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

Heinz H Bauschke1, Patrick L Combettes, D Russell Luke

  • 1Department of Mathematics and Statistics, University of Guelph, Ontario, Canada. bauschke@cecm.sfu.ca

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|July 4, 2002
PubMed
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Phase retrieval, crucial in applied physics, uses algorithms whose success lacks theoretical explanation. This study connects these methods to convex optimization, offering a new framework for understanding and improving phase recovery.

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.

Related Experiment Videos

  • 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.