In many imaging applications, including sensor arrays, MRI and CT,data is often sampled on
non-rectangular point sets with non-uniform density. Moreover, in image and video processing,
a mix of non-rectangular sampling structures naturally arise. Multirate processing typically
utilizes a normalized integer indexing scheme, which masks the true physical dimensions of the
points. However, the spatial correlation of such signals often contains important
information.This paper presents a theory of signals defined on regular discrete sets called
lattices, and presents an associated form of a finite Fourier transform denoted here as
multiresolution lattice discrete Fourier transform (MRL-DFT). Multirate processing techniques
such as decimation, interpolation and polyphase representations are presented in a context
which preserves the true spatial dimensions of the sampling structure.Moreover, the polyphase
formulation enables systematic representation and processing for sampling patterns with
variable spatial density, and provides a framework for developing generalized FFT and
regridding algorithms.