Adversarial Regularizers in Inverse Problems (1805.11572v2)

Abstract: Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying data-driven approaches to inverse problems, using a neural network as a regularization functional. The network learns to discriminate between the distribution of ground truth images and the distribution of unregularized reconstructions. Once trained, the network is applied to the inverse problem by solving the corresponding variational problem. Unlike other data-based approaches for inverse problems, the algorithm can be applied even if only unsupervised training data is available. Experiments demonstrate the potential of the framework for denoising on the BSDS dataset and for computed tomography reconstruction on the LIDC dataset.

Adversarial Regularizers in Inverse Problems: An Overview

The paper "Adversarial Regularizers in Inverse Problems" presents a novel framework for addressing inverse problems in areas such as medical imaging and computer vision. Traditionally, these problems have been tackled using variational regularization models, which rely on combining forward operator knowledge with hand-crafted regularization functionals that encode prior information about the solution. The authors propose using neural networks as a flexible and data-driven alternative to these handcrafted functionals, allowing for potentially more adaptive and accurate reconstructions.

Motivations and Methodology

The key motivation behind this research is the potential of neural networks to excel in tasks where traditionally crafted models fall short, particularly in high-dimensional and complex inverse problem settings. The main contributions of the paper can be summarized as follows:

1. Learning Regularization Functionals: A neural network is employed as a regularization functional, learning to distinguish between the distributions of true images and their noisy reconstructions. This is conceptually achieved by employing aspects of adversarial learning, inspired by GANs (Generative Adversarial Networks).
2. Algorithm for High-dimensional Parameter Spaces: The authors develop a training algorithm suitable for high-dimensional data. This leverages the Wasserstein GAN framework to train the network to function as a "critic," discerning between ground truth data and noisy approximations.
3. Unsupervised Learning Potential: Unlike conventional data-based inversion techniques requiring supervised training examples (pairs of measurements and ground truths), this approach can be trained with unsupervised data. This reduces the dependency on scarce labeled datasets, particularly beneficial in medical imaging contexts.

Theoretical and Practical Implications

Incorporating neural networks in the regularization process theoretically enriches the solution space by embedding data priors implicitly learned from distributions. The paper discusses that, under a weak data manifold assumption, the learned regularization function aligns with desirable characteristics such as being Lipschitz continuous. This implies that the method enhances the probability of recovering noise-free images closely resembling their respective ground truths.

Moreover, the paper dives into a distributional analysis, proving beneficial convergence properties in Wasserstein distance norms, indicating the method's capability to shrink the statistical gap between reconstructed and true distributions.

Experimental Results

The framework is tested on denoising tasks using the BSDS dataset and CT reconstruction tasks using the LIDC/IDRI dataset. In these experiments, adversarial regularizers demonstrated superior performance in terms of standard image quality metrics like PSNR and SSIM compared to model-based techniques like total variation, even matching closely with supervised learning-based methods.

For instance, in CT reconstructions with high noise, adversarial regularizers achieved a PSNR of 30.5 dB and an SSIM of 0.927, compared to post-processing techniques which slightly exceed this. However, it remains impressive that such results are obtained with unsupervised training alone.

Future Prospects and Conclusion

The development of adversarial regularizers for inverse problems paves the way for enhanced, more adaptable inverse problem-solving methods. This could be especially impactful where ground truth data is difficult to acquire or where the complexity of the forward operator precludes straightforward application of traditional model-based solutions.

Potential future work could explore different neural network architectures or hybrid approaches that integrate both data-driven and model-based insights. Additionally, further work might investigate broader classes of inverse problems and real-world applications, enhancing robustness and understanding fundamental limitations.

In conclusion, this framework represents a promising stride forward in the domain of inverse problems, offering a flexible solution strategy that leverages the power of deep learning while maintaining the rigor of variational problem formulations.