Aliasing and oblique dual pair designs for consistent sampling

TL;DR

This paper investigates the structure and optimization of oblique dual frames in finite-dimensional spaces, focusing on eigenvalues, convex potentials, and aliasing, with applications to minimizing duality-related distortions.

Contribution

It provides a comprehensive spectral and geometric analysis of oblique duals, introduces a notion of aliasing, and characterizes optimal rotations for dual frame minimization.

Findings

Eigenvalue lists of oblique duals are characterized.

Spectral and geometric structures of convex potential minimizers are described.

Conditions for rotations minimizing aliasing are established.

Abstract

In this paper we study some aspects of oblique duality between finite sequences of vectors $\cF$ and $\cG$ lying in finite dimensional subspaces $\cW$ and $\cV$, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to $\cF$ lying in $\cV$; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for $\cF$ under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces $\cV$ and $\cW$ has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual of $U\cdot \cF$ minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual pair associated to $U\cdot \cF$ minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.