Moments Matter: Posterior Recovery in Poisson Denoising via Log-Networks

Poisson denoising plays a central role in photon-limited imaging applications such as microscopy, astronomy, and medical imaging. It is common to train deep learning models for denoising using the mean-squared error (MSE) loss, which corresponds to computing the posterior mean ${\mathbb{E}}[{\mathbf{x}}\mid {\mathbf{y}}]$. When the noise is Gaussian, the Tweedie’s formula enables approximation of the posterior distribution through its higher-order moments. However, this connection no longer holds for Poisson denoising: while ${\mathbb{E}}[{\mathbf{x}}\mid {\mathbf{y}}]$ still minimizes MSE, it fails to capture posterior uncertainty. We propose a new strategy for Poisson denoising based on training a log-network. Instead of predicting the posterior mean ${\mathbb{E}}[{\mathbf{x}}\mid {\mathbf{y}}]$, the log-network is trained to learn ${\mathbb{E}}\left[ {\log {\mathbf{x}}|{\mathbf{y}}} \right]$, leveraging the logarithm as a convenient parameterization for the Poisson distribution. We provide a theoretical proof that the proposed log-network enables recovery of higher-order posterior moments and, thus supports posterior approximation. Experiments on simulated data show that our method matches the denoising performance of standard MMSE models, while providing access to the posterior.