标题： 扩散先验景观在盲解卷积中如何影响后验 显示英文标题

标题： How Diffusion Prior Landscapes Shape the Posterior in Blind Deconvolution

摘要： 最大后验（MAP）估计是在盲去卷积中广泛使用的框架，用于从模糊观测中恢复清晰图像。 估计的图像和模糊滤波器被定义为后验分布的最大化器。 然而，当与促进稀疏性的图像先验结合时，MAP估计已被证明倾向于模糊解，限制了其有效性。 在本文中，我们使用基于扩散的先验重新审视这一结果，这是一类捕捉真实图像分布的模型。 通过对先验似然景观的实证检查，我们发现了两个关键特性：首先，模糊图像往往具有更高的似然性；其次，景观中包含许多对应于自然图像的局部最小值。 基于这些见解，我们提供了对盲去模糊后验的理论分析。 这表明，MAP估计器倾向于产生尖锐的滤波器（接近狄拉克δ函数）和模糊解。 然而，后验的局部最小值，可以通过梯度下降获得，对应于现实的自然图像，有效地解决了盲去卷积问题。 我们的发现表明，克服MAP的局限性需要在后验景观中的局部最小值处进行良好的局部初始化。 我们通过数值实验验证了我们的分析，展示了我们的见解在设计改进先验和优化技术方面的实际意义。 显示英文摘要

摘要： The Maximum A Posteriori (MAP) estimation is a widely used framework in blind deconvolution to recover sharp images from blurred observations. The estimated image and blur filter are defined as the maximizer of the posterior distribution. However, when paired with sparsity-promoting image priors, MAP estimation has been shown to favors blurry solutions, limiting its effectiveness. In this paper, we revisit this result using diffusion-based priors, a class of models that capture realistic image distributions. Through an empirical examination of the prior's likelihood landscape, we uncover two key properties: first, blurry images tend to have higher likelihoods; second, the landscape contains numerous local minimizers that correspond to natural images. Building on these insights, we provide a theoretical analysis of the blind deblurring posterior. This reveals that the MAP estimator tends to produce sharp filters (close to the Dirac delta function) and blurry solutions. However local minimizers of the posterior, which can be obtained with gradient descent, correspond to realistic, natural images, effectively solving the blind deconvolution problem. Our findings suggest that overcoming MAP's limitations requires good local initialization to local minima in the posterior landscape. We validate our analysis with numerical experiments, demonstrating the practical implications of our insights for designing improved priors and optimization techniques.