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.
Highlights
It is based on Fourier transforms, which is something that we want to avoid. If, as it is the case in our application domains, the problem to solve is purely discrete, the multi-scale Radon transform performs better than any other discrete Radon transform [24], including those based on pseudo-polar FFT
The three selected methods for comparison will be: the conventional multi-scale discrete Radon Transform, which is the direct ancestor of our propositions; a method representing those based on Fourier projection-slice theorem, namely the pseudo-polar Fourier-based Radon transform, PPFRT; and one covering the main alternative, whose lines can dispense with interpolation: the Mojette transform, in particular, the fast Mojette transform, FMT
There is a sort of vignetting in conventional discrete Radon transform (DRT), due to the different length of discrete lines that are considered in this method: every line traverses N values, but depending on the slope and displacement, more or less of those values correspond to real pixels on the input image or zeroes in the padded zone
Summary
The Radon transform, as originally envisioned [1], describes a bi-dimensional function, in terms of the set of line integrals through its domain. We will consider a family of methods, which we will call multi-scale, that compute the forward transform with a divide-and-conquer approach, and so they exhibit the same complexity as those based on Fourier, O( N 2 log( N )), but without requiring interpolation or the use of floating point numbers. When they were initially proposed, they were not considered invertible, but it was later shown that such methods, operating on integer mathematics, had a fast and exact inverse [16]. Let us begin by explaining the multi-scale methods
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