Aliasing and oblique dual pair designs for consistent sampling

MARÍA JOSÉ BENAC; PEDRO MASSEY; DEMETRIO STOJANOFF

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Lugar: Amsterdam; Año: 2015 vol. 487 p. 112 - 145

0024-3795

In this paper we study some aspects of oblique duality between finite sequences of vectors F and G lying in finite dimensional subspaces W and V, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to F lying in V. We compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for F 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 V and W has in oblique duality. We apply this analysis to compute those rigid rotations U for W such that the canonical oblique dual of U⋅F minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations U for W such that the canonical oblique dual pair associated to U⋅F minimize the aliasing. We point out that these two last problems are intrinsic to oblique duality, within the context of consistent sampling.