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Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview

Alexander Auras, Kanchana Vaishnavi Gandikota, Hannah Droege, Michael Moeller

TL;DR

The paper analyzes inverse problems in imaging from both variational and learning-based perspectives, focusing on robustness of point estimators to adversarial perturbations and the exploration of the data-consistent solution subspace. It distinguishes direct DL point estimators from learned priors, unrolled methods, and Bayesian approaches that sample or align with posterior distributions, and it surveys generative priors and diffusion-based strategies for flexible reconstructions. Through a 1D toy-problem study, it contrasts classical stability notions (data fidelity plus symmetric Bregman distance) with empirical neural-network robustness, and it discusses defenses (e.g., adversarial training) and the impact of distribution shifts and forward-model changes. Finally, it highlights explorability mechanisms for guiding reconstructions toward semantically meaningful or texture-controlled outcomes, including diffusion-guided and text-guided approaches, underscoring the potential of hybrid methods to deliver robust and controllable imaging solutions.

Abstract

This paper provides an overview of current approaches for solving inverse problems in imaging using variational methods and machine learning. A special focus lies on point estimators and their robustness against adversarial perturbations. In this context results of numerical experiments for a one-dimensional toy problem are provided, showing the robustness of different approaches and empirically verifying theoretical guarantees. Another focus of this review is the exploration of the subspace of data-consistent solutions through explicit guidance to satisfy specific semantic or textural properties.

Robustness and Exploration of Variational and Machine Learning Approaches to Inverse Problems: An Overview

TL;DR

The paper analyzes inverse problems in imaging from both variational and learning-based perspectives, focusing on robustness of point estimators to adversarial perturbations and the exploration of the data-consistent solution subspace. It distinguishes direct DL point estimators from learned priors, unrolled methods, and Bayesian approaches that sample or align with posterior distributions, and it surveys generative priors and diffusion-based strategies for flexible reconstructions. Through a 1D toy-problem study, it contrasts classical stability notions (data fidelity plus symmetric Bregman distance) with empirical neural-network robustness, and it discusses defenses (e.g., adversarial training) and the impact of distribution shifts and forward-model changes. Finally, it highlights explorability mechanisms for guiding reconstructions toward semantically meaningful or texture-controlled outcomes, including diffusion-guided and text-guided approaches, underscoring the potential of hybrid methods to deliver robust and controllable imaging solutions.

Abstract

This paper provides an overview of current approaches for solving inverse problems in imaging using variational methods and machine learning. A special focus lies on point estimators and their robustness against adversarial perturbations. In this context results of numerical experiments for a one-dimensional toy problem are provided, showing the robustness of different approaches and empirically verifying theoretical guarantees. Another focus of this review is the exploration of the subspace of data-consistent solutions through explicit guidance to satisfy specific semantic or textural properties.
Paper Structure (20 sections, 24 equations, 9 figures, 1 table)

This paper contains 20 sections, 24 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of Deep Learning for Inverse Problems in Imaging
  3. Deep Learning for Point Estimates to Inverse Problems
  4. Direct Approaches
  5. Neural Network Priors
  6. Deep Learning for Bayesian Approaches to Inverse Problems
  7. Stability and Robustness
  8. Adversarial Robustness
  9. Different Notions of Robustness - Numerical Experiments on a Toy Problem.
  10. Targeted changes
  11. Defense
  12. Robustness to distribution shifts
  13. Robustness to changes in forward measurement operator
  14. Robustness in recovering fine details
  15. Robustness for Bayesian Methods
  16. ...and 5 more sections

Figures (9)

  • Figure 1: The approach \ref{['eq:ganprior']} corresponds to a posterior that is the product of the likelihood $p(f|u)$ and a prior $p(u)$ that restricts $u$ to the range of a generator $\mathcal{G}_\theta(\hat{z})$. The left plot illustrates level lines of $-\log(p(f|u))$ in 2d along with a lower dimensional manifold that is the range of $\mathcal{G}_\theta(\hat{z})$ as a dashed red line. The resulting costs are shown on the right. Running gradient descent from different starting points can merely sample local minima (dashed green lines) that correspond to local maxima of the posterior.
  • Figure 2: Convex regularizers rate convex combinations (a) of images (b) as at least as "natural" as one of the images itself.
  • Figure 3: The measurement (and its adversarial version) we attempt to reconstruct in our experiments. As is obvious, the difference between both is negligible.
  • Figure 4: Results of the total-variation-based reconstruction of 1D-signals in a compressed sensing setting.
  • Figure 5: Results of the empirical evaluation of the bound in equation \ref{['eq:stability_tv']} for different variational reconstruction methods using Gaussian noise. Violations are artifacts of limited precision calculations.
  • ...and 4 more figures