Abstract
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.
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References
Beylkin, G.: Discrete radon transform. IEEE Transactions on Acoustics, Speech, & Signal Processing 35, 162–172 (1987)
Matus, F., Flusser, J.: Image representation via a finite Radon transform. IEEE Transactions on Pattern Analysis & Machine Intelligence 15, 996–1006 (1993)
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)
Svalbe, I., Kingston, A.: Farey sequences and discrete Radon transform projection angles. Electronic Notes in Discrete Mathematics, vol. 12. Elsevier, Amsterdam (2003)
Hardy, G., Wright, E.: An introduction to the theory of numbers, 4th edn. Clarendon Press, Oxford (1960)
Acketa, D., Zunic, J.: On the number of linear partitions of the (m, n)-grid. Information Processing Letters 38, 163–168 (1991)
Svalbe, I., van der Spek, D.: Reconstruction of tomographic images using analog projections and the digital Radon transform. Linear Algebra and Its Applications 339, 125–145 (2001)
Kingston, A., Svalbe, I.: Adaptive discrete Radon transforms for grayscale images. Electronic Notes in Discrete Mathematics, vol. 12. Elsevier, Amsterdam (2003)
Boca, F., Cobeli, C., Zaharescu, A.: Distribution of lattice points visible from the origin. Commun. Math. Phys. 213, 433–470 (2000)
Hsung, T., Lun, D., Siu, W.: The discrete periodic Radon transform. IEEE Transactions on Signal Processing 44, 2651–2657 (1996)
Lun, D., Hsung, T., Shen, T.: Orthogonal discrete periodic Radon transform. Part I: theory and realization. Signal Processing 83, 941–955 (2003)
Kingston, A.: Orthogonal discrete Radon transform over p n. Signal Processing (November 2003) (submitted)
Kingston, A., Svalbe, I.: A discrete Radon transform for square arrays of arbitrary size. Submitted to DGCI 2005 (2004)
Augustin, V., Boca, F., Cobeli, C., Zaharescu, A.: The h-spacing distribution between Farey points. Math. Proc. Cambridge Phil. Soc. 131, 23–38 (2001)
Boca, F., Zaharescu, A.: The correlations of Farey fractions (2004), http://arxiv.org/ps/math.NT/0404114 (preprint)
Apostol, T.: Introduction to analytic number theory. Springer, New York (1976)
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Svalbe, I., Kingston, A. (2004). On Correcting the Unevenness of Angle Distributions Arising from Integer Ratios Lying in Restricted Portions of the Farey Plane. In: Klette, R., Žunić, J. (eds) Combinatorial Image Analysis. IWCIA 2004. Lecture Notes in Computer Science, vol 3322. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30503-3_9
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DOI: https://doi.org/10.1007/978-3-540-30503-3_9
Publisher Name: Springer, Berlin, Heidelberg
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