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1. Acceleration and Implicit Regularization in Gaussian Phase Retrieval

   Maunu, Tyler; Molina-Fructuoso, Martin (May 2024, Proceedings of Machine Learning Research)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   We study accelerated optimization methods in the Gaussian phase retrieval problem. In this setting, we prove that gradient methods with Polyak or Nesterov momentum have similar implicit regularization to gradient descent. This implicit regularization ensures that the algorithms remain in a nice region, where the cost function is strongly convex and smooth despite being nonconvex in general. This ensures that these accelerated methods achieve faster rates of convergence than gradient descent. Experimental evidence demonstrates that the accelerated methods converge faster than gradient descent in practice. 

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2. Phase contrast imaging of non-collinear spin textures with Lorentz microscopy

   https://doi.org/10.1557/s43578-023-01216-1  

   Streubel, Robert (December 2023, Journal of Materials Research)

   This article discusses the physical and mathematical background of phase contrast imaging with in‑line electron holography from a physics rather than a microscopy perspective and showcases the strength of non‑iterative and iterative approaches by application to magnetism research. A comprehensive derivation of magnetic and electric phase shift due to electromagnetic interaction with matter and electron wave propagation is presented as the foundation for phase retrieval algorithms based on the transport‑of‑intensity equation and Gerchberg–Saxton—an iterative exit wave reconstruction algorithm. The strength and potential of both algorithms are highlighted by experimental and numerical quantitative comparison using non‑collinear spin textures. Although the focus of this work is on magnetism research, the indifference of the exit wave reconstruction to the origin of the phase shift ensures applicability to study spatial variations in both electric and spin distributions in quantum, energy, and magnetic materials. 

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3. Spatial light interference microscopy: principle and applications to biomedicine

   https://doi.org/10.1364/AOP.417837  

   Chen, Xi; Kandel, Mikhail_E; Popescu, Gabriel (May 2021, Advances in Optics and Photonics)

   In this paper, we review spatial light interference microscopy (SLIM), a common-path, phase-shifting interferometer, built onto a phase-contrast microscope, with white-light illumination. As one of the most sensitive quantitative phase imaging (QPI) methods, SLIM allows for speckle-free phase reconstruction with sub-nanometer path-length stability. We first review image formation in QPI, scattering, and full-field methods. Then, we outline SLIM imaging from theory and instrumentation to diffraction tomography. Zernike’s phase-contrast microscopy, phase retrieval in SLIM, and halo removal algorithms are discussed. Next, we discuss the requirements for operation, with a focus on software developed in-house for SLIM that enables high-throughput acquisition, whole slide scanning, mosaic tile registration, and imaging with a color camera. We introduce two methods for solving the inverse problem using SLIM, white-light tomography, and Wolf phase tomography. Lastly, we review the applications of SLIM in basic science and clinical studies. SLIM can study cell dynamics, cell growth and proliferation, cell migration, mass transport, etc. In clinical settings, SLIM can assist with cancer studies, reproductive technology, blood testing, etc. Finally, we review an emerging trend, where SLIM imaging in conjunction with artificial intelligence brings computational specificity and, in turn, offers new solutions to outstanding challenges in cell biology and pathology. 

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4. Compressive phase retrieval: Optimal sample complexity with deep generative priors

   https://doi.org/10.1002/cpa.22155  

   Hand, Paul; Leong, Oscar; Voroninski, Vladislav (September 2023, Communications on Pure and Applied Mathematics)

   Abstract Advances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data‐driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non‐convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements. 

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5. Computing high quality phase-only holograms for holographic displays

   https://doi.org/10.1117/12.2547647  

   Chakravarthula, Praneeth Kumar; Peng, Yifan; Kollin, Joel; Heide, Felix; Fuchs, Henry (February 2020, Optical Architectures for Displays and Sensing in Augmented, Virtual, and Mixed Reality (AR, VR, MR), International Society for Optics and Photonics, 2020)

   Holography has demonstrated potential to achieve a wide field of view, focus supporting, optical see-through augmented reality display in an eyeglasses form factor. Although phase modulating spatial light modulators are becoming available, the phase-only hologram generation algorithms are still imprecise resulting in severe artifacts in the reconstructed imagery. Since the holographic phase retrieval problem is non-linear and non-convex and computationally expensive with the solutions being non-unique, the existing methods make several assumptions to make the phase-only hologram computation tractable. In this work, we deviate from any such approximations and solve the holographic phase retrieval problem as a quadratic problem using complex Wirtinger gradients and standard first-order optimization methods. Our approach results in high-quality phase hologram generation with at least an order of magnitude improvement over existing state-of-the-art approaches. 

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