Dual frames compensating for erasures -- non-canonical case

TL;DR

This paper investigates methods for reconstructing signals from frame coefficients with erasures, focusing on constructing dual frames in non-canonical cases and proposing efficient iterative algorithms.

Contribution

It provides sufficient conditions for dual frame construction in non-canonical cases and introduces an iterative method for faster computation.

Findings

Iterative algorithm can outperform other methods in speed.

Conditions established for dual frame construction after erasures.

Analysis of differences between canonical and non-canonical dual frames.

Abstract

In this paper we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set $E$. Starting from a frame $(x_n)_{n=1}^\infty$ and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of $(x_n)_{n\in E^c}$ so that the perfect reconstruction can be obtained from the preserved frame coefficients. The work is motivated by methods using the canonical dual frame of $(x_n)_{n=1}^\infty$, which however do not extend automatically to the case when the canonical dual is replaced with another dual frame. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated. We also give several ways of computing a dual of the reduced frame, among which we are the most interested in the iterative procedure for computing this dual frame. Computational tests show that in certain cases the iterative algorithm performs faster than the other considered procedures.