A discrete modulo N projective radon transform for N × N images

  • Authors:

  • Andrew Kingston;Imants Svalbe

  • Affiliations:

  • Centre for X-ray Physics and Imaging, School of Physics and Materials Engineering, Monash University, VIC, AUS;Centre for X-ray Physics and Imaging, School of Physics and Materials Engineering, Monash University, VIC, AUS

  • Venue:
  • DGCI'05 Proceedings of the 12th international conference on Discrete Geometry for Computer Imagery
  • Year:
  • 2005

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Abstract

This paper presents a Discrete Radon Transform (DRT) based on congruent mathematics that applies to N × N arrays where $N \in \mathcal{N}$. This definition incorporates and is a natural extension of the more restricted cases of the finite Radon transform [1] where N must be prime, the discrete periodic Radon transform [2] where N must be a power of 2, and the DRT over pn [3], where N must be a power of a single prime. The DRT exactly and invertibly maps a 2-D image to a set of 1-D projections of length N. Projections are found as the sum of the pixels centred on a parallel set of discrete lines. The image is assumed to be periodic and these lines wrap around the array under modulo N arithmetic. Properties of the continuous Radon transform are preserved in the DRT; a discrete form of the Fourier slice theorem applies, as does the convolution property. A formula is given to find the projection set required to be exactly invertible for arrays with N any composite number, as well as a means to determine the level of redundancy in sampling that is introduced on such composite arrays.