A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Israeli, Y. Shkolnisky
2008 SIAM Journal on Scientific Computing  
The Radon transform is a fundamental tool in many areas. For example, in reconstruction of an image from its projections (CT scanning). Although it is situated in the core of many modern physical computations, the Radon transform lacks a coherent discrete definition for 2D discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for discrete 2D images, which is based on summation along lines of absolute slope less than
more » ... . Values at non-grid locations are defined using trigonometric interpolation on a zero-padded grid. The discrete 2D definition of the Radon transform is shown to be geometrically faithful as the lines used for summation exhibit no wraparound effects. There exists a special set of lines in R 2 for which the transform is rapidly computable and invertible. We describe an algorithm that computes the 2D discrete Radon transform and uses O(N log N ) operations, where N = n 2 is the number of pixels in the image. The algorithm relies on a discrete Fourier slice theorem, which associates the discrete Radon transform with the pseudo-polar Fourier transform [14] . Finally, we prove that our definition provides a faithful description of the continuum, as it converges to the continuous Radon transform as the discretization step goes to zero.
doi:10.1137/060650283 fatcat:4d5rzevp4refvkbahnwegersze