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 HM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}<^>{M}$$\end{document}. We obtain a sufficient condition on phase retrieval frames for quaternionic left Hilbert module (HM,(center dot,center dot))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,(\cdot ,\,\cdot )\big )$$\end{document} of the form {emTng}m,n is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}T_{n}g\}_{m,\,n\in {\mathbb {N}}_{M}}$$\end{document}, where {em}m is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}\}_{m\in {\mathbb {N}}_{M}}$$\end{document} is an orthonormal basis for (HM,(center dot,center dot))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,(\cdot ,\,\cdot )\big )$$\end{document} and (center dot,center dot)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\cdot ,\,\cdot )$$\end{document} is the Euclidean inner product on HM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {H}}<^>{M}$$\end{document}. It is worth noting that {em}m is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}\}_{m\in {\mathbb {N}}_{M}}$$\end{document} is not necessarily 1Me2 pi im center dot Mm is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \frac{1}{\sqrt{M}}e<^>{\frac{2\pi im\cdot }{M}}\right\} _{m\in {\mathbb {N}}_{M}}$$\end{document}, and that our method also applies to phase retrievability in CM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}<^>{M}$$\end{document}. For the real Hilbert space (HM,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,\langle \cdot ,\,\cdot \rangle \big )$$\end{document} induced by (HM,(center dot,center dot))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,(\cdot ,\,\cdot )\big )$$\end{document}, we present a sufficient condition on phase retrieval frames {emTng}m is an element of N4M,n is an element of NM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}T_{n}g\}_{m\in {\mathbb {N}}_{4M},\,n\in {\mathbb {N}}_{M}}$$\end{document}, where {em}m is an element of N4M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{e_{m}\}_{m\in {\mathbb {N}}_{4M}}$$\end{document} is an orthonormal basis for (HM,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,\langle \cdot ,\,\cdot \rangle \big )$$\end{document}. We also give a method to construct and verify general phase retrieval frames for (HM,⟨center dot,center dot⟩)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\big ({\mathbb {H}}<^>{M},\,\langle \cdot ,\,\cdot \rangle \big )$$\end{document}. Finally, some examples are provided to illustrate the generality of our theory.
