Expansions from frame coefficients with erasures

TL;DR

This paper introduces a novel method for perfect signal reconstruction from frame coefficients with erasures by constructing dual frames, and also characterizes full spark frames and methods for frame transformation.

Contribution

It presents a new approach to erasure recovery using dual frames, characterizes full spark frames, and offers methods for constructing totally positive matrices and transforming frames.

Findings

Constructed dual frames enable perfect reconstruction despite erasures.

Characterized all full spark frames in finite-dimensional Hilbert spaces.

Provided methods for creating totally positive matrices and converting frames to Parseval frames.

Abstract

We propose a new approach to the problem of recovering signal from frame coefficients with erasures. Such problems arise naturally from applications where some of the coefficients could be corrupted or erased during the data transmission. Provided that the erasure set satisfies the minimal redundancy condition, we construct a suitable synthesizing dual frame which enables us to perfectly reconstruct the original signal without recovering the lost coefficients. Such dual frames which compensate for erasures are described from various viewpoints. In the second part of the paper frames robust with respect to finitely many erasures are investigated. We characterize all full spark frames for finite-dimensional Hilbert spaces. In particular, we show that each full spark frames is generated by a matrix whose all square submatrices are nonsingular. In addition, we provide a method for constructing totally positive matrices. Finally, we give a method, applicable to a large class of frames, for transforming general frames into Parseval ones.