3D discrete Radon transform here based on summation over planes with absolute slopes less than 1 in each direction. This 3D discrete definition is shown to be algebraically exact, invertible and rapidly computable. An algorithm is described that relies on the 3D discrete slice theorem that associates the Radon transform with the pseudo-polar Fourier transform. With respect to the invertibility, the authors point out that the lack of a reconstruction algorithm will be treated in another work. On the other hand, to have Radon transforms in higher dimensions, they also announce the generalization of the above methodology in a future paper, by means of a representation of hyperplanes at any dimension (and not just planes as in the 3D case).