Unsupervised PSNR (uPSNR) for Denoising

• Unsupervised PSNR (uPSNR) is an estimator that computes the peak signal-to-noise ratio for denoising methods using several independent noisy observations, eliminating the need for clean data.
• It leverages an unbiased uMSE derived by subtracting noise variance estimates from squared residuals, ensuring consistency under standard statistical assumptions.
• Practical applications include scientific imaging, video processing, and raw data analysis, where uPSNR closely approximates supervised PSNR in both synthetic and real-world experiments.

Unsupervised Peak Signal-to-Noise Ratio (uPSNR) is an estimator for evaluating denoising algorithms in the absence of ground-truth clean data, relying solely on multiple independent noisy observations of the same signal. It addresses the challenge of performance assessment for unsupervised denoising methods in domains where clean images or signals are unavailable, such as raw imaging, real-world videos, and scientific datasets. uPSNR was formalized by Krull, Krishnan, Buchholz, and Weigert for scenarios typical in scientific imaging and live video, providing a statistically rigorous, unbiased, and consistent alternative to classical supervised PSNR (Marcos-Morales et al., 2022).

1. Mathematical Definition

Given a clean unknown vector $x \in \mathbb{R}^n$, a noisy observation $y = x + z$, and a denoised output $f(y)$, uPSNR requires access to three additional independent noisy reference vectors $a = x + w$, $b = x + v$, and $c = x + u$, where $w, v, u$ are independent zero-mean noise realizations (mutually independent and also independent of $z$). The classical mean squared error (MSE) for a denoiser $f$ is

$$\mathrm{MSE} = \frac{1}{n} \sum_{i=1}^n (x_i - f(y)_i)^2.$$

The unsupervised MSE (uMSE) is defined as

$$\mathrm{uMSE} := \frac{1}{n} \sum_{i=1}^n \left[ (a_i - f(y)_i)^2 - \frac{1}{2} (b_i - c_i)^2 \right].$$

The unsupervised PSNR (uPSNR) for dynamic range $M$ is then

$$\mathrm{uPSNR} := 10 \log_{10}\left(\frac{M^2}{\text{uMSE}}\right).$$

uPSNR only assumes knowledge of the noisy instances and does not require direct access to $x$.

2. Underlying Statistical Properties and Consistency

uMSE is constructed to be an asymptotically unbiased estimator of the classical MSE. Consider a single pixel $i$:

• The term $E[(a_i - f(y)_i)^2]$ decomposes as $E[(x_i - f(y)_i)^2] + \mathrm{Var}[w_i]$, since $E[w_i(x_i-f(y)_i)] = 0$ (zero-mean noise).
• The quadratic residual $(b_i - c_i)^2$ serves as an independent estimator of $2\mathrm{Var}[v_i]$; hence, $E[\frac{1}{2}(b_i - c_i)^2]=\mathrm{Var}[v_i]$. Subtracting the average of $(b_i-c_i)^2/2$ debiases the estimate, yielding $E[\mathrm{uMSE}] = \mathrm{MSE}$ under the independence and zero-mean assumptions (Theorem 4.1). The variance of uMSE decays as $O(1/n)$:

$\mathrm{Var[uMSE]} \leq \frac{C}{n},$

for some finite $C$ depending on the bounded moments of the noise and the denoising error. Under suitable regularity conditions, a central limit theorem holds: with bounded sixth moments and independence, $$\sqrt{n}\,(\mathrm{uMSE} - \mathrm{MSE}) \rightarrow \mathcal{N}(0, \sigma_u^2)$$ for large $n$ (Theorem 4.4). Since uPSNR is a continuous mapping of uMSE, uPSNR converges in probability to the supervised PSNR as $n \to \infty$ (Corollary 4.3) (Marcos-Morales et al., 2022).

3. Assumptions and Validity Conditions

The validity of uPSNR relies on the following conditions:

• Independence: All vectors ($y, a, b, c$) have pixel-wise independent entries, and all noise realizations ($z, w, v, u$) are independent.
• Zero-mean: $E[a_i] = E[b_i] = E[c_i] = x_i$ for all $i$.
• Bounded moments: The second through sixth central moments of both the noise and the per-entry denoising error $|x_i-f(y)_i|$ are uniformly bounded. These are satisfied for standard i.i.d. additive Gaussian or Poisson shot noise.

4. Algorithmic and Practical Workflow

The following outlines the computation and usage of uPSNR:

1. Input: One noisy vector $y$ and three independent noisy references $a, b, c$ (all length $n$), denoiser $f(\cdot)$, and dynamic range $M$.
2. Compute squared residuals: For each $i$, set $r_i = (a_i - f(y)_i)^2$, $s_i = (b_i - c_i)^2$.
3. Form uMSE: $$\mathrm{uMSE} = \frac{1}{n} \sum_{i=1}^n [r_i - \frac{1}{2} s_i].$$
4. Get uPSNR: $\mathrm{uPSNR} = 10 \log_{10}(M^2/\mathrm{uMSE}).$
5. Confidence intervals (bootstrap): Generate $K$ bootstrap samples by resampling with replacement, recompute uMSE for each, and transform percentiles into uPSNR intervals via $10\log_{10}(M^2/ \cdot)$.

Reference generation methods:

• Multiple-frame (video/temporal): Use additional frames of the same static scene, captured with minimal delay to preserve content.
• Spatial subsampling (single image): Partition a $2N \times 2N$ image into four $N \times N$ subimages by cyclically or randomly assigning pixels of each $2 \times 2$ block to $a, b, c, y$. This pseudo-randomization approximates independence when the image is smooth at the pixel scale. For single-image settings, spatial subsampling introduces a small bias if the underlying image is not locally smooth.

5. Empirical Performance and Experimental Validation

Synthetic Benchmarks

uPSNR has been validated on natural images corrupted with i.i.d. Gaussian noise ($\sigma=25,50,75,100$) and Poisson-corrupted simulated transmission electron microscopy (TEM) images. Denoisers include DnCNN, UNet, Noise2Noise, Noise2Self, Neighbor2Neighbor, and classical bilateral filtering. When reference images are generated by adding noise to the same ground-truth $x$, uPSNR matches supervised PSNR to within 0.1 dB. For spatial subsampling, the bias remains below 0.5 dB for smooth images (e.g., Poisson-TEM), though it may exceed 2 dB for natural images with high PSNR.

Raw-video (Real Camera) Data

Evaluations on surveillance video sequences with 10 realizations per frame and ISO=800–6400, using wavelet, temporal averaging, UDVD-image-CNN, and UDVD-video-CNN denoisers, reveal that uPSNR (with three independent references) tracks the PSNR averaged over 10-clean-frame averages within approximately 0.2 dB.

Transmission Electron Microscopy Movies

For 18,000 frames of Pt/CeO₂ nanoparticles measured with pixel-correlated Poisson noise, uPSNR was evaluated across moderate-SNR and low-SNR regimes, comparing Neighbor2Neighbor-UNet, Noise2Self-UNet, BlindSpot-UNet, and Gaussian smoothing. uPSNR correctly orders denoisers and SNR levels and distinctly separates per-image uMSE distributions for both method and SNR regimes (Marcos-Morales et al., 2022).

6. Practical Considerations, Limitations, and Use Cases

uPSNR achieves high accuracy (deviation $\leq$0.1 dB from supervised PSNR on synthetic data, $\approx$0.2 dB on real video) when supplied with three independent references. For single-image settings, spatially subsampled pseudo-references may introduce bias in the presence of high-frequency image details, making uPSNR best suited for relatively smooth scientific images. Quantification and empirical correction of such biases can be performed via smoothing or held-out calibration data.

Unlike classical SURE, which requires precise knowledge of the noise model and often expensive Monte Carlo divergence estimation, uPSNR imposes weaker distributional assumptions—requiring only independent noise realizations with bounded moments. Recommended application scenarios include:

• Absence of ground-truth clean images.
• Availability of multiple captures of the same scene (video, burst images).
• Locally smooth images permitting credible spatial subsampling.

Estimates should always be accompanied by bootstrap-derived confidence intervals to quantify the uncertainty resulting from finite sampling and the specifics of the reference generation strategy.

7. Theoretical and Applied Significance

uPSNR, enabled by its core uMSE estimator, allows for principled model selection, hyperparameter optimization, and unbiased benchmarking in unsupervised denoising contexts across real-world and scientific imaging applications where clean reference images do not exist. The methodology's unbiasedness, consistency, and asymptotic normality under general assumptions enable statistically sound evaluation and comparison of modern denoisers, from deep learning models to classical algorithms. The separation of performance obtainable by uPSNR across noise levels and algorithms in both synthetic and real data demonstrates its effectiveness for both algorithm development and real-data deployment (Marcos-Morales et al., 2022).