[Paper Review] Phase Retrieval using Expectation Consistent Signal Recovery Algorithm based on Hypernetwork

Chang-Jen Wang, Chao-Kai Wen+4 more|arXiv (Cornell University)|Jan 12, 2021

Advanced X-ray Imaging TechniquesPhysics and Astronomy73 references11 citations

TL;DR

This paper proposes a dynamic, adaptive deep unfolding framework for phase retrieval using a hypernetwork to generate damping factors in the generalized expectation consistent signal recovery (GEC-SR) algorithm. By integrating a recurrent hypernetwork with self-attention, the method achieves superior convergence speed, robustness, and adaptability across varying measurement conditions and iteration counts, outperforming existing state-of-the-art methods, especially under harsh or mismatched settings.

ABSTRACT

Phase retrieval (PR) is an important component in modern computational imaging systems. Many algorithms have been developed over the past half-century. Recent advances in deep learning have introduced new possibilities for a robust and fast PR. An emerging technique called deep unfolding provides a systematic connection between conventional model-based iterative algorithms and modern data-based deep learning. Unfolded algorithms, which are powered by data learning, have shown remarkable performance and convergence speed improvement over original algorithms. Despite their potential, most existing unfolded algorithms are strictly confined to a fixed number of iterations when layer-dependent parameters are used. In this study, we develop a novel framework for deep unfolding to overcome existing limitations. Our development is based on an unfolded generalized expectation consistent signal recovery (GEC-SR) algorithm, wherein damping factors are left for data-driven learning. In particular, we introduce a hypernetwork to generate the damping factors for GEC-SR. Instead of learning a set of optimal damping factors directly, the hypernetwork learns how to generate the optimal damping factors according to the clinical settings, thereby ensuring its adaptivity to different scenarios. To enable the hypernetwork to adapt to varying layer numbers, we use a recurrent architecture to develop a dynamic hypernetwork that generates a damping factor that can vary online across layers. We also exploit a self-attention mechanism to enhance the robustness of the hypernetwork. Extensive experiments show that the proposed algorithm outperforms existing ones in terms of convergence speed and accuracy and still works well under very harsh settings, even under which many classical PR algorithms are unstable.

Motivation & Objective

• To address the limitations of fixed-iteration, non-adaptive unfolded algorithms in phase retrieval.
• To enable learned parameters to generalize across varying forward models, noise levels, and measurement sizes without retraining.
• To develop a flexible architecture that supports variable iteration counts while maintaining performance.
• To improve robustness under mismatched distributions, such as binary measurement matrices or low SNR.
• To enhance the adaptability and stability of the GEC-SR algorithm through data-driven damping factor generation.

Proposed method

• A hypernetwork is introduced to generate damping factors for the GEC-SR algorithm, replacing direct learning of fixed parameters.
• A recurrent architecture (GEC-SR-HyperGRU) enables dynamic generation of damping factors that vary online across layers, supporting variable iteration counts.
• Self-attention mechanisms are incorporated into the hypernetwork to improve feature representation and robustness across diverse input scenarios.
• The framework retains all original mathematical functions of GEC-SR but replaces only the damping factors with learnable parameters generated by the hypernetwork.
• The hypernetwork is trained end-to-end using data from various forward models, enabling generalization to unseen settings.
• The method is evaluated on both synthetic and real image reconstructions under diverse conditions, including Gaussian and binary measurement matrices.

Experimental results

Research questions

• RQ1 Can a hypernetwork be used to generate damping factors in an unfolded GEC-SR algorithm to improve adaptability across different imaging scenarios?
• RQ2 How does a recurrent hypernetwork architecture enable dynamic damping factor generation across variable iteration counts?
• RQ3 To what extent does self-attention in the hypernetwork enhance robustness under distribution shifts?
• RQ4 How does the proposed method compare to existing state-of-the-art phase retrieval algorithms in terms of convergence speed and accuracy?
• RQ5 Can the framework maintain performance without retraining when applied to mismatched or harsh conditions?

Key findings

• GEC-SR-HyperGRU with self-attention achieved −21.07 dB reconstruction MSE on a 50×50 smiley face image under SNR=15 dB, significantly outperforming GEC-SR-Net (−13.59 dB).
• Under binary measurement matrices, GEC-SR-HyperGRU maintained stable convergence and achieved −15 dB MSE at M/N = 2.5 with only 10 iterations, while GEC-SR-Net converged to only −9 dB under the same conditions.
• For sparse signals (ρ=0.2), GEC-SR-HyperGRU outperformed GEC-SR-HyperNet, demonstrating the advantage of dynamic, state-aware damping factor adjustment.
• In low-SNR and high-sparsity regimes, GEC-SR-HyperGRU showed superior robustness and maintained high performance without retraining, even under mismatched distributions.
• The method achieved faster convergence and better stability than prVAMP and GEC-SR-Net, especially in challenging settings like binary transform matrices and low measurement ratios.
• The hypernetwork-based approach generalized well across diverse measurement ratios (R=M/N), with GEC-SR-HyperGRU achieving the best MSE performance across all tested R values.