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

Abstract

The digitization D(Ra, b)) of a real disc D(Ra, b)) having radius R and the centre (a, b) consists of all integer points inside of D(Ra, b)), i.e., D(Ra, b)) = D(Ra, b)) ∩ Z 2. In this paper we show that that there are 3πR 2 + O (R 339/208 . (logR) 18627/8320) different (up to translations) digitizations of discs having the radius R. More formally, #{D(Ra, b)) | a and b vary through [0,1)} = 3πR 2 + O (R 339/208 . (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.