Robust dual reconstruction systems and fusion frames

TL;DR

This paper investigates duality and optimality in finite-dimensional reconstruction systems, focusing on projective systems analogous to fusion frames, and explores error measures, approximations, and existence conditions for dual systems.

Contribution

It introduces the concept of dual projective reconstruction systems, analyzes their optimality under erasures, and studies approximation and existence problems in this context.

Findings

Dual systems can be optimized for erasure resilience.

Approximation of arbitrary systems by projective systems is characterized.

Existence conditions for dual projective systems are identified.

Abstract

We study the duality of reconstruction systems, which are $g$-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.