A framework for discrete integral transformations II - The 2D discrete Radon transform

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Israeli, Y. Shkolnisky, I. Sedelnikov

Research output: Contribution to journalArticlepeer-review

Abstract

Although naturally at the heart of many fundamental physical computations, and potentially useful in many important image processing tasks, the Radon transform lacks a coherent discrete definition for two-dimensional (2D) discrete images which is algebraically exact, invertible, and rapidly computable. We define a notion of 2D discrete Radon transforms for 2D discrete images, which is based on summation along lines of absolute slope less than 1. Values at nongrid locations are defined using trigonometric interpolation on a zero-padded grid. Our definition is shown to be geometrically faithful: the summation avoids wrap-around effects. Our proposal uses a special collection of lines in ℝ2 for which the transform is rapidly computable and invertible. We describe a fast algorithm using O(N log N) operations, where N = n 2 is the number of pixels in the image. The fast algorithm exploits a discrete projection-slice theorem, which associates the discrete Radon transform with the pseudopolar Fourier transform [A. Averbuch et al., SIAM J. Sci. Comput., 30 (2008), pp. 764-784]. Our definition for discrete images converges to a natural continuous counterpart with increasing refinement.

Original languageEnglish
Pages (from-to)785-803
Number of pages19
JournalSIAM Journal of Scientific Computing
Volume30
Issue number2
DOIs
StatePublished - 2007

Keywords

  • Linogram
  • Projection-slice theorem
  • Pseudopolar Fourier transform
  • Radon transform
  • Shearing of digital images
  • Slant stack
  • Trigonometric interpolation

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