Sampling below the Nyquist density using spectral subtiles

An image is assumed to have a spectrum zero outside of a defined support. To avoid aliasing, the replicated support due to sampling cannot overlap. The minimum sampling density corresponding to nonoverlapping supports is the Nyquist density. Replication often necessitate gaps. Support shapes that fill the frequency plane without gaps are tiles. We offer a strategy for achieving minimum sampling density when the spectrum is confined to a subtile. Cookie cutter versions of the subtile shape, when rotated, translated, and/or flipped, result in a tile. The composite signal can have symmetric redundancies that allow reduction of the sampling density to the area of the subtile. We analyze the cases for tiles with twofold point symmetry and mirror symmetry. Two subtiles are required to construct a tile. Threefold, fourfold, and sixfold symmetry is also considered. In the cases considered, the overall sampling density in terms of the samples' required storage is reduced to the area of the support of the subtile.