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

Published 19 Feb 2024 in eess.IV, cs.CV, cs.LG, cs.NA, and math.NA | (2402.12072v2)

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.

Summary

  • The paper presents a systematic comparison of deterministic and stochastic deep learning methods, highlighting their roles in achieving robustness in imaging reconstructions.
  • The paper demonstrates that variational approaches provide inherent stability while deep learning models need specialized architectures to handle adversarial and distribution shifts.
  • The paper emphasizes guided explorability through generative models to produce diverse, semantically meaningful reconstructions in under-determined imaging scenarios.

Exploring Robustness and Explorability in AI-driven Imaging Reconstruction

Overview of Deep Learning in Imaging Inverse Problems

Deep Learning (DL) techniques have significantly advanced the field of inverse problems in imaging by offering models that outperform traditional algorithms on tasks like super-resolution, denoising, and tomographic reconstruction. These models primarily fall into two categories: deterministic models that provide point estimates similar to the Maximum A Posteriori (MAP) estimates, and stochastic models that allow sampling from the posterior distribution of the solution space, catering especially to under-determined problems.

Deterministic Approaches

Deterministic methods encompass direct prediction models and network-prior-based approaches. Direct models are trained end-to-end to map noisy or incomplete observations to high-quality reconstructions. Conversely, network priors learn a distribution of high-quality images and leverage this knowledge in a variational framework to find the best matching reconstruction for given observations. These methods notably include architectures for direct inversion, learned post-processors, and unrolled optimization approaches, each offering a blend of model-inherent regularization and data-driven flexibility.

Stochastic Sampling and Neural Network Priors

The stochastic approaches provide a mechanism to explore the subspace of solutions that are consistent with the observed data. Techniques like conditional generative models allow for sampling multiple reconstructions, providing insights into the uncertainty and variability of solutions. On the other hand, using neural networks as priors within variational models enables a more flexible and powerful means to incorporate image priors, significantly enhancing the quality of reconstructive solutions.

Stability and Robustness Concerns

Despite their success, DL-based inverse problem solutions have faced criticism for their potential vulnerabilities, especially in medical imaging and other critical applications where errors or instabilities can have severe consequences. The paper focuses on the robustness of such models against adversarial attacks, distribution shifts, and changes in measurement models. These concerns entail the need for models to not only provide accurate reconstructions under ideal conditions but also maintain their performance in the presence of real-world imperfections and deliberate perturbations.

Numerical Experiments

Using a one-dimensional toy problem, the study demonstrates the varying degrees of robustness across classical variational methods and modern deep learning approaches. The variational methods exhibit inherent stability properties governed by the condition number of the forward operator, whereas DL models require careful consideration of their architecture and training process to achieve similar robustness.

Explorability

The capacity to explore the space of plausible reconstructions—especially in under-determined situations where multiple solutions fit the observed data—emerges as a vital attribute of next-generation inverse problem solvers. The paper discusses methods for guided exploration, such as using text descriptions to steer the generation process of super-resolution images, highlighting the potential of deep generative models to synthesize diverse solutions that meet specific semantic criteria.

Future Directions and Conclusion

The interplay between robustness and explorability in DL-based inverse problem solutions underscores the necessity for ongoing research to enhance the stability, interpretability, and flexibility of these models. Future developments must aim towards models that not only excel in reconstruction performance but also offer guarantees against instabilities and facilitate user-guided exploration of the solution space. This balance will be crucial in expanding the applicability of DL techniques to a wider range of real-world and critical applications in imaging and beyond.

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