Solving large-scale general phase retrieval problems via a sequence of convex relaxations

Area of Science:

• Optics and Photonics
• Computational Imaging
• Applied Mathematics

Background:

• Phase retrieval is essential for reconstructing optical fields from intensity measurements.
• Traditional methods often struggle with large-scale, general problems due to non-convexity.
• Accurate phase estimation is vital in fields like microscopy and astronomy.

Purpose of the Study:

• To introduce a novel convex relaxation-based algorithm for large-scale general phase retrieval.
• To address the computational challenges associated with existing phase retrieval techniques.
• To enhance the reliability and speed of phase retrieval algorithms.

Main Methods:

• Reformulated the non-convex phase retrieval problem into a rank-constrained problem.
• Employed nuclear norm minimization for convex relaxation.
• Developed an iterative algorithm, convex optimization-based phase retrieval (COPR).
• Utilized the alternating direction method of multipliers (ADMM) for efficient computation.

Main Results:

• Demonstrated linear or faster convergence for certain phase retrieval problems in noise-free conditions.
• The COPR algorithm, accelerated by ADMM, shows significant improvements in speed and reliability.
• Numerical simulations confirm the algorithm's effectiveness compared to state-of-the-art methods.

Conclusions:

• The proposed COPR algorithm provides an efficient and reliable solution for large-scale general phase retrieval.
• ADMM integration overcomes scalability limitations of semidefinite programming.
• This work advances the practical application of phase retrieval in various scientific domains.