Abstract
The digitization D(R,(a,b)) of a real disc D(R, (a ,b)) having radius R and the centre (a, b) consists of all integer points inside of D(R, (a,b)), i.e., \(D(R,(a,b))=D(R,(a,b))\cap \mathcal{Z}^{2}\). In this paper we show that that there are
3πR 21O(R 339/208 ·(log R)18627/8320)
different (up to translations) digitizations of discs having the radius R. More formally,
#D(R, (a, b)) | a and b vary through [0, 1)
3πR 21O(R 339/208 ·(log R)18627/8320)
The result is of an interest in the area of digital image processing because it describes (in, let say, a combinatorial way) how big the impact of the object position on its digitization can be.
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References
Corteel, S., Rémond, G., Schaeffer, G., Thomas, H.: The Number of Plane Corner Cuts. Advances in Applied Mathematics 23, 49–53 (1999)
Fisk, S.: Separating point sets by circles, and the recognition of digital discs. IEEE Trans. on Pattern Analysis and Machine Intelligence 8, 554–556 (1984)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 4th edn. Oxford University Press, Oxford (1959)
Huxley, M.N.: Area, Lattice Points, and Exponential Sums. London Math. Soc. Monographs 13 (1996)
Huxley, M.N.: Exponential Sums and Lattice Points III. Proc. London Math. Soc. 87, 591–609 (2003)
Klette, R., Žunić, J.: Multigrid convergence of calculated features in image analysis. Journal of Mathematical Imaging and Vision 13, 173–191 (2000)
Krätzel, E.: Lattice Points. VEB Deutscher Verlag der Wissenschaften, Berlin (1988)
Koplowitz, J., Lindenbaum, M., Bruckstein, A.: On the number of digital straight lines on a squared grid. IEEE Trans. Information Theory 15, 949–953 (1993)
Lindenbaum, M., Koplowitz, J.: A new parametrization of digital straight lines. IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 847–852 (1991)
Wagner, U.: On the Number of Corner Cuts. Advances in Applied Mathematics 29, 152–161 (2002)
Žunić, J.: Cutting Corner with Spheres in d-dimensions. Advances in Applied Mathematics 32, 609–614 (2004)
Žunić, J.: On the Number of Digital Discs. Journal of Mathematical Imaging and Vision (accepted)
Žunić, J., Sladoje, N.: Efficiency of Characterizing Ellipses and Ellipsoids by Discrete Moments. IEEE Trans. on Pattern Analysis and Machine Intelligence 22, 407–414 (2000)
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Huxley, M.N., Žunić, J. (2004). On the Number of Digitizations of a Disc Depending on Its Position. 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_17
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DOI: https://doi.org/10.1007/978-3-540-30503-3_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23942-0
Online ISBN: 978-3-540-30503-3
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