# Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images

arXiv: https://arxiv.org/abs/2106.07009 · HTML: https://mlsanity.com/paper/2106.07009/

## Key facts

- Cited by 113 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 30 references that are in the map.
- First posted on arXiv 13 Jun 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Noise2Void - Learning Denoising from Single Noisy Images](https://mlsanity.com/paper/1811.10980/) (2018)
- [Noise2Self: Blind Denoising by Self-Supervision](https://mlsanity.com/paper/1901.11365/) (2019)
- [High-Quality Self-Supervised Deep Image Denoising](https://mlsanity.com/paper/1901.10277/) (2019)
- [Noisier2Noise: Learning to Denoise from Unpaired Noisy Data](https://mlsanity.com/paper/1910.11908/) (2019)
- [Noise2Same: Optimizing A Self-Supervised Bound for Image Denoising](https://mlsanity.com/paper/2010.11971/) (2020)
- [AR-DAE: Towards Unbiased Neural Entropy Gradient Estimation](https://mlsanity.com/paper/2006.05164/) (2020)

## Built on by (most cited first, 30 of 113)

- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022)
- [Diffusion Models for Medical Image Analysis: A Comprehensive Survey](https://mlsanity.com/paper/2211.07804/) (2022)
- [Image Restoration with Mean-Reverting Stochastic Differential Equations](https://mlsanity.com/paper/2301.11699/) (2023)
- [Diffusion-based Image Translation using Disentangled Style and Content Representation](https://mlsanity.com/paper/2209.15264/) (2022)
- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [MR Image Denoising and Super-Resolution Using Regularized Reverse Diffusion](https://mlsanity.com/paper/2203.12621/) (2022)
- [Learning Diffusion Priors from Observations by Expectation Maximization](https://mlsanity.com/paper/2405.13712/) (2024)
- [Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistent](https://mlsanity.com/paper/2302.09057/) (2023)
- [Improving 3D Imaging with Pre-Trained Perpendicular 2D Diffusion Models](https://mlsanity.com/paper/2303.08440/) (2023)
- [DDM2: Self-Supervised Diffusion MRI Denoising with Generative Diffusion Models](https://mlsanity.com/paper/2302.03018/) (2023)
- [Solving Inverse Problems with Score-Based Generative Priors learned from Noisy Data](https://mlsanity.com/paper/2305.01166/) (2023)
- [Don't Play Favorites: Minority Guidance for Diffusion Models](https://mlsanity.com/paper/2301.12334/) (2023)
- [DiffFashion: Reference-based Fashion Design with Structure-aware Transfer by Diffusion Models](https://mlsanity.com/paper/2302.06826/) (2023)
- [Spatially Adaptive Self-Supervised Learning for Real-World Image Denoising](https://mlsanity.com/paper/2303.14934/) (2023)
- [Ambient Diffusion Posterior Sampling: Solving Inverse Problems with Diffusion Models Trained on Corrupted Data](https://mlsanity.com/paper/2403.08728/) (2024)
- [General Image-to-Image Translation with One-Shot Image Guidance](https://mlsanity.com/paper/2307.14352/) (2023)
- [A theoretical framework for self-supervised MR image reconstruction using sub-sampling via variable density Noisier2Noise](https://mlsanity.com/paper/2205.10278/) (2022)
- [It's All About Your Sketch: Democratising Sketch Control in Diffusion Models](https://mlsanity.com/paper/2403.07234/) (2024)
- [Transfer CLIP for Generalizable Image Denoising](https://mlsanity.com/paper/2403.15132/) (2024)
- [Controllable Inversion of Black-Box Face Recognition Models via Diffusion](https://mlsanity.com/paper/2303.13006/) (2023)
- [MagiCapture: High-Resolution Multi-Concept Portrait Customization](https://mlsanity.com/paper/2309.06895/) (2023)
- [Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation](https://mlsanity.com/paper/2507.02608/) (2025)
- [Noise Distribution Adaptive Self-Supervised Image Denoising using Tweedie Distribution and Score Matching](https://mlsanity.com/paper/2112.03696/) (2021)
- [Unleashing the Power of Self-Supervised Image Denoising: A Comprehensive Review](https://mlsanity.com/paper/2308.00247/) (2023)
- [Diffusion models for inverse problems](https://mlsanity.com/paper/2508.01975/) (2025)
- [Falcon: Fast Visuomotor Policies via Partial Denoising](https://mlsanity.com/paper/2503.00339/) (2025)
- [Physics-/Model-Based and Data-Driven Methods for Low-Dose Computed Tomography: A survey](https://mlsanity.com/paper/2203.15725/) (2022)
- [Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems](https://mlsanity.com/paper/2306.03466/) (2023)
- [TweedieMix: Improving Multi-Concept Fusion for Diffusion-based Image/Video Generation](https://mlsanity.com/paper/2410.05591/) (2024)

## Abstract (arXiv)

Recently, there has been extensive research interest in training deep networks to denoise images without clean reference. However, the representative approaches such as Noise2Noise, Noise2Void, Stein's unbiased risk estimator (SURE), etc. seem to differ from one another and it is difficult to find the coherent mathematical structure. To address this, here we present a novel approach, called Noise2Score, which reveals a missing link in order to unite these seemingly different approaches. Specifically, we show that image denoising problems without clean images can be addressed by finding the mode of the posterior distribution and that the Tweedie's formula offers an explicit solution through the score function (i.e. the gradient of log likelihood). Our method then uses the recent finding that the score function can be stably estimated from the noisy images using the amortized residual denoising autoencoder, the method of which is closely related to Noise2Noise or Nose2Void. Our Noise2Score approach is so universal that the same network training can be used to remove noises from images that are corrupted by any exponential family distributions and noise parameters. Using extensive experiments with Gaussian, Poisson, and Gamma noises, we show that Noise2Score significantly outperforms the state-of-the-art self-supervised denoising methods in the benchmark data set such as (C)BSD68, Set12, and Kodak, etc.