This article deals with ambiguous surface digitizations by dilation in n-dimensional space. The digitization of a sufficiently regular surface is separating but not necessarily minimal. We will determine conditions under which the supercover and the grid intersection digitizations are discrete surfaces. It will also be proven that non-overlapping domains do not solve the problem of simple point in digitizations. No matter how the digitization domain is chosen there will will occur ambiguous cases which have to be treated differently.