On Correcting the Unevenness of Angle Distributions Arising from Integer Ratios Lying in Restricted Portions of the Farey Plane
In 2D discrete projective transforms, projection angles correspond to lines linking pixels at integer multiples of the x and y image grid spacing. To make the projection angle set non-redundant, the integer ratios are chosen from the set of relatively prime fractions given by the Farey sequence. To sample objects uniformly, the set of projection angles should be uniformly distributed. The unevenness function measures the deviation of an angle distribution from a uniformly increasing sequence of angles. The allowed integer multiples are restricted by the size of the discrete image array or by functional limits imposed on the range of x and y increments for a particular transform. This paper outlines a method to compensate the unevenness function for the geometric effects of different restrictions on the ranges of integers selected to form these ratios. This geometric correction enables a direct comparison to be made of the effective uniformity of an angle set formed over selected portions of the Farey Plane. This result has direct application in comparing the smoothness of digital angle sets.
Keywordsdiscrete image processing discrete Radon transforms Farey sequences and digital angles
Unable to display preview. Download preview PDF.
- 3.Guedon, J., Normand, N.: The Mojette transform: applications in image analysis and coding. In: The International Society for Optical Engineering. SPIE-Int. Soc. Opt. Eng., USA, vol. 3024, pp. 873–884 (1997)Google Scholar
- 4.Svalbe, I., Kingston, A.: Farey sequences and discrete Radon transform projection angles. Electronic Notes in Discrete Mathematics, vol. 12. Elsevier, Amsterdam (2003)Google Scholar
- 8.Kingston, A., Svalbe, I.: Adaptive discrete Radon transforms for grayscale images. Electronic Notes in Discrete Mathematics, vol. 12. Elsevier, Amsterdam (2003)Google Scholar
- 12.Kingston, A.: Orthogonal discrete Radon transform over p n. Signal Processing (November 2003) (submitted)Google Scholar
- 13.Kingston, A., Svalbe, I.: A discrete Radon transform for square arrays of arbitrary size. Submitted to DGCI 2005 (2004)Google Scholar
- 15.Boca, F., Zaharescu, A.: The correlations of Farey fractions (2004), http://arxiv.org/ps/math.NT/0404114 (preprint)