Poisson Phase Retrieval in Very Low-count Regimes.

Area of Science:

• Computational imaging
• Optimization algorithms
• Statistical signal processing

Background:

• Phase retrieval is crucial for reconstructing images from indirect measurements.
• Maximum Likelihood (ML) estimation is sensitive to noise models, especially in low-count regimes.
• Existing algorithms struggle with Poisson noise in very low photon count scenarios.

Purpose of the Study:

• To develop advanced phase retrieval algorithms for maximum likelihood estimation under Poisson distributions in low-count regimes.
• To enhance reconstruction quality and convergence speed compared to current methods.
• To eliminate parameter tuning in optimization algorithms, except for the number of iterations.

Main Methods:

• Proposed a modified Wirtinger flow (WF) algorithm utilizing a step size derived from observed Fisher information.
• Introduced a novel curvature for majorize-minimize (MM) algorithms with a quadratic majorizer, theoretically proven to be sharper.
• Compared proposed WF and MM algorithms against various optimization techniques (other WF schemes, LBFGS, ADMM).

Main Results:

• Poisson ML-based algorithms yield superior reconstruction quality over Gaussian ML models for low-count data.
• Regularization techniques like anisotropic total variation (TV) further enhance reconstruction.
• The proposed WF algorithm with Fisher information step size demonstrates faster convergence (cost function and PSNR vs. time) in both unregularized and regularized cases compared to LBFGS, MM, and ADMM.

Conclusions:

• The developed phase retrieval algorithms significantly improve image reconstruction accuracy and efficiency for low-count Poisson measurements.
• The modified WF algorithm with Fisher information step size represents a state-of-the-art approach for this challenging problem.
• The findings have implications for various imaging applications dealing with photon-starved data.