Linear Transformations and Restricted Isometry Property

TL;DR

This paper investigates how linear transformations impact the Restricted Isometry Property (RIP) in compressed sensing, with applications to extending sensing matrices and using different bases, including redundant dictionaries.

Contribution

It provides new insights into the effect of linear transformations on RIP, extending compressed sensing theory to more general settings.

Findings

Linear transformations can preserve or alter RIP under certain conditions

Results apply to extending sensing matrices and different basis representations

Application to redundant dictionary settings enhances compressed sensing flexibility

Abstract

The Restricted Isometry Property (RIP) introduced by Cand\'es and Tao is a fundamental property in compressed sensing theory. It says that if a sampling matrix satisfies the RIP of certain order proportional to the sparsity of the signal, then the original signal can be reconstructed even if the sampling matrix provides a sample vector which is much smaller in size than the original signal. This short note addresses the problem of how a linear transformation will affect the RIP. This problem arises from the consideration of extending the sensing matrix and the use of compressed sensing in different bases. As an application, the result is applied to the redundant dictionary setting in compressed sensing.