Inverse problems are prevalent across various disciplines in science and engineering. In the field of computer vision, tasks such as inpainting, deblurring, and super-resolution are commonly formulated as inverse problems. Recently, diffusion models (DMs) have emerged as a promising approach for addressing noisy linear inverse problems, offering effective solutions without requiring additional task-specific training. Specifically, with the prior provided by DMs, one can sample from the posterior by finding the likelihood. Since the likelihood is intractable, it is often approximated in the literature. However, this approximation compromises the quality of the generated images. To overcome this limitation and improve the effectiveness of DMs in solving inverse problems, we propose an information-theoretic approach. Specifically, we maximize the conditional mutual information $\mathrm{I}(\boldsymbol{x}_0; \boldsymbol{y} | \boldsymbol{x}_t)$, where $\boldsymbol{x}_0$ represents the reconstructed signal, $\boldsymbol{y}$ is the measurement, and $\boldsymbol{x}_t$ is the intermediate signal at stage $t$. This ensures that the intermediate signals $\boldsymbol{x}_t$ are generated in a way that the final reconstructed signal $\boldsymbol{x}_0$ retains as much information as possible about the measurement $\boldsymbol{y}$. We demonstrate that this method can be seamlessly integrated with recent approaches and, once incorporated, enhances their performance both qualitatively and quantitatively.

翻译：逆问题在科学与工程的多个学科中普遍存在。在计算机视觉领域，图像修复、去模糊和超分辨率等任务通常被表述为逆问题。近年来，扩散模型已成为解决含噪线性逆问题的一种有前景的方法，无需额外任务特定训练即可提供有效解决方案。具体而言，利用扩散模型提供的先验，可以通过寻找似然函数从后验分布中采样。由于似然函数难以精确计算，文献中常采用近似方法。然而，这种近似会损害生成图像的质量。为克服这一局限并提升扩散模型解决逆问题的效能，我们提出一种信息论方法。具体来说，我们最大化条件互信息 $\mathrm{I}(\boldsymbol{x}_0; \boldsymbol{y} | \boldsymbol{x}_t)$，其中 $\boldsymbol{x}_0$ 表示重建信号，$\boldsymbol{y}$ 为观测测量值，$\boldsymbol{x}_t$ 是第 $t$ 阶段的中间信号。这确保了中间信号 $\boldsymbol{x}_t$ 的生成方式能使最终重建信号 $\boldsymbol{x}_0$ 尽可能保留关于测量值 $\boldsymbol{y}$ 的信息。我们证明该方法可与现有最新方案无缝集成，且一经融合即可在定性与定量层面提升其性能。