Compressed sensing for real measurements of quaternion signals

TL;DR

This paper demonstrates that sparse quaternion signals can be uniquely reconstructed from limited real measurements using compressed sensing techniques, with guarantees provided by the restricted isometry property.

Contribution

It extends compressed sensing theory to quaternion signals, proving unique reconstruction via  minimization and providing error estimates for noisy and noiseless cases.

Findings

Sparse quaternion signals can be reconstructed from limited measurements.

Reconstruction guarantees depend on the measurement matrix satisfying the restricted isometry property.

Error bounds are established for both noisy and noiseless scenarios.

Abstract

The article concerns compressed sensing methods in the quaternion algebra. We prove that it is possible to uniquely reconstruct - by $\ell_1$ norm minimization - a sparse quaternion signal from a limited number of its real linear measurements, provided the measurement matrix satisfies so-called restricted isometry property with a sufficiently small constant. We also provide error estimates for the reconstruction of a non-sparse quaternion signal in the noisy and noiseless cases.