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On the Number of Digitizations of a Disc Depending on Its Position

  • Martin N. Huxley
  • Joviša Žunić
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)

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.

Keywords

Digital disc lattice points enumeration 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Martin N. Huxley
    • 1
  • Joviša Žunić
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffU.K.
  2. 2.Computer Science DepartmentExeter UniversityExeterU.K.

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