Stochastic Deep Restoration Priors for Imaging Inverse Problems (2410.02057v1)

Abstract: Deep neural networks trained as image denoisers are widely used as priors for solving imaging inverse problems. While Gaussian denoising is thought sufficient for learning image priors, we show that priors from deep models pre-trained as more general restoration operators can perform better. We introduce Stochastic deep Restoration Priors (ShaRP), a novel method that leverages an ensemble of such restoration models to regularize inverse problems. ShaRP improves upon methods using Gaussian denoiser priors by better handling structured artifacts and enabling self-supervised training even without fully sampled data. We prove ShaRP minimizes an objective function involving a regularizer derived from the score functions of minimum mean square error (MMSE) restoration operators, and theoretically analyze its convergence. Empirically, ShaRP achieves state-of-the-art performance on tasks such as magnetic resonance imaging reconstruction and single-image super-resolution, surpassing both denoiser-and diffusion-model-based methods without requiring retraining.

• The paper introduces ShaRP, a novel method that leverages an ensemble of pre-trained deep restoration models to regularize imaging inverse problems.
• It presents a rigorous mathematical framework demonstrating convergence stability and superior performance compared to traditional Gaussian denoiser priors.
• Empirical validation on MRI reconstruction and single-image super-resolution shows ShaRP's robust ability to reduce structured artifacts without retraining models.

Stochastic Deep Restoration Priors for Imaging Inverse Problems

The paper "Stochastic Deep Restoration Priors for Imaging Inverse Problems" introduces the concept of Stochastic deep Restoration Priors (ShaRP) to improve the field of imaging inverse problems. The authors propose a novel method leveraging an ensemble of restoration models to enhance the process of regularizing inverse problems, surpassing traditional approaches that utilize Gaussian denoiser priors.

Main Contributions

1. Introduction of ShaRP: ShaRP is designed to employ deep models pre-trained as diverse restoration operators. By sampling from a set of degradation operators, ShaRP can adaptively utilize these as priors, leading to improved handling of structured artifacts in images.
2. Mathematical Framework: The paper provides a theoretical foundation for ShaRP, proving that it minimizes an objective function built on a regularizer derived from the score functions of MMSE restoration operators. The paper explores the convergence analysis of ShaRP iterations, with results showing that ShaRP maintains convergence stability using exact and approximate MMSE operators.
3. Empirical Validation: ShaRP demonstrates state-of-the-art performance in challenging tasks such as magnetic resonance imaging (MRI) reconstruction and single-image super-resolution (SISR), clearly outperforming current denoiser- and diffusion-based methods without the need for retraining the models.

Numerical Results & Theoretical Analysis

• Numerical Results:

The empirical results, particularly on tasks involving MRI reconstruction with varied undersampling patterns and single-image super-resolution, show that using ShaRP leads to enhanced restoration performances. ShaRP proficiently adapts pre-trained restoration models, providing a notable improvement over existing approaches, especially in resilience against structured artifacts.

• Theoretical Insights:

The theoretical formulations establish ShaRP as a stochastic gradient method aimed at minimizing a composite objective that includes a regularizer promoting degraded images consistent with realistic observations. This approach encourages solutions closely aligned with true image characteristics.

Implications and Future Directions

The implications of ShaRP are significant, given its theoretical backing and empirical success. It provides a robust alternative to Gaussian denoiser priors, accommodating broader image degradation models. This offers a more flexible and adaptable framework for imaging inverse problems, particularly in scenarios where only subsampled data is available, a setting where ShaRP demonstrates its unique self-supervised training advantage.

In terms of future developments, ShaRP opens up potential exploration areas in AI-driven imaging solutions across diverse applications. The framework could be extended to integrate additional degradation operators, enhancing its adaptability to various inverse problems. Further, the convergence results lay the groundwork for potential enhancements in stochastic optimization methods applied to image restoration.

In conclusion, ShaRP sets a precedent for leveraging restoration networks more broadly and stochastically as priors, offering substantial enhancements over traditional methods and making a compelling case for further advancements in the AI landscape of imaging inverse problems.