Fourier phase retrieval with a single mask by Douglas-Rachford algorithms

Area of Science:

• Optics and Photonics
• Computational Imaging
• Signal Processing

Background:

• Phase retrieval is crucial for reconstructing objects from intensity measurements.
• Traditional methods often require multiple measurements or specific object constraints for unique solutions.
• The Fourier-domain Douglas-Rachford (FDR) algorithm offers a novel approach to phase retrieval.

Purpose of the Study:

• To analyze the Fourier-domain Douglas-Rachford (FDR) algorithm for phase retrieval with a single random mask.
• To investigate conditions ensuring the uniqueness of phase retrieval solutions.
• To establish convergence properties of the FDR algorithm.

Main Methods:

• Analysis of the FDR algorithm under two settings: object sector condition and additional diffraction patterns.
• Proof of uniqueness for the projected fixed point.
• Derivation of local, geometric convergence rate using a spectral gap condition.
• Numerical experiments to demonstrate convergence behavior.

Main Results:

• Uniqueness of the projected fixed point is proven for both analyzed settings.
• Local, geometric convergence is established for the second setting.
• Numerical experiments show global, power-law convergence from arbitrary initialization.
• Effective reconstruction is achieved even without oversampling, using 3 or more coded diffraction patterns.
• A projection trick recovers geometric convergence from the power-law regime.

Conclusions:

• The FDR algorithm provides a robust framework for unique phase retrieval.
• The algorithm demonstrates reliable convergence properties, adaptable for practical applications.
• Accurate phase retrieval is achievable from generic initializations, enhancing its utility.