What conjugate phase retrieval complex vectors can do in quaternion Euclidean spaces

Abstract

Quaternion algebra ℍ {\mathbb{H}} is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses conjugate phase retrieval problem in the quaternion Euclidean space ℍ M {\mathbb{H}^{M}} with M ≥ 2 {M\geq 2} . Write ℂ η = { ξ : ξ = ξ 0 + β ⁢ η , ξ 0 , β ∈ ℝ } {\mathbb{C}_{\eta}=\{\xi:\xi=\xi_{0}+\beta\eta,\,\xi_{0},\,\beta\in\mathbb{R}\}} for η ∈ { i , j , k } {\eta\in\{i,\,j,\,k\}} . We remark that not only ℂ η M {\mathbb{C}_{\eta}^{M}} -vectors cannot allow traditional conjugate phase retrieval in ℍ M {\mathbb{H}^{M}} , but also ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors cannot allow phase retrieval in ℍ M {\mathbb{H}^{M}} . We are devoted to conjugate phase retrieval of ℂ i M ∪ ℂ j M {\mathbb{C}_{i}^{M}\cup\mathbb{C}_{j}^{M}} -complex vectors in ℍ M {\mathbb{H}^{M}} , where “conjugate” is not the traditional conjugate. We introduce the notions of conjugation, maximal commutative subset and conjugate phase retrieval. Using the phase lifting techniques, we present some sufficient conditions on complex vectors allowing conjugate phase retrieval. And some examples are also provided to illustrate the generality of our theory.