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UNSURE: self-supervised learning with Unknown Noise level and Stein's Unbiased Risk Estimate
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Proposes UNSURE, a SURE-style self-supervised loss for image reconstruction that drops the need to know the noise level by constraining only the expected divergence of the estimator to zero, with extensions to correlated Gaussian and Poisson-Gaussian noise.
Problem
Self-supervised learning of image reconstruction from noisy measurements when the noise level (or correlation structure) is unknown; SURE needs the exact noise distribution while cross-validation methods are suboptimal.
Approach
Relax the zero-derivative constraint of cross-validation to zero expected divergence (ZED) and optimize a SURE-like loss with a Lagrange multiplier that estimates the noise level, giving a closed-form optimal ZED estimator f(y)=y+eta_hat*score(y) that is more expressive than cross-validation while not needing sigma.
Key results
- On MNIST Gaussian denoising, UNSURE with unknown noise level performs only slightly worse than supervised learning, while SURE and R2R are highly sensitive to misspecified noise level.
- For correlated Gaussian noise on DIV2K, UNSURE (unknown covariance) reaches 28.72 dB vs 29.77 dB for SURE with known covariance and 23.61 dB for Neighbor2Neighbor.
- For accelerated MRI (FastMRI, sigma=0.03), UNSURE + EI reaches 35.73 dB, above EI and CV + EI, and about 1 dB below supervised learning (36.63 dB).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use UNSURE with unknown covariance instead of Noise2Void | Self-supervised denoising on DIV2K · DIV2K validation, colored Gaussian noise | PSNR: 19.09 ± 1.79 -> 28.72± 1.03 | Helped |
| use UNSURE with unknown covariance instead of Neighbor2Neighbor | Self-supervised denoising on DIV2K · DIV2K validation, colored Gaussian noise | PSNR: 23.61 ± 0.13 -> 28.72± 1.03 | Helped |
| use UNSURE with unknown covariance instead of SURE with known covariance | Self-supervised denoising on DIV2K · DIV2K validation, colored Gaussian noise | PSNR: 29.77 ± 1.22 -> 28.72± 1.03 | Hurt |
| use PG-UNSURE with unknown noise instead of Noise2Inverse | Poisson-Gaussian CT reconstruction · LIDC computed tomography | PSNR: 32.54 ± 0.71 -> 33.31± 0.57 | Helped |
| use PG-UNSURE with unknown noise instead of PG-SURE with known noise | Poisson-Gaussian CT reconstruction · LIDC computed tomography | PSNR: 33.76 ± 0.61 -> 33.31± 0.57 | Hurt |
| use UNSURE + EI with unknown noise instead of CV + EI | Self-supervised MRI reconstruction · FastMRI, single-coil 2× accelerated MRI | PSNR: 33.25 ± 1.14 -> 35.73± 1.45 | Helped |
| use UNSURE + EI with unknown noise instead of EI assuming σ = 0 | Self-supervised MRI reconstruction · FastMRI, single-coil 2× accelerated MRI | PSNR: 34.32 ± 0.91 -> 35.73± 1.45 | Helped |
| use UNSURE + EI with unknown noise instead of SURE + EI assuming σ = 0.05 | Self-supervised MRI reconstruction · FastMRI, single-coil 2× accelerated MRI | PSNR: 28.05 ± 4.73 -> 35.73± 1.45 | Helped |
| use a 3×3 covariance kernel instead of 1×1 (ablation) | UNSURE on DIV2K · DIV2K validation, colored Gaussian noise | PSNR: 23.62 -> 28.72± 1.03 | Helped |
| use a 5×5 covariance kernel instead of 3×3 (ablation) | UNSURE on DIV2K · DIV2K validation, colored Gaussian noise | PSNR: 28.72± 1.03 -> 27.38± 0.88 | Hurt |
Surprising findings
- The optimal ZED estimator can have a negative slope, unlike the MMSE estimator, which always has non-negative derivatives.
- Removing the expected divergence from pretrained DRUNet costs less than 1 dB for most noise levels on DIV2K.
Limitations
- Analysis restricted to the l2 loss, so perceptual reconstructions are not addressed.
- Training is more computationally intensive than supervised learning because of the extra estimator evaluation for the divergence term.
- The ZED estimator fails for high-entropy signal distributions such as isotropic Gaussian signals.
Experimental setup
Datasets: MNIST DIV2K FastMRI LIDC Topaz-EM Cryo-EM data
Benchmarks: MNIST DIV2K LIDC FastMRI
Metrics: PSNR
Baselines: Noise2Score Neighbor2Neighbor Recorrupted2Recorrupted SURE Noise2Void Noise2Inverse PG-SURE SSDU Noise2Noise Equivariant imaging
Builds on: SURE Noise2Self Noise2Score Neighbor2Neighbor Equivariant imaging
Closest papers by meaning
- Analyzing the Sample Complexity of Self-Supervised Image Reconstruction Methods (NeurIPS 2023 · similarity 0.90)
- Ditch the Denoiser: Emergence of Noise Robustness in Self-Supervised Learning from Data Curriculum (NeurIPS 2025 · similarity 0.89)
- Extending Stein's unbiased risk estimator to train deep denoisers with correlated pairs of noisy images (NeurIPS 2019 · similarity 0.89)
- Positive2Negative: Breaking the Information-Lossy Barrier in Self-Supervised Single Image Denoising (CVPR 2025 · similarity 0.86)
- Score-based Self-supervised MRI Denoising (ICLR 2025 · similarity 0.85)
- Self-diffusion for Solving Inverse Problems (NeurIPS 2025 · similarity 0.85)
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