Abstract
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.
Abstract (translated)
最大后验估计（MAP）是盲去卷积中广泛使用的一种框架，用于从模糊观测中恢复清晰图像。在该框架下，估计的图像和模糊滤波器被定义为后验分布的最大值。然而，当与促进图像稀疏性的先验条件结合时，MAP估计会倾向于产生模糊解，从而限制了其有效性。 本文重新审视了这一结果，并使用基于扩散的先验模型进行分析。这是一种捕捉现实图像分布的模型类型。通过实证检查先验似然景观，我们发现了两个关键特性：首先，模糊图像往往具有更高的可能性；其次，该景观包含大量局部极小值点，这些极小值对应于自然图像。 基于这些见解，我们对盲去卷积后验进行了理论分析。这揭示了MAP估计器倾向于产生锐利的滤波器（接近狄拉克δ函数）和模糊解。然而，通过梯度下降等方法可以获取到后验中的局部极小值点，这些局部极小值对应于真实的自然图像，从而有效解决了盲去卷积问题。 我们的研究结果表明，克服MAP局限性的关键是找到一个好的局部初始化以到达后验景观中的局部最小值点。我们通过数值实验验证了我们的分析，并展示了我们的见解对设计改进的先验条件和优化技术的实际意义。
URL
https://arxiv.org/abs/2508.02923
