This paper presents a stochastic differential equation (SDE) approach for general-purpose image restoration. The key construction consists in a mean-reverting SDE that transforms a high-quality image into a degraded counterpart as a mean state with fixed Gaussian noise. Then, by simulating the corresponding reverse-time SDE, we are able to restore the origin of the low-quality image without relying on any task-specific prior knowledge. Crucially, the proposed mean-reverting SDE has a closed-form solution, allowing us to compute the ground truth time-dependent score and learn it with a neural network. Moreover, we propose a maximum likelihood objective to learn an optimal reverse trajectory which stabilizes the training and improves the restoration results. In the experiments, we show that our proposed method achieves highly competitive performance in quantitative comparisons on image deraining, deblurring, and denoising, setting a new state-of-the-art on two deraining datasets. Finally, the general applicability of our approach is further demonstrated via qualitative results on image super-resolution, inpainting, and dehazing. Code is available at \url{https://github.com/Algolzw/image-restoration-sde}.

翻译：本文提出了一种面向通用图像复原的随机微分方程方法。其核心构造在于一个均值回归随机微分方程，该方程能将高质量图像转化为以固定高斯噪声为均值状态的退化版本。通过模拟相应的逆向时间随机微分方程，我们无需依赖任何特定任务的先验知识即可恢复低质量图像的原始状态。关键之处在于，所提出的均值回归随机微分方程具有闭式解，使我们能够计算真实的时间相关分数函数，并通过神经网络对其进行学习。此外，我们提出了一个最大似然目标函数来学习最优逆向轨迹，该目标函数能稳定训练过程并提升复原效果。实验表明，所提出的方法在图像去雨、去模糊和去噪的定量比较中取得了极具竞争力的性能，并在两个去雨数据集上刷新了最优结果。最后，通过图像超分辨率、修复和去雾的定性实验结果，进一步证明了该方法具有广泛的适用性。代码见 \url{https://github.com/Algolzw/image-restoration-sde}。