3D Fourier based discrete Radon transform

Abstract

The Radon transform is a fundamental tool in many areas. For example, in reconstruction of an image from its projections (CT scanning). Recently A. Averbuch et al. [SIAM J. Sci. Comput., submitted for publication] developed a coherent discrete definition of the 2D discrete Radon transform for 2D discrete images. The definition in [SIAM J. Sci. Comput., submitted for publication] is shown to be algebraically exact, invertible, and rapidly computable. We define a notion of 3D Radon transform for discrete 3D images (volumes) which is based on summation over planes with absolute slopes less than 1 in each direction. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. The 3D discrete definition of the Radon transform is shown to be geometrically faithful as the planes used for summation exhibit no wraparound effects. There exists a special set of planes in the 3D case for which the transform is rapidly computable and invertible. We describe an algorithm that computes the 3D discrete Radon transform which uses O( Nlog N) operations, where N= n 3 is the number of pixels in the image. The algorithm relies on the 3D discrete slice theorem that associates the Radon transform with the pseudo-polar Fourier transform. The pseudo-polar Fourier transform evaluates the Fourier transform on a non-Cartesian pointset, which we call the pseudo-polar grid. The rapid exact evaluation of the Fourier transform at these non-Cartesian grid points is possible using the fractional Fourier transform.