Phase Retrieval in Quaternion Euclidean Spaces

Abstract

Quaternion algebra is a noncommutative associative algebra. Noncommutativity limits the flexibility of computation and makes analysis related to quaternions nontrivial and challenging. Due to its applications in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention in recent years. This paper addresses phase retrievability in quaternion Euclidean spaces ${\mathbb {H}}^{M}$. We obtain a sufficient condition on phase retrieval frames for quaternionic left Hilbert module $\big ({\mathbb {H}}^{M},\,(\cdot ,\,\cdot )\big )$ of the form $\{e_{m}T_{n}g\}_{m,\,n\in {\mathbb {N}}_{M}}$, where $\{e_{m}\}_{m\in {\mathbb {N}}_{M}}$ is an orthonormal basis for $\big ({\mathbb {H}}^{M},\,(\cdot ,\,\cdot )\big )$ and $(\cdot ,\,\cdot )$ is the Euclidean inner product on ${\mathbb {H}}^{M}$. It is worth noting that $\{e_{m}\}_{m\in {\mathbb {N}}_{M}}$ is not necessarily $\left\{ \frac{1}{\sqrt{M}}e^{\frac{2\pi im\cdot }{M}}\right\} _{m\in {\mathbb {N}}_{M}}$, and that our method also applies to phase retrievability in ${\mathbb {C}}^{M}$. For the real Hilbert space $\big ({\mathbb {H}}^{M},\,\langle \cdot ,\,\cdot \rangle \big )$ induced by $\big ({\mathbb {H}}^{M},\,(\cdot ,\,\cdot )\big )$, we present a sufficient condition on phase retrieval frames $\{e_{m}T_{n}g\}_{m\in {\mathbb {N}}_{4M},\,n\in {\mathbb {N}}_{M}}$, where $\{e_{m}\}_{m\in {\mathbb {N}}_{4M}}$ is an orthonormal basis for $\big ({\mathbb {H}}^{M},\,\langle \cdot ,\,\cdot \rangle \big )$. We also give a method to construct and verify general phase retrieval frames for $\big ({\mathbb {H}}^{M},\,\langle \cdot ,\,\cdot \rangle \big )$. Finally, some examples are provided to illustrate the generality of our theory.