Denoisers play a central role in many applications, from noise suppression in low-grade imaging sensors, to empowering score-based generative models. The latter category of methods makes use of Tweedie's formula, which links the posterior mean in Gaussian denoising (\ie the minimum MSE denoiser) with the score of the data distribution. Here, we derive a fundamental relation between the higher-order central moments of the posterior distribution, and the higher-order derivatives of the posterior mean. We harness this result for uncertainty quantification of pre-trained denoisers. Particularly, we show how to efficiently compute the principal components of the posterior distribution for any desired region of an image, as well as to approximate the full marginal distribution along those (or any other) one-dimensional directions. Our method is fast and memory-efficient, as it does not explicitly compute or store the high-order moment tensors and it requires no training or fine tuning of the denoiser. Code and examples are available on the project webpage in https://hilamanor.github.io/GaussianDenoisingPosterior/ .

翻译：去噪器在许多应用中扮演核心角色，从低质量成像传感器的噪声抑制，到赋能基于分数的生成模型。后者这类方法利用了Tweedie公式，该公式将高斯去噪中的后验均值（即最小MSE去噪器）与数据分布的分数联系起来。本文推导了后验分布的高阶中心矩与后验均值的高阶导数之间的基本关系。我们利用这一结果对预训练去噪器进行不确定性量化。特别地，我们展示了如何高效计算图像任意感兴趣区域后验分布的主成分，以及如何近似沿这些（或任何其他）一维方向的全边际分布。我们的方法快速且内存高效，因为它无需显式计算或存储高阶矩张量，且无需对去噪器进行训练或微调。代码和示例可在项目网页 https://hilamanor.github.io/GaussianDenoisingPosterior/ 获取。