# Stochastic Image Denoising by Sampling from the Posterior Distribution

arXiv: https://arxiv.org/abs/2101.09552 · HTML: https://mlsanity.com/paper/2101.09552/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 59 references that are in the map.
- First posted on arXiv 23 Jan 2021.

## Builds on (most cited first, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising](https://mlsanity.com/paper/1608.03981/) (2016)
- [LSUN: Construction of a Large-scale Image Dataset using Deep Learning with Humans in the Loop](https://mlsanity.com/paper/1506.03365/) (2015)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [The Perception-Distortion Tradeoff](https://mlsanity.com/paper/1711.06077/) (2017)
- [FFDNet: Toward a Fast and Flexible Solution for CNN based Image Denoising](https://mlsanity.com/paper/1710.04026/) (2017)
- [PULSE: Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models](https://mlsanity.com/paper/2003.03808/) (2020)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [MemNet: A Persistent Memory Network for Image Restoration](https://mlsanity.com/paper/1708.02209/) (2017)
- [Trainable Nonlinear Reaction Diffusion: A Flexible Framework for Fast and Effective Image Restoration](https://mlsanity.com/paper/1508.02848/) (2015)
- [Residual Dense Network for Image Restoration](https://mlsanity.com/paper/1812.10477/) (2018)
- [Non-Local Recurrent Network for Image Restoration](https://mlsanity.com/paper/1806.02919/) (2018)
- [Solving Linear Inverse Problems Using the Prior Implicit in a Denoiser](https://mlsanity.com/paper/2007.13640/) (2020)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [Explorable Super Resolution](https://mlsanity.com/paper/1912.01839/) (2019)
- [Inverting Deep Generative models, One layer at a time](https://mlsanity.com/paper/1906.07437/) (2019)
- [A Review Paper: Noise Models in Digital Image Processing](https://mlsanity.com/paper/1505.03489/) (2015)
- [Class-Aware Fully-Convolutional Gaussian and Poisson Denoising](https://mlsanity.com/paper/1808.06562/) (2018)
- [Fast Mixing of Multi-Scale Langevin Dynamics under the Manifold Hypothesis](https://mlsanity.com/paper/2006.11166/) (2020)
- [When and How Can Deep Generative Models be Inverted?](https://mlsanity.com/paper/2006.15555/) (2020)
- [Deep-Energy: Unsupervised Training of Deep Neural Networks](https://mlsanity.com/paper/1805.12355/) (2018)

## Built on by (most cited first, 30 of 45)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Deblurring via Stochastic Refinement](https://mlsanity.com/paper/2112.02475/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [SNIPS: Solving Noisy Inverse Problems Stochastically](https://mlsanity.com/paper/2105.14951/) (2021)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Inversion by Direct Iteration: An Alternative to Denoising Diffusion for Image Restoration](https://mlsanity.com/paper/2303.11435/) (2023)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [PriorGrad: Improving Conditional Denoising Diffusion Models with Data-Dependent Adaptive Prior](https://mlsanity.com/paper/2106.06406/) (2021)
- [Provably Robust Score-Based Diffusion Posterior Sampling for Plug-and-Play Image Reconstruction](https://mlsanity.com/paper/2403.17042/) (2024)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [MR Image Denoising and Super-Resolution Using Regularized Reverse Diffusion](https://mlsanity.com/paper/2203.12621/) (2022)
- [A Theory of the Distortion-Perception Tradeoff in Wasserstein Space](https://mlsanity.com/paper/2107.02555/) (2021)
- [How to Trust Your Diffusion Model: A Convex Optimization Approach to Conformal Risk Control](https://mlsanity.com/paper/2302.03791/) (2023)
- [JPEG Artifact Correction using Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2209.11888/) (2022)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learning](https://mlsanity.com/paper/2409.06219/) (2024)
- [Posterior-Mean Rectified Flow: Towards Minimum MSE Photo-Realistic Image Restoration](https://mlsanity.com/paper/2410.00418/) (2024)
- [Enhancing Diffusion-Based Image Synthesis with Robust Classifier Guidance](https://mlsanity.com/paper/2208.08664/) (2022)
- [Principal Uncertainty Quantification with Spatial Correlation for Image Restoration Problems](https://mlsanity.com/paper/2305.10124/) (2023)
- [Threat Model-Agnostic Adversarial Defense using Diffusion Models](https://mlsanity.com/paper/2207.08089/) (2022)
- [Boomerang: Local sampling on image manifolds using diffusion models](https://mlsanity.com/paper/2210.12100/) (2022)
- [Deep Optimal Transport: A Practical Algorithm for Photo-realistic Image Restoration](https://mlsanity.com/paper/2306.02342/) (2023)
- [Score Priors Guided Deep Variational Inference for Unsupervised Real-World Single Image Denoising](https://mlsanity.com/paper/2308.04682/) (2023)
- [Conditional Injective Flows for Bayesian Imaging](https://mlsanity.com/paper/2204.07664/) (2022)
- [Reasons for the Superiority of Stochastic Estimators over Deterministic Ones: Robustness, Consistency and Perceptual Quality](https://mlsanity.com/paper/2211.08944/) (2022)
- [From Posterior Sampling to Meaningful Diversity in Image Restoration](https://mlsanity.com/paper/2310.16047/) (2023)
- [Restoration by Generation with Constrained Priors](https://mlsanity.com/paper/2312.17161/) (2023)
- [SVNR: Spatially-variant Noise Removal with Denoising Diffusion](https://mlsanity.com/paper/2306.16052/) (2023)

## Abstract (arXiv)

Image denoising is a well-known and well studied problem, commonly targeting a minimization of the mean squared error (MSE) between the outcome and the original image. Unfortunately, especially for severe noise levels, such Minimum MSE (MMSE) solutions may lead to blurry output images. In this work we propose a novel stochastic denoising approach that produces viable and high perceptual quality results, while maintaining a small MSE. Our method employs Langevin dynamics that relies on a repeated application of any given MMSE denoiser, obtaining the reconstructed image by effectively sampling from the posterior distribution. Due to its stochasticity, the proposed algorithm can produce a variety of high-quality outputs for a given noisy input, all shown to be legitimate denoising results. In addition, we present an extension of our algorithm for handling the inpainting problem, recovering missing pixels while removing noise from partially given data.