Equivariant plug-and-play image reconstruction (2312.01831v2)

Abstract: Plug-and-play algorithms constitute a popular framework for solving inverse imaging problems that rely on the implicit definition of an image prior via a denoiser. These algorithms can leverage powerful pre-trained denoisers to solve a wide range of imaging tasks, circumventing the necessity to train models on a per-task basis. Unfortunately, plug-and-play methods often show unstable behaviors, hampering their promise of versatility and leading to suboptimal quality of reconstructed images. In this work, we show that enforcing equivariance to certain groups of transformations (rotations, reflections, and/or translations) on the denoiser strongly improves the stability of the algorithm as well as its reconstruction quality. We provide a theoretical analysis that illustrates the role of equivariance on better performance and stability. We present a simple algorithm that enforces equivariance on any existing denoiser by simply applying a random transformation to the input of the denoiser and the inverse transformation to the output at each iteration of the algorithm. Experiments on multiple imaging modalities and denoising networks show that the equivariant plug-and-play algorithm improves both the reconstruction performance and the stability compared to their non-equivariant counterparts.

• The paper introduces an equivariant framework that improves the stability and convergence of plug-and-play algorithms using invariant denoisers.
• It presents theoretical insights on symmetrizing the Jacobian matrix and reducing the Lipschitz constant to bolster reconstruction robustness.
• Experimental results across tasks like deblurring, super-resolution, and MRI demonstrate superior performance over non-equivariant methods.

Equivariant Plug-and-Play Image Reconstruction: An Analytical Perspective

The paper "Equivariant Plug-and-Play Image Reconstruction" presents an insightful exploration into the stability and performance enhancements achievable through the integration of equivariance in plug-and-play (PnP) algorithms for image reconstruction. This work provides a detailed theoretical and experimental analysis, focusing on how enforcing equivariance to group transformations enhances both algorithmic stability and image quality.

The crux of PnP algorithms lies in their ability to leverage powerful pre-trained denoisers to solve inverse imaging problems, bypassing the need to design dedicated models for each task. Nevertheless, these denoising-based approaches have historically suffered from issues of instability, often resulting in suboptimal reconstructions. This research posits that integrating equivariance—that is, ensuring the denoiser is invariant to certain transformations like rotations and translations—can significantly mitigate these issues.

Theoretical Insights

The paper builds upon the premise that many imaging tasks contain inherent equivariances which can be capitalized upon. By employing operations invariant to these transformations, the authors propose that the stability and convergence of PnP algorthims can be enhanced. A key contribution of the work is the illustration of how this equivariant approach can symmetrize the Jacobian matrix of a denoiser, which is crucial for the existence of a well-behaved implicit regularizer or prior.

Furthermore, the authors demonstrate that the Lipschitz constant of the denoiser—a significant factor in ensuring algorithm convergence—can be made smaller or equal when averaged over the set of transformation, which is a pivotal property for enhancing the stability of the algorithms.

Experimental Validation

The empirical evaluation covers multiple inverse imaging tasks such as deblurring and super-resolution, alongside magnetic resonance imaging (MRI). Across these tasks, the equivariant versions of the PnP and RED algorithms consistently outperform their non-equivariant counterparts in terms of stability and reconstruction quality. For instance, the reduction in the required tuning and mitigation of unwanted artifacts in the reconstructions underscores the practical utility of the approach.

The experiments also offer insight into how the proposed method helps in sampling problems addressed through Langevin dynamics. The equivariant plug-and-play framework demonstrated superior performance in producing high-quality samples compared to non-equivariant baselines, attesting to the robustness introduced by the equivariant framework.

Broader Implications and Future Directions

The implications of this work are twofold. Practically, this method provides a way to meaningfully incorporate denoisers that were traditionally hampered by stability issues into a consistent framework amenable to a wide array of imaging modalities. Theoretically, it opens avenues for further exploration into other forms of transformation-invariance that might be integrated with denoising processes. Future work may likely explore adaptive ways to tune the level of equivariance dynamically, tailoring the approach to the specific topology of the problem space.

In summary, the integration of equivariance into PnP reconstruction frameworks emerges as a compelling direction for research that harmonizes the power of state-of-the-art denoisers with the organizational principles of imaging tasks. This paper elucidates a pathway toward more stable and robust plug-and-play methods in image reconstruction, promising practical advances in diverse applications spanning medical imaging to astrophysical image processing.