Structured Random Models for Phase Retrieval with Optical Diffusers

TL;DR

This work tackles phase retrieval by replacing costly dense random sensing with a structured cascade of transforms and random diagonals, achieving comparable reconstruction with log-linear forward complexity. A two-layer architecture suffices to emulate i.i.d. randomness, and reconstruction is robust when combined with spectral initialization and gradient descent, achieving perfect recovery at $OR=2.8$; the key determinant of performance is the singular-value spectrum of the forward operator. The approach is physically realizable with simple optical components (lenses and diffusers) and is compatible with alternative unitary transforms, offering scalable imaging potential and a practical testbed for advanced reconstruction techniques. Overall, the paper provides a principled, hardware-friendly framework for large-scale phase imaging with provable guarantees and practical efficiency.

Abstract

Phase retrieval is a nonlinear inverse problem that arises in a wide range of imaging modalities, from electron microscopy to Fourier ptychography. In particular, the reconstruction is facilitated when the sensing matrix is i.i.d. random, enabling strong theoretical guarantees and efficient reconstruction algorithms. However, its applicability is restricted by excessive computational costs. In this paper, we propose structured random models for phase retrieval, where we emulate a dense random matrix by a cascade of structured transforms and random diagonal matrices. We reduce the complexity from quadratic to log-linear at no cost in reconstruction performance. Through a spectral method initialization followed by gradient descent, robust reconstruction is obtained at an oversampling ratio as low as 2.8. Moreover, we observe that the reconstruction performance is solely determined by the singular-value distribution of the forward matrix. This class of models can directly be implemented with basic optical elements such as lenses and diffusers, paving the way for large-scale phase imaging with robust reconstruction guarantees.

Figures (11)

• Figure 1: Optical implementations of structured random models with 1.5 and 2 layers, with each structured transform corresponding to a lens and each random diagonal matrix to a diffuser.
• Figure 2: Covariance matrices of ($4 \times 4$) structured random models with different layers using the Fourier transform. Covariance is gradually removed by adding more depth, which produces uncorrelated elements with 2 layers.
• Figure 3: Reconstruction Comparison. Structured random models obtains the same accuracy as dense random models for each benchmarked algorithm, achieving a perfect recovery at an OR of 2.8.
• Figure 4: Time benchmark of the forward pass of dense and structured random models under an OR of 1. Structured random models reach nearly constant forward time on the GPU for moderate image sizes, whereas dense random models quickly become prohibitively slow.
• Figure 5: Performance of structured random models with different depths. A 1-layer model is unable to reconstruct, regardless of its OR, while a 1.5-layer model yields suboptimal performance. A model with more than 2 layers produces the same performance as a dense random model.

Theorems & Definitions (7)

• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof