[Paper Review] Phase Retrieval With One or Two Diffraction Patterns by Alternating Projections of the Null Vector

Pengwen Chen, Albert Fannjiang+1 more|arXiv (Cornell University)|Oct 26, 2015

Advanced X-ray Imaging Techniques33 references3 citations

TL;DR

This paper proposes parallel (PAP) and serial (SAP) alternating projection algorithms for phase retrieval using one or two coded diffraction patterns, leveraging the null vector method for accurate initialization. It proves geometric convergence with sharp spectral gap bounds and demonstrates faster, more accurate recovery than Wirtinger flow and other non-convex solvers, with SAP outperforming PAP in convergence speed.

ABSTRACT

Two versions of alternating projection (AP), the parallel alternating projection (PAP) and the serial alternating projection (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null vector method is proposed for initialization and proved to produce asymptotically accurate initialization for the Gaussian case. Extensive numerical experiments are performed to show that the null vector method produces more accurate initialization than the spectral vector method and that PAP/SAP converge faster to more accurate solutions than other iterative schemes for non-convex optimization such as the Wirtinger flow. Moreover, SAP converges still faster than PAP. In practice AP and the null vector method together produce globally convergent iterates to the true object.

Motivation & Objective

• Address the challenge of phase retrieval from a single or dual coded diffraction pattern, where phase information is missing and the problem is non-convex.
• Overcome the local convergence limitations of iterative phase retrieval methods by introducing a robust initialization strategy.
• Develop and analyze two novel alternating projection schemes—parallel (PAP) and serial (SAP)—to improve convergence speed and accuracy.
• Establish theoretical convergence guarantees with sharp bounds based on spectral gap conditions.
• Demonstrate empirically that the proposed method outperforms existing non-convex optimization techniques like Wirtinger flow.

Proposed method

• Propose the parallel alternating projection (PAP) and serial alternating projection (SAP) algorithms to iteratively refine estimates of the object from intensity-only measurements.
• Introduce the null vector method as a novel initialization technique that asymptotically approaches the true object for Gaussian signals, improving upon the spectral vector method.
• Establish geometric convergence of PAP and SAP using a spectral gap condition, providing sharp upper bounds on convergence rates.
• Use the null vector to initialize the alternating projection process, ensuring global convergence to the true solution in practice.
• Formulate the phase retrieval problem as a feasibility problem over the intersection of two sets: the measured intensity constraints and the null space of the measurement operator.
• Leverage the structure of the measurement model to derive convergence rates dependent on the spectral properties of the measurement matrix.

Experimental results

Research questions

• RQ1 Can alternating projection methods with proper initialization achieve global convergence in phase retrieval with only one or two diffraction patterns?
• RQ2 How do the convergence rates of PAP and SAP compare under spectral gap conditions, and which scheme converges faster?
• RQ3 Does the null vector method provide more accurate initialization than the spectral vector method for Gaussian signals?
• RQ4 To what extent do PAP and SAP outperform existing non-convex solvers like Wirtingger flow in terms of accuracy and speed?
• RQ5 What theoretical guarantees can be established for the convergence of these alternating projection schemes in the context of phase retrieval?

Key findings

• The null vector method produces asymptotically accurate initialization for Gaussian signals, significantly outperforming the spectral vector method in numerical experiments.
• PAP and SAP both exhibit geometric convergence with sharp bounds derived from the spectral gap condition, ensuring predictable and fast convergence rates.
• SAP converges faster than PAP, demonstrating the advantage of sequential updates in the alternating projection framework.
• The combination of the null vector initialization and alternating projection schemes leads to globally convergent iterates that recover the true object with high accuracy.
• Numerical results show that PAP and SAP achieve higher solution accuracy and faster convergence than Wirtinger flow and other non-convex optimization methods.
• The proposed method is effective even with only one or two diffraction patterns, making it suitable for applications with limited measurement resources.