Compressed sensing in the quaternion algebra

TL;DR

This paper extends compressed sensing theory to quaternion algebra, demonstrating that sparse quaternion signals can be uniquely reconstructed via -minimization under certain matrix conditions, with error estimates provided.

Contribution

It introduces the application of restricted isometry property to quaternion measurement matrices and proves unique reconstruction of sparse quaternion signals.

Findings

Unique reconstruction of sparse quaternion signals via -minimization.

Reconstruction guarantees under restricted isometry property.

Error bounds for noisy and noiseless cases.

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 linear measurements, provided the quaternion 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.