Phase Retrieval of Quaternion Signal via Wirtinger Flow

The main aim of this paper is to study quaternion phase retrieval (QPR), i.e., the recovery of quaternion signal from the magnitude of quaternion linear measurements. We show that all $d$-dimensional quaternion signals can be reconstructed up to a global right quaternion phase factor from $O(d)$ phaseless measurements. We also develop the scalable algorithm quaternion Wirtinger flow (QWF) for solving QPR, and establish its linear convergence guarantee. Compared with the analysis of complex Wirtinger flow, a series of different treatments are employed to overcome the difficulties of the non-commutativity of quaternion multiplication. Moreover, we develop a variant of QWF that can effectively utilize a pure quaternion priori (e.g., for color images) by incorporating a quaternion phase factor estimate into QWF iterations. The estimate can be computed efficiently as it amounts to finding a singular vector of a $4\times 4$ real matrix. Motivated by the variants of Wirtinger flow in prior work, we further propose quaternion truncated Wirtinger flow (QTWF), quaternion truncated amplitude flow (QTAF) and their pure quaternion versions. Experimental results on synthetic data and color images are presented to validate our theoretical results. In particular, for pure quaternion signal recovery, our quaternion method often succeeds with measurements notably fewer than real methods based on monochromatic model or concatenation model.