Compressed sensing for real measurements of quaternion signals

Compressed sensing in the quaternion algebra

“…We also estimated the error of the approximated reconstruction of a non-sparse quaternion signal from exact and noisy data. This improves our previous result for real measurement matrices and sparse quaternion vectors [1] and explains success of various numerical experiments in the quaternion setup [4,17,27].…”

“…[14,Chapter 6] for references). On the other hand, the authors of [6] constructed examples of s-sparse real signals which can not be uniquely reconstructed via ℓ 1 -norm minimization for δ s > 1 3 . This gives an obvious upper bound for δ s also for the general quaternion case.…”

“…In [1] we presented results of numerical experiments of sparse quaternion vector Ü reconstruction (by ℓ 1 -norm minimization) from its linear measurements Ý = ΦÜ for the case of real-valued measurement matrix Φ. Those experiments were inspired by the articles [4,17,27] and involved expressing the ℓ 1 quaternion norm minimization problem in terms of the second-order cone programming (SOCP).…”

Quaternion Gaussian matrices satisfy the RIP

“…The notion of restricted isometry constants was introduced by Candès and Tao in [4] and repeatedly considered afterwards. The concept was also generalized to quaternion signals [1,2]. Definition 1.1.…”

“…real Gaussian and Bernoulli random matrices, also partial Discrete Fourier Transform matrices satisfy the RIP (with overwhelming probability) [5], however, until recently there were no known examples of quaternion matrices satisfying this condition. In [1,Lemma 3.2] we proved that if a real matrix Φ ∈ R m×n satisfies the inequalities (1.1) for real s-sparse vectors Ü ∈ R n , then it also satisfies it -with the same constant δ s -for s-sparse quaternion vectors Ü ∈ H n . This was a first step towards developing theoretical background of compressed sensing methods in the quaternion algebra, since we also showed that it is possible to reconstruct sparse quaternion vectors from a small number of their linear measurements if an appropriate restricted isometry constant of a real measurement matrix Φ ∈ R m×n is sufficiently small [1,Corollary 5.1].…”