Aliasing and oblique dual pair designs for consistent sampling

Abstract

In this paper we study some aspects of oblique duality between finite sequences of vectors FF and GG lying in finite dimensional subspaces WW and VV, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to FF lying in VV. We compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for FF with norm restrictions; as an application, we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible, among oblique duals with norm restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces VV and WW has in oblique duality. We apply this analysis to compute those rigid rotations U for WW such that the canonical oblique dual of U⋅FU⋅F minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations U for WW such that the canonical oblique dual pair associated to U⋅FU⋅F minimize the aliasing. We point out that these two last problems are intrinsic to oblique duality, within the context of consistent sampling.