Different digitisations of displaced discs

The digitisation D(R, (a, b)) of areal 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)) boolean AND Z(2). In this paper we show that there are 4 pi R-2 + O(R-339/208 . (log R)(18627/8320)) different (up to translations) digitisations of discs having radius R. More formally, #(D(R, (a, b)) vertical bar a and b vary through [0, 1)} = 4 pi R-2 + O(R-339/208. (log R)(18627/8320)). 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.