Central and Periodic Multi-Scale Discrete Radon Transforms

Abstract

The multi-scale discrete Radon transform (DRT) calculates, with linearithmic complexity, the summation of pixels, through a set of discrete lines, covering all possible slopes and intercepts in an image, exclusively with integer arithmetic operations. An inversion algorithm exists and is exact and fast, in spite of being iterative. In this work, the DRT forward and backward pair is evolved to propose two faster algorithms: central DRT, which computes only the central portion of intercepts; and periodic DRT, which computes the line integrals on the periodic extension of the input. Both have an output of size N×4N, instead of 3N×4N, as in the original algorithm. Periodic DRT is proven to have a fast inversion, whereas central DRT does not. An interesting application of periodic DRT is its use as building a block of discrete curvelet transform. Central DRT can provide almost a 2× speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners.

Considerations on the search of a fast non-iterative inverse discrete Radon transform

“…Some of the applications of the Radon transform are the computed axial tomography, where the inverse of the two dimensional transform is computed, 2 barcode scanners, 3,4 reflection seismology 5 and image analysis. 6 The discrete Radon transform was proposed simultaneously by several authors in the 1990s. [7][8][9] The algorithm for computing the transform is of linearithmic complexity: O(N log N ).…”