The bitmap obtained by scanning a printed pattern depends on the exact location of the scanning grid relative to the pattern. We consider ideal sampling with a regular lattice of delta functions. The displacement of the lattice relative to the pattern is random and obeys a uniform probability density function defined over a unit cell of the lattice. Random-phase sampling affects the edge-pixels of sampled patterns. The resulting number of distinct bitmaps and their relative frequencies can be predicted from a mapping of the original pattern boundary to the unit cell (called a modulo-grid diagram). The theory is supported by both simulated and experimental results. The modulo-grid diagram may be useful in helping to understand the effects of edge-pixel variation on Optical Character Recognition.