Phase Retrieval: An Overview of Recent Developments (1510.07713v1)

Abstract: The problem of phase retrieval is a classic one in optics and arises when one is interested in recovering an unknown signal from the magnitude (intensity) of its Fourier transform. While there have existed quite a few approaches to phase retrieval, recent developments in compressed sensing and convex optimization-based signal recovery have inspired a host of new ones. This work presents an overview of these approaches. Since phase retrieval, by its very nature, is ill-posed, to make the problem meaningful one needs to either assume prior structure on the signal (e.g., sparsity) or obtain additional measurements (e.g., masks, structured illuminations). For both the cases, we review conditions for the identifiability of the signal, as well as practical algorithms for signal recovery. In particular, we demonstrate that it is possible to robustly and efficiently identify an unknown signal solely from phaseless Fourier measurements, a fact with potentially far-reaching implications.

• The paper presents a comprehensive survey of phase retrieval, framing it as a constrained optimization problem crucial for signal reconstruction.
• It analyzes classical methods like oversampling and alternating projections while spotlighting advanced techniques such as SDP-based PhaseLift and compressed sensing.
• Innovative strategies including structured masks and STFT-based redundancy are introduced to mitigate ambiguities and enhance unique signal recovery.

An Overview of Recent Developments in Phase Retrieval

The paper "Phase Retrieval: An Overview of Recent Developments" offers a comprehensive survey of theoretical advancements and algorithmic strategies in the field of phase retrieval. This problem is pivotal across various engineering and applied physics domains, including optical imaging, X-ray crystallography, and signal processing, given the prevalent challenge of reconstructing a signal from its Fourier magnitude alone due to the unavailability of direct phase measurements.

The crux of the paper is the mathematical formulation of phase retrieval as a constrained optimization problem, emphasizing the reconstruction of a signal subject to the magnitude square of its Fourier transform. Traditional approaches such as the oversampling method and alternating projection techniques are evaluated. Algorithms like the Gerchberg-Saxton and Fienup's Hybrid Input-Output have been instrumental yet limited by their dependence on initial conditions and convergence to local minima.

Recent advancements have leveraged more sophisticated mathematical frameworks, notably semidefinite programming (SDP) and sparsity constraints, the latter inspired by compressed sensing paradigms. Algorithms like PhaseLift demonstrate robust recovery under certain conditions by lifting the problem to higher-dimensional spaces. Nonetheless, computational demands remain significant, and the problem's inherent ill-posedness persists in high-dimensional settings without additional information.

An emerging direction is the integration of masks and structured illuminations in optical systems to augment the information content captured during data acquisition. The paper outlines how strategically designed masks, akin in function to structured diffractive elements, substantially mitigate ambiguities and enable unique signal recovery, even under scenarios plagued by traditional phase retrieval challenges. The computational strategies herein employ optimization techniques and combinatorial algorithms, optimizing for practical implementation and efficiency.

Furthermore, phase retrieval from Short-Time Fourier Transform (STFT) magnitudes introduces redundancy that aids in overcoming the shortcomings of classic approaches. The exploration of STFT phase retrieval for unique signal reconstruction demonstrates the potential of creating robust algorithms by systematically leveraging overlap between short-time signal sections. The success of these methods is contingent upon carefully chosen parameters for window length and shift intervals, with significant influence on uniqueness and stability.

The paper also broaches sparse phase retrieval, highlighting how integrating sparsity constraints has revolutionized the understanding of underlying signal structures and provided avenues for solving otherwise intractable problems. Techniques influenced by compressed sensing, such as TSPR and GESPAR, are lauded for their potential to efficiently recover signals amidst noise and ill-posed conditions, outperforming traditional iterative methods.

In summary, this paper delineates the multifaceted progress within phase retrieval, underscoring both theoretical insights and practical algorithms. The implications of this research stretch beyond traditional optical setups, hinting at potential breakthroughs in imaging, communications, and signal processing. As emerging techniques refine, one can anticipate the convergence of more efficient methodologies enabling high-fidelity reconstructions from purely magnitude-based measurements. Future research avenues may further explore the integration of advanced priors, enhanced computational strategies, and interdisciplinary approaches to phase retrieval, underscoring its fundamental role in advanced technological applications.