Different Digitisations of Displaced Discs

AbstractThe digitisation ${\bf D}(R, (a, b))$ 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., ${\bf D}(R, (a,b)) = D(R, (a, b)) \cap {\bf Z}^2.$ In this paper we show that there are $4 \pi R^2 +{\cal O}(R^{339/208}\cdot (\log R)^{18627/8320})$ different (up to translations) digitisations of discs having radius $R$. More formally, $\#\{{\bf D}(R, (a, b)) \;|\; \;a\; \makebox{and} \;b\; \makebox{vary through} \;[0,1)\} = 4 \pi R^2 +{\cal O}(R^{339/208}\cdot (\log R)^{18627/8320}).$ 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.