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Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties

Clemens Arndt, Judith Nickel

TL;DR

This work extends the theory of invertible residual networks (iResNets) for solving ill-posed inverse imaging problems by scaling to real-world linear and nonlinear forward operators (linear Gaussian blur and nonlinear Perona–Malik diffusion). It formalizes an invertible reconstruction scheme where the forward operator is learned with iResNets and the inverse is used for reconstruction, accompanied by a convergence guarantee under a Lipschitz schedule $L(\delta)\to1$ and $\delta/(1-L(\delta))\to0$, yielding a convergent regularization in practice. Empirically, iResNets achieve competitive PSNR/SSIM with state-of-the-art methods like ConvResNet, U-Net, and DiffNet, at the cost of longer training times due to inverse computations and Lipschitz constraints. Beyond performance, the paper highlights robustness to adversarial perturbations, test-time noise variations, and small training datasets, and leverages invertibility to interpret the learned forward operator and its regularization, thereby enhancing interpretability. The authors also discuss extensions to averaged-operator Plug-and-Play formulations and outline avenues to reduce training time and broaden applicability to additional forward models. Overall, the results provide promising evidence that fully learned, provably regularizing iResNets can match high-performing reconstructions while offering stability and interpretability advantages in real-world inverse imaging tasks.

Abstract

Learning-based methods have demonstrated remarkable performance in solving inverse problems, particularly in image reconstruction tasks. Despite their success, these approaches often lack theoretical guarantees, which are crucial in sensitive applications such as medical imaging. Recent works by Arndt et al addressed this gap by analyzing a data-driven reconstruction method based on invertible residual networks (iResNets). They revealed that, under reasonable assumptions, this approach constitutes a convergent regularization scheme. However, the performance of the reconstruction method was only validated on academic toy problems and small-scale iResNet architectures. In this work, we address this gap by evaluating the performance of iResNets on two real-world imaging tasks: a linear blurring operator and a nonlinear diffusion operator. To do so, we compare the performance of iResNets against state-of-the-art neural networks, revealing their competitiveness at the expense of longer training times. Moreover, we numerically demonstrate the advantages of the iResNet's inherent stability and invertibility by showcasing increased robustness across various scenarios as well as interpretability of the learned operator, thereby reducing the black-box nature of the reconstruction scheme.

Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties

TL;DR

This work extends the theory of invertible residual networks (iResNets) for solving ill-posed inverse imaging problems by scaling to real-world linear and nonlinear forward operators (linear Gaussian blur and nonlinear Perona–Malik diffusion). It formalizes an invertible reconstruction scheme where the forward operator is learned with iResNets and the inverse is used for reconstruction, accompanied by a convergence guarantee under a Lipschitz schedule and , yielding a convergent regularization in practice. Empirically, iResNets achieve competitive PSNR/SSIM with state-of-the-art methods like ConvResNet, U-Net, and DiffNet, at the cost of longer training times due to inverse computations and Lipschitz constraints. Beyond performance, the paper highlights robustness to adversarial perturbations, test-time noise variations, and small training datasets, and leverages invertibility to interpret the learned forward operator and its regularization, thereby enhancing interpretability. The authors also discuss extensions to averaged-operator Plug-and-Play formulations and outline avenues to reduce training time and broaden applicability to additional forward models. Overall, the results provide promising evidence that fully learned, provably regularizing iResNets can match high-performing reconstructions while offering stability and interpretability advantages in real-world inverse imaging tasks.

Abstract

Learning-based methods have demonstrated remarkable performance in solving inverse problems, particularly in image reconstruction tasks. Despite their success, these approaches often lack theoretical guarantees, which are crucial in sensitive applications such as medical imaging. Recent works by Arndt et al addressed this gap by analyzing a data-driven reconstruction method based on invertible residual networks (iResNets). They revealed that, under reasonable assumptions, this approach constitutes a convergent regularization scheme. However, the performance of the reconstruction method was only validated on academic toy problems and small-scale iResNet architectures. In this work, we address this gap by evaluating the performance of iResNets on two real-world imaging tasks: a linear blurring operator and a nonlinear diffusion operator. To do so, we compare the performance of iResNets against state-of-the-art neural networks, revealing their competitiveness at the expense of longer training times. Moreover, we numerically demonstrate the advantages of the iResNet's inherent stability and invertibility by showcasing increased robustness across various scenarios as well as interpretability of the learned operator, thereby reducing the black-box nature of the reconstruction scheme.
Paper Structure (20 sections, 3 theorems, 32 equations, 17 figures, 5 tables)

This paper contains 20 sections, 3 theorems, 32 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. iResNets for inverse problems
  3. Appoximating diffusion processes with iResNets
  4. Numerical Results
  5. Architecture
  6. Performance and comparison to other models
  7. Why stability matters
  8. Adversarial examples
  9. Test-time noise level variations
  10. Small training dataset
  11. Why invertibility matters
  12. Investigation of the learned forward operator by linearization
  13. Investigation of regularization properties
  14. Discussion and Outlook
  15. Transforming the inverse of iResNets into averaged operators
  16. ...and 5 more sections

Key Result

Lemma 2.1

For $x^\dagger\in X$, let $z^\delta \in X$ satisfy $\|z^\delta - F(x^\dagger)\| \leq \delta$. Moreover, assume that $x^\dagger$ and the network $\varphi_{\theta(L)}$ with network parameters $\theta(L)$ for $L \in [0,1)$ satisfy If the Lipschitz parameter $L:(0,\infty) \rightarrow [0,1)$ is chosen such that then it holds

Figures (17)

  • Figure 1: Architecture of the iResNets used in our numerical experiments.
  • Figure 1: Comparison of reconstructions for both forward operators with $\delta = 0.025$ on the LoDoPaB-CT dataset. All models were optimized on 64 samples from the STL-10 dataset with $\delta = 0.025$.
  • Figure 1: Reconstruction performance on test data with noise level $\delta_{\text{test}} = c \cdot 0.025$ for the nonlinear diffusion operator and noise amplification factor $c \in [0.7, 1.8]$. All networks and hyperparameters were optimized on training data with $\delta=0.025$.
  • Figure 1: Clustering of the saliency maps of the operator and the network trained with noise level $\delta=0.05$ for different Lipschitz parameters. The first row visualizes the clustering with the ground truth image and the second row with the corresponding edges (weak edges are gray and strong edges are black).
  • Figure 2: Reconstruction performance of iResNets dependent on the Lipschitz parameter $L$ for the nonlinear diffusion operator and different noise levels ($\delta = 0.01, 0.025, 0.05$).
  • ...and 12 more figures

Theorems & Definitions (7)

  • Lemma 2.1: Convergence, cf. iresnet_01_regtheory
  • Remark 2.2
  • Lemma 3.1
  • Proof 1
  • Proof 2
  • Lemma C.1
  • Proof 3