On the Posterior Distribution in Denoising: Application to Uncertainty Quantification

TL;DR

This work addresses uncertainty quantification in Gaussian denoising by linking higher-order posterior central moments to derivatives of the posterior mean, yielding a simple recursion that starts from the MSE-optimal denoiser $\mu_1$. In the univariate case, $\mu_2(y)=\sigma^2\,\mu_1'(y)$ and $\mu_{k+1}(y)=\sigma^2\,\mu_k'(y) + k\,\mu_{k-1}(y)\mu_2(y)$, with a multivariate extension involving Jacobians and higher-order tensor recursions. The approach enables computing posterior PCs and directional marginals without storing high-order moment tensors or retraining denoisers, by leveraging Jacobian-vector products and forward-difference approximations. It is demonstrated across natural images, faces, handwriting (MNIST), and microscopy, revealing meaningful uncertainty directions and providing improved marginal estimates via maximum-entropy fits. The method is training-free, fast, and scalable, with code available for reproducibility and practical deployment in uncertainty visualization tasks.

Abstract

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/ .

Figures (12)

• Figure 1: Recovering posteriors in univariate denoising. The left pane shows the posterior distribution $p_{{\textnormal{x}}|{\textnormal{y}}}(\cdot|\cdot)$ and the posterior mean function $\mu_1(\cdot)$ for the scalar Gaussian denoising task (\ref{['eq:scalarDenoising']}). On the right we plot the posterior distribution of ${\textnormal{x}}$ given that ${\textnormal{y}}=y^*$, along with an estimate of that distribution, which we obtain by analyzing the denoiser function $\mu_1(\cdot)$ at the vicinity of $y^*$. Specifically, this estimate corresponds to the maximum entropy distribution that matches the first four moments, which are obtained from Theorem \ref{['thm:univariateDenoising']} by numerically approximating $\mu'_1(y^*),\mu"_1(y^*),\mu"'_1(y^*)$.
• Figure 2: Computing posterior principal components for a pre-trained face denoising model. For each noisy image ${\bm{y}}$, we depict one of the posterior PCs obtained with Alg. \ref{['Alg:EfficientEVsCalculation']}. To the right of that PC, we show the denoiser's output, ${\bm{\mu}}_1({\bm{y}})$, and its perturbation along that PC. As can be seen, this visualization captures the denoiser's uncertainty along semantically meaningful directions, such as the color of the moustache, the thickness of the lips, and the extent to which the mouth is open.
• Figure 3: Computing marginals along principal components. On the left, we show the prior $p_{\mathbf{x}}$ as a heatmap, a noisy sample ${\bm{y}}$ (red), the corresponding MSE-optimal estimate ${\bm{\mu}}_1({\bm{y}})$ (black), and the two principal axes, computed using Alg. \ref{['Alg:EfficientEVsCalculation']}. Here, we used the closed form for ${\bm{\mu}}_1({\bm{y}})$. The second pane shows the marginal posterior distribution along the first principal component, computed both using our proposed procedure (dashed red), and by using the closed-form solution (solid blue). On the right we show the same experiment, but with a simple neural network trained on data samples.
• Figure 4: Uncertainty quantification for denoising a handwritten digit. The first three PCs corresponding to the noisy image are shown on the left. On the right, images along the third PC, marked in blue, are shown, together with the marginal posterior distribution we estimated for that direction. The two modes of the possible restoration, corresponding to the digits 4 and 9, can clearly be seen as peaks in the marginal posterior distribution, whereas the MSE-optimal restoration in the middle is obviously less likely.
• Figure 5: Uncertainty quantification for natural image denoising using SwinIR (top) and microscopy image denoising using N2V (bottom). In each row, the first three PCs corresponding to the noisy image are shown on the left, and one is marked in blue. On the right, images along the marked PC are shown above the marginal posterior distribution estimated for this direction. The PCs show the uncertainty along meaningful directions, such as the existence of cracks on an old vase and changes in the tiger's stripes, as well as the sizes of cells and the existence of septum, which constitute important geometric features in cellular analysis.

Theorems & Definitions (4)

• Theorem 1: Posterior moments in univariate denoising
• Corollary 1
• Theorem 2: Posterior moments in multivariate denoising
• Theorem 3: Directional posterior moments in multivariate denoising