Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization.

Area of Science:

• Applied Physics
• Computational Mathematics
• Signal Processing

Background:

• The phase retrieval problem is critical in applied physics and engineering.
• Current state-of-the-art algorithms by Gerchberg, Saxton, and Fienup lack theoretical explanation for their success.
• The error reduction algorithm was previously linked to nonconvex alternating projections.

Purpose of the Study:

• To mathematically formulate the phase retrieval problem.
• To establish novel connections between numerical phase retrieval schemes and convex optimization.
• To provide a theoretical framework for understanding and improving phase recovery algorithms.

Main Methods:

• Identification of phase retrieval algorithms as convex optimization counterparts.
• Demonstration that Fienup's basic input-output algorithm is equivalent to Dykstra's algorithm.
• Characterization of Fienup's hybrid input-output algorithm as an instance of the Douglas-Rachford algorithm.

Main Results:

• Established a connection between Fienup's basic input-output algorithm and Dykstra's algorithm.
• Showed Fienup's hybrid input-output algorithm as a specific case of the Douglas-Rachford algorithm.
• Developed a theoretical framework linking established phase retrieval methods to convex optimization.

Conclusions:

• The study provides a new theoretical lens for analyzing phase retrieval algorithms.
• Connections to convex optimization offer potential avenues for algorithm improvement.
• Enhanced understanding of algorithm behavior, shortcomings, and performance is achieved.