Different digitisations of displaced discs

Abstract

The digitisation D(R, (a, b)) of a real disc D(R, (a,b)) having radius R and centre (a, b) consists of all integer points inside D(R, (a,b)) , i.e., D}(R, (a,b)) = D(R, (a, b)) ∩ Z2. In this paper we show that there are 4 π R2 + script O sign(R339/208 · R)18627/8320) different (up to translations) digitisations of discs having radius R. More formally, #{D(R,(a, b)) | a and b vary through [0,1)} = 4 π R2 + script O sign(R339/208). The result is of interest in the area of digital image processing because it describes how large the impact of the object position can be on its digitisation.