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Learning Regularization Functionals for Inverse Problems: A Comparative Study

Johannes Hertrich, Hok Shing Wong, Alexander Denker, Stanislas Ducotterd, Zhenghan Fang, Markus Haltmeier, Željko Kereta, Erich Kobler, Oscar Leong, Mohammad Sadegh Salehi, Carola-Bibiane Schönlieb, Johannes Schwab, Zakhar Shumaylov, Jeremias Sulam, German Shâma Wache, Martin Zach, Yasi Zhang, Matthias J. Ehrhardt, Sebastian Neumayer

TL;DR

This study systematically compares a wide range of learned regularization functionals for inverse problems in imaging under a unified framework. By unifying code and evaluation, it benchmarks eight architectures (e.g., FoE, CNN-based, ICNN/IDCNN, patch-based, TDV, LSR, LPNs) across three training paradigms (bilevel, contrastive, distribution matching) and two forward models (denoising and CT). The results show that learned regularizers generally outperform handcrafted ones, with bilevel training delivering the strongest reconstructions for large architectures, albeit at higher computational cost; more efficient training schemes can closely match performance for simpler convex models. Importantly, many learned regularizers generalize to CT without operator-specific data, underscoring their potential as universal priors, while task-specific fine-tuning further improves quality. The work provides a practical blueprint for reproducible evaluation and highlights open questions about generalization, stability, and uncertainty quantification in learned priors for inverse problems.

Abstract

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

Learning Regularization Functionals for Inverse Problems: A Comparative Study

TL;DR

This study systematically compares a wide range of learned regularization functionals for inverse problems in imaging under a unified framework. By unifying code and evaluation, it benchmarks eight architectures (e.g., FoE, CNN-based, ICNN/IDCNN, patch-based, TDV, LSR, LPNs) across three training paradigms (bilevel, contrastive, distribution matching) and two forward models (denoising and CT). The results show that learned regularizers generally outperform handcrafted ones, with bilevel training delivering the strongest reconstructions for large architectures, albeit at higher computational cost; more efficient training schemes can closely match performance for simpler convex models. Importantly, many learned regularizers generalize to CT without operator-specific data, underscoring their potential as universal priors, while task-specific fine-tuning further improves quality. The work provides a practical blueprint for reproducible evaluation and highlights open questions about generalization, stability, and uncertainty quantification in learned priors for inverse problems.

Abstract

In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines.

Paper Structure

This paper contains 39 sections, 49 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Overview of Regularizer Architectures
  3. Overview of Regularizer Architectures
  4. Fields-of-Experts Regularizer
  5. Convolutional Neural Network
  6. Input Convex Neural Network
  7. Input Difference-of-Convex Neural Network
  8. Patch-Based Architectures
  9. Total Deep Variation
  10. Least Squares Residual Regularizer
  11. Learned Proximal Networks
  12. Overview of Training Methods
  13. Bilevel Learning
  14. Implicit Differentiation
  15. Unrolling
  16. ...and 24 more sections

Figures (8)

  • Figure 1: Left: The 64 filter impulse responses of a learned RR. Right: The two discussed potentials $\psi^1$ and their derivative $\varphi^1=(\psi^1)'$.
  • Figure 2: Visualization of $_3^3$. On the highest level, an energy value is assigned to every pixel by applying a . The $\Psi$ (blue) is composed of three U-Net-like macro-blocks (gray). Each macro-blocks consist of five micro-blocks (yellow) with residual connections.
  • Figure 3: Denoising result for Experiment 1 on test image 'castle' from BSD68.
  • Figure 4: Experiment 1: Comparison of training schemes on 'lion' image from BSD68.
  • Figure 5: CT reconstructions for Experiment 2 on test image from LoDoPaB-CT.
  • ...and 3 more figures