Quaternion Phase Retrieval via the Alternating Direction Method of Multipliers

Phase retrieval in quaternion domains presents unique computational challenges due to the non-commutative nature of quaternion algebra. Although several methods have been developed to address this issue, there remains significant potential for improvement. In this paper, we develop a novel quaternion-based alternating direction method of multipliers framework for quaternion phase retrieval from magnitude-only measurements based on the generalized Hamilton-real calculus, and provide a rigorous convergence analysis. Through extensive numerical simulations, we demonstrate that the proposed quaternion-based alternating direction method of multipliers framework outperforms existing methods, such as quaternion Wirtinger flow, quaternion truncated Wirtinger flow, and quaternion truncated amplitude flow, thereby establishing a new benchmark for quaternion phase retrieval problems. The theoretical framework and empirical results demonstrate that quaternion-based alternating direction method of multipliers offers a novel and effective approach for solving quaternion phase retrieval problems.