Variable Splitting and Fusing for Image Phase Retrieval.

Area of Science:

• Signal Processing
• Optimization
• Computational Imaging

Background:

• Phase Retrieval is crucial for reconstructing signals from intensity measurements of their Fourier Transforms.
• It is a challenging non-linear, non-convex optimization problem with applications in X-ray crystallography, microscopy, and blind deconvolution.
• Existing analytical methods for Phase Retrieval face limitations in convergence speed and performance.

Purpose of the Study:

• To develop novel algorithms for Phase Retrieval of real signals with improved convergence properties.
• To address the non-linear and non-convex nature of Phase Retrieval using variable splitting and alternating minimization.
• To enhance the efficiency and accuracy of signal reconstruction in Phase Retrieval applications.

Main Methods:

• The study employs variable splitting and alternating minimization techniques tailored for real signals.
• A novel algorithmic step involving recombination of separated variables was conceptualized based on geometric relations.
• Theoretical analysis was conducted to validate the convergence properties and the efficacy of the recombination step.

Main Results:

• The proposed Phase Retrieval method demonstrates substantially faster convergence rates compared to state-of-the-art analytical methods.
• Experimental results show equivalent or superior reconstruction quality across various setups.
• The algorithm effectively handles the complexities of non-linear and non-convex optimization in Phase Retrieval.

Conclusions:

• The developed Phase Retrieval algorithm offers a significant improvement in convergence speed for real signal recovery.
• The novel recombination step enhances the performance and reliability of the alternating minimization approach.
• This work provides a more efficient and accurate solution for Phase Retrieval problems in diverse scientific and engineering fields.