On the number of digitizations of a disc depending on its position

Authors:
Martin N. Huxley;Joviša Žunić
Affiliations:
School of Mathematics, Cardiff University, Cardiff, U.K.;Computer Science Department, Exeter University, Exeter, U.K.
Venue:
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
Year:
2004

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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 3πR21O(R339/208 ·(logR)18627/8320) different (up to translations) digitizations of discs having the radius R. More formally, #D(R, (a, b)) | aandbvarythrough [0, 1) 3πR21O(R339/208 ·(logR)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.