[Paper Review] Numerical Optimization Algorithm of Wavefront Phase Retrieval from Multiple Measurements

Ji Li, Tie Zhou|arXiv (Cornell University)|Jul 7, 2016

Advanced X-ray Imaging Techniques31 references3 citations

TL;DR

This paper proposes a numerical optimization framework for wavefront phase retrieval from multiple intensity measurements using nonconvex objective functions based on maximum likelihood (MLP) and least squares (LS) models. It develops complex-variable optimization techniques, showing LS outperforms MLP due to better Hessian eigenvalue clustering, with LBFGS proving most efficient and robust in simulations.

ABSTRACT

Wavefront phase retrieval from a set of intensity measurements can be formulated as an optimization problem. Two nonconvex objective models (MLP and its variants LS) based on maximum likelihood estimation are investigated. We develop numerical optimization algorithms for real-valued function of complex variables and apply them to solve the wavefront phase retrieval problem efficiently. Numerical simulation is given with application to three wavefront phase retrieval problems. LS model shows better numerical performances than MLP model. An explanation for this is that the distribution of the eigenvalues of Hessian matrix of LS model is more clustered than MLP model. LBFGS shows more robust performance and takes fewer calculations than other line search methods.

Motivation & Objective

• Address the ill-posedness and nonuniqueness of wavefront phase retrieval in adaptive optics and astronomy.
• Formulate phase retrieval as a nonconvex optimization problem using multiple intensity measurements (phase diversity).
• Develop numerical optimization algorithms tailored for real-valued functions of complex variables to solve the phase retrieval problem.
• Compare performance of maximum likelihood (MLP) and least squares (LS) models in terms of convergence and robustness.
• Evaluate the efficiency and stability of various line search methods, particularly LBFGS, in solving the optimization problem.

Proposed method

• Formulate two nonconvex objective functions: MLP based on Poisson likelihood estimation and LS as a Gaussian approximation.
• Derive gradient and Hessian operators for both models using complex calculus, enabling efficient optimization in complex-valued domains.
• Implement line search methods for real-valued functions of complex variables, directly operating on complex-valued parameters.
• Apply the LBFGS algorithm to minimize the objective functions, leveraging limited-memory Hessian approximations for scalability.
• Use discrete Fourier transforms to model wavefront intensity measurements and define the forward operator F(u).
• Incorporate a small regularization parameter ε² to stabilize the Hessian computation and avoid division by zero in iterative updates.

Experimental results

Research questions

• RQ1 How do the MLP and LS models compare in terms of numerical performance and convergence speed for wavefront phase retrieval?
• RQ2 What is the impact of Hessian eigenvalue distribution on the convergence behavior of phase retrieval algorithms?
• RQ3 Which line search method—LBFGS or others—demonstrates superior robustness and computational efficiency in this context?
• RQ4 Can the LS model serve as an effective and computationally efficient approximation to the more statistically principled MLP model?
• RQ5 How does the use of multiple measurements (phase diversity) improve the uniqueness and stability of the phase retrieval solution?

Key findings

• The LS model consistently outperforms the MLP model in numerical simulations, achieving faster convergence and higher accuracy.
• The Hessian matrix of the LS model exhibits a more clustered eigenvalue distribution compared to MLP, which contributes to its superior convergence behavior.
• LBFGS demonstrates the most robust performance and requires fewer iterations and function evaluations than other line search methods.
• The LS model is computationally more efficient than the MLP model while maintaining high reconstruction fidelity.
• The use of multiple measurements (phase diversity) enables unique recovery of the wavefront phase, mitigating the nonuniqueness issue inherent in single-intensity measurements.
• The derived gradient and Hessian operators for complex-valued functions are essential for accurate and efficient optimization in the complex domain.