Image restoration aims to recover high-quality images from degraded observations. When the degradation process is known, the recovery problem can be formulated as an inverse problem, and in a Bayesian context, the goal is to sample a clean reconstruction given the degraded observation. Recently, modern pretrained diffusion models have been used for image restoration by modifying their sampling procedure to account for the degradation process. However, these methods often rely on certain approximations that can lead to significant errors and compromised sample quality. In this paper, we provide the first rigorous analysis of this approximation error for linear inverse problems under distributional assumptions on the space of natural images, demonstrating cases where previous works can fail dramatically. Motivated by our theoretical insights, we propose a simple modification to existing diffusion-based restoration methods. Our approach introduces a time-varying low-pass filter in the frequency domain of the measurements, progressively incorporating higher frequencies during the restoration process. We develop an adaptive curriculum for this frequency schedule based on the underlying data distribution. Our method significantly improves performance on challenging image restoration tasks including motion deblurring and image dehazing.

翻译：图像恢复旨在从退化的观测中恢复高质量图像。当退化过程已知时，恢复问题可表述为一个逆问题；在贝叶斯框架下，目标是在给定退化观测的情况下采样得到干净的复原图像。近年来，现代预训练的扩散模型通过修改其采样过程以考虑退化过程，已被用于图像恢复。然而，这些方法通常依赖于某些近似，可能导致显著误差并损害样本质量。本文首次在自然图像空间分布假设下，对线性逆问题的此类近似误差进行了严格分析，证明了先前工作可能严重失效的情况。基于我们的理论见解，我们提出对现有基于扩散的恢复方法进行一项简单修改。我们的方法在测量值的频域中引入一个时变的低通滤波器，在恢复过程中逐步纳入更高频率成分。我们基于底层数据分布为此频率调度设计了一种自适应课程。我们的方法在包括运动去模糊和图像去雾在内的挑战性图像恢复任务上显著提升了性能。