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prDeep: Robust Phase Retrieval with a Flexible Deep Network
Christopher A. Metzler, Philip Schniter, Ashok Veeraraghavan, Richard G. Baraniuk
TL;DR
Phase retrieval must recover signals from amplitude or intensity measurements, while noise-robust prior-based methods have been limited in supported measurement models. The paper combines RED with the DnCNN denoiser to create prDeep, which is reported to be robust to noise and flexible across measurement systems. Its main stated limitation is that it currently supports amplitude-only targets.
Problem
Existing phase-retrieval algorithms struggle with noise, while robust prior-based methods can be restricted to Gaussian or coded diffraction measurements.
Method
prDeep adapts Regularization by Denoising and incorporates the DnCNN convolutional neural network within a phase-retrieval optimization framework.
Results
prDeep is reported to be exceptionally robust to noise, handle a wide range of measurement models, and provide the best reconstructions in the reported comparison with Plug-and-Play ADMM.
Takeaways & Limitations
By integrating optimization with a learned prior, prDeep combines measurement-model flexibility with the use of powerful denoisers for phase retrieval.
Takeaways & Limitations
prDeep is presently restricted to amplitude-only targets and does not yet handle complex-valued targets.
Abstract
from arXiv · showhide
Phase retrieval algorithms have become an important component in many modern computational imaging systems. For instance, in the context of ptychography and speckle correlation imaging, they enable imaging past the diffraction limit and through scattering media, respectively. Unfortunately, traditional phase retrieval algorithms struggle in the presence of noise. Progress has been made recently on more robust algorithms using signal priors, but at the expense of limiting the range of supported measurement models (e.g., to Gaussian or coded diffraction patterns). In this work we leverage the regularization-by-denoising framework and a convolutional neural network denoiser to create prDeep, a new phase retrieval algorithm that is both robust and broadly applicable. We test and validate prDeep in simulation to demonstrate that it is robust to noise and can handle a variety of system models. A MatConvNet implementation of prDeep is available at https://github.com/ricedsp/prDeep.
1. Introduction
Phase retrieval recovers signals from amplitude or intensity measurements, but noise-robust prior-based methods have faced measurement-model restrictions and high computational cost. prDeep combines flexible RED optimization with DnCNN denoising to improve robustness while supporting varied systems.
- Motivation: Phase retrieval recovers a vectorized signal from only the amplitude or intensity of a linear system’s output.The measurement model includes a forward operator and noise.
- Limitations of prior methods: Natural-image priors improve noise robustness, but existing methods can be restricted to Gaussian or coded diffraction measurements and remain computationally demanding.SPAR and BM3D-prGAMP have the stated measurement restrictions, while all three reviewed methods are computationally demanding.
- Contributions: prRED adapts Regularization by Denoising into a general optimization problem that supports diverse measurements, including Fourier measurements.Its flexibility comes from using a general optimization formulation rather than a specific algorithm.
- Contributions: prDeep combines prRED with the DnCNN convolutional neural network denoiser.This integrates a learned denoising prior into the RED-based phase retrieval framework.
- Evaluation: Simulation results report that prDeep offers excellent performance with reasonable run times and compares favorably with existing algorithms in computation time and noise robustness.The evaluation applies simulated data to assess these properties.
2. Related Work
Prior work uses denoisers and learned priors to improve inverse-problem recovery, but phase-retrieval methods have often been tied to specific measurement models or applications. This work instead uses a neural network as a regularizer within an optimization framework, preserving broader applicability.
- Plug-and-Play Regularization for Linear Inverse Problems: Plug-and-play regularization uses advanced denoisers as implicit priors for linear inverse problems such as deblurring, superresolution, and compressed sensing.The denoiser is treated as a regularizer associated with an implicit cost function.
- Plug-and-Play Regularization for Linear Inverse Problems: Advanced denoisers such as BM3D have produced state-of-the-art recovery accuracy in the linear inverse problems where they have been applied.The stated explanation is that these priors model the distribution of natural images accurately.
- Plug-and-Play Regularization for PR: Earlier plug-and-play phase-retrieval methods improved robustness to noise but were limited by specific measurement assumptions, including coded diffraction or nearly i.i.d. Gaussian operators.ADMM-based approaches support generic measurement matrices, including Fourier measurements.
- Deep Learning for PR: Deep-learning phase-retrieval methods had been designed for particular applications such as ptychography or holography.The cited networks were tailored to combine filtered images or remove holographic twin-image components.
- This Work: The paper uses a neural network as a regularizer inside an optimization framework rather than designing it for one specific phase-retrieval problem.This approach is intended to make the network applicable to numerous phase-retrieval problems.
3. PR via Regularization by Denoising
This section adapts RED to phase retrieval, using an explicit denoiser-based regularizer with FASTA and DnCNN to support flexible measurements and robust denoising.
- prRED: prRED adapts the RED framework to phase retrieval by formulating an optimization problem that can handle varied measurements, including Fourier measurements.Unlike methods tied to a specific algorithm, prRED is designed around a general optimization problem.
- RED: RED uses a denoiser to construct an explicit regularizer rather than minimizing an implicit cost function associated with the denoiser.The regularizer penalizes differences between an image and its denoised version, as well as correlations between the image and residual.
- Optimization: The RED proximal mapping is implemented recursively under homogeneity and passivity conditions, with experiments indicating that one denoiser call often performs well.The recursion is an approximation because these conditions do not hold exactly in practice.
- PR formulation: The RED regularizer can be applied to phase retrieval with a data-fidelity term encouraging Ax to match phaseless measurements y.The paper adopts an amplitude loss, while noting that the tested Poisson log-likelihood performed slightly worse.
- Optimization: FASTA solves the resulting problem using forward-backward splitting, adaptive step sizes, a proximal mapping for regularization, and a data-fidelity subgradient.In a typical trajectory, the cost decreases monotonically and converges after about 200 iterations.
- prDeep: prDeep is the special case of prRED that uses the DnCNN denoiser, which is trained with residual learning and outperforms BM3D while running hundreds of times faster with parallelization and GPUs.The training used four noise levels and 300,000 overlapping patches from 400 Berkeley Segmentation Dataset images.
4. Experimental Results
The experiments evaluate prDeep and competing phase-retrieval methods on noisy coded-diffraction and Fourier measurements using natural and unnatural test images. prDeep performs especially well for noisy Fourier measurements, while plug-and-play methods are similar for coded-diffraction measurements.
- Experimental setup: The experiments test prDeep and competing algorithms on simulated coded-diffraction and Fourier measurements with Poisson noise.Both image sets contain six natural and six unnatural real, nonnegative test images.
- Coded-diffraction measurements: Four intensity-only coded-diffraction measurements use a phase-only spatial light modulator followed by far-field Fourier diffraction.The measurement operator applies distinct random phase patterns before recording intensity measurements.
- Coded-diffraction measurements: At low SNRs, all five plug-and-play methods produce similar reconstructions from coded-diffraction measurements.Figure 5 likewise reports good reconstructions from noisy coded-diffraction measurements at α = 81.
- Fourier measurements: With Fourier measurements and large amounts of noise, prDeep is superior to existing phase-retrieval algorithms.The Fourier reconstructions are less accurate than their coded-diffraction counterparts at a given noise level.
- Fourier measurements: For noisy Fourier measurements, prDeep produces fewer artifacts than competing methods in visual reconstructions.The experiments use 4× oversampled intensity-only Fourier measurements and assume the image support is known.
5. Conclusions and Future Work
prDeep extends RED with a DnCNN denoiser for phase retrieval, producing robust reconstructions across measurement models including Fourier measurements. Its major current limitation is support only for amplitude-only targets, leaving complex-valued targets for future work.
- Contributions: prDeep combines RED with the DnCNN neural-network denoiser for phase retrieval.The method integrates a learned image prior into the RED framework.
- Experimental result: In Fourier-measurement reconstructions, prDeep produces fewer artifacts than competing methods.The comparison uses noisy, 4× oversampled intensity-only Fourier measurements.
- Experimental result: Plug and Play ADMM and prDeep provide the best reconstructions among the compared algorithms.The reported comparison accounts for translation and reflection ambiguities associated with Fourier measurements.
- Measurement scope: prDeep handles measurement models ranging from intensity-only coded diffraction patterns to Fourier measurements.This flexibility is enabled by the general optimization formulation used by prRED.
- Future work: prDeep is presently restricted to amplitude-only targets and does not yet handle complex-valued targets.Extending the method to complex-valued targets is identified as an important direction for future research.