  In this paper, we provide a comprehensive theory of anti-aliasing sampling
patterns that explains and revises known results, and show how patterns as
predicted by the theory can be generated via a variational optimization
framework. We start by deriving the exact spectral expression for expected
error in reconstructing an image in terms of power spectra of sampling
patterns, and analyzing how the shape of power spectra is related to
anti-aliasing properties. Based on this analysis, we then formulate the problem
of generating anti-aliasing sampling patterns as constrained variational
optimization on power spectra. This allows us to not rely on any parametric
form, and thus explore the whole space of realizable spectra. We show that the
resulting optimized sampling patterns lead to reconstructions with less visible
aliasing artifacts, while keeping low frequencies as clean as possible.
