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HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training

Markus Haltmeier, Lukas Neumann, Nadja Gruber, Johannes Schwab, Gyeongha Hwang

TL;DR

HyDRA tackles ill-posed inverse problems when ground-truth data are unavailable by introducing a measurement-only training framework for Deep Equilibrium (DEQ) models. It hybridizes a data-consistency term with an adaptive, trainable denoising regularizer within a fixed-point architecture, and adds a data-driven early stopping criterion to prevent overfitting. The approach is validated on sparse-view CT (LoDoPaB) and achieves competitive PSNR/SSIM with notably faster inference than iterative baselines, while avoiding ground-truth supervision. Theoretical guarantees of existence and uniqueness of the fixed point are provided via contraction properties, and the method shows promise for broader adoption in MRI and computational microscopy due to its efficiency and ground-truth-free training paradigm.

Abstract

Solving image reconstruction problems of the form \(\mathbf{A} \mathbf{x} = \mathbf{y}\) remains challenging due to ill-posedness and the lack of large-scale supervised datasets. Deep Equilibrium (DEQ) models have been used successfully but typically require supervised pairs \((\mathbf{x},\mathbf{y})\). In many practical settings, only measurements \(\mathbf{y}\) are available. We introduce HyDRA (Hybrid Denoising Regularization Adaptation), a measurement-only framework for DEQ training that combines measurement consistency with an adaptive denoising regularization term, together with a data-driven early stopping criterion. Experiments on sparse-view CT demonstrate competitive reconstruction quality and fast inference.

HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training

TL;DR

HyDRA tackles ill-posed inverse problems when ground-truth data are unavailable by introducing a measurement-only training framework for Deep Equilibrium (DEQ) models. It hybridizes a data-consistency term with an adaptive, trainable denoising regularizer within a fixed-point architecture, and adds a data-driven early stopping criterion to prevent overfitting. The approach is validated on sparse-view CT (LoDoPaB) and achieves competitive PSNR/SSIM with notably faster inference than iterative baselines, while avoiding ground-truth supervision. Theoretical guarantees of existence and uniqueness of the fixed point are provided via contraction properties, and the method shows promise for broader adoption in MRI and computational microscopy due to its efficiency and ground-truth-free training paradigm.

Abstract

Solving image reconstruction problems of the form remains challenging due to ill-posedness and the lack of large-scale supervised datasets. Deep Equilibrium (DEQ) models have been used successfully but typically require supervised pairs \((\mathbf{x},\mathbf{y})\). In many practical settings, only measurements are available. We introduce HyDRA (Hybrid Denoising Regularization Adaptation), a measurement-only framework for DEQ training that combines measurement consistency with an adaptive denoising regularization term, together with a data-driven early stopping criterion. Experiments on sparse-view CT demonstrate competitive reconstruction quality and fast inference.
Paper Structure (15 sections, 1 theorem, 6 equations, 3 figures, 1 table)

This paper contains 15 sections, 1 theorem, 6 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Single-shot reconstruction
  4. Supervised reconstruction
  5. Unsupervised reconstruction (measurements-only)
  6. HyDRA Framework
  7. HyDRA architecture
  8. HyDRA loss
  9. Automated Stopping
  10. Numerical experiments
  11. Data set
  12. Compared methods
  13. Implementation Details
  14. Results
  15. Conclusion

Key Result

Proposition 3.2

For any $\mathbf{y} \in \mathbb{R}^m$ and $\theta \in \Theta$, the map $\mathcal{N}_\theta \circ \mathcal{G}(\cdot,\mathbf{y}) \colon \mathbb{R}^n \to \mathbb{R}^n$ is a contraction and thus has exactly one fixed point. In particular, $\mathcal{B}_\theta$ is a well-defined implicit network.

Figures (3)

  • Figure 1: Example reconstructions for 16 views.
  • Figure 2: Example reconstructions for 32 views.
  • Figure 3: Example reconstructions for 64 views.

Theorems & Definitions (3)

  • Definition 3.1: HyDRA architecture
  • Proposition 3.2: Existence and uniqueness
  • proof