# Phase Retrieval of Quaternion Signal via Wirtinger Flow

25 Oct 2022  ·  , ·

The main aim of this paper is to open the study of quaternion phase retrieval (QPR), i.e., the recovery of quaternion signal from 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 arising in the quaternion setting, especially those from 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 is obtained by finding a singular vector of a $4\times 4$ real matrix constructed from the QWF iterate, and hence can be computed efficiently. Experimental results on synthetic data and color images are presented to validate our theoretical results.

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