Image Restoration Learning via Noisy Supervision in the Fourier Domain

Edmund Y. Lam; Haosen Liu; Jiahao Liu; Shan Tan

REVIEW 3 major objections 6 minor 56 references

Image Restoration Learning via Noisy Supervision in the Fourier Domain

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Training with one noisy target per scene reproduces clean-target training when the loss is computed on Fourier coefficients, including for spatially correlated noise that Noise2Noise cannot handle.

desk verdict The Fourier-domain noisy-supervision theorem is sound and useful; the independence claim is false but not load-bearing. read the letter →

arxiv 2506.00564 v1 pith:LNNJMTXC submitted 2025-05-31 eess.IV cs.CV

Haosen Liu , Jiahao Liu , Shan Tan , Edmund Y. Lam This is my paper · ORCID

classification eess.IVcs.CV MSC 94A0860F0562E20

keywords noisysupervisionimagerestorationFourierdomainself-superviseddenoisingspatiallycorrelatednoisestriperemovalstatisticalequivalenceweaklysupervisedlearning

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that image-restoration networks can be trained with noisy images as targets, rather than clean ones, if the loss is computed in the Fourier domain. Its central theorem states that, when the noise's Fourier coefficients are zero-mean Gaussian, the expected Fourier-domain loss with noisy targets is identical to a clean-target loss with a slightly blurred penalty, and both have the same optimum: the clean image. If this is right, practitioners can collect one noisy observation per scene and still match clean-target training, including for spatially correlated noise such as stripes and periodic patterns, where the standard Noise2Noise argument fails. The authors support the theorem with a proof that Fourier coefficients of a wide range of noise converge to independent Gaussians, and with experiments on denoising, super-resolution, and deblurring.

What carries the argument

The 2D discrete Fourier transform is the central object: its coefficients are global weighted sums of all pixels, and for spatially correlated noise they are sparse, affecting only a few frequencies. Two supporting mechanisms carry the argument: the convolution model n = h * η, whose transform variance is |F(h)|^2 var[F(η)], showing which coefficients are corrupted; and the identity ϕ = φ * p, which converts the expected noisy-target loss into a clean-target loss with a blurred penalty. The paper proves coefficient-wise Gaussian convergence using cumulants and the central limit theorem, and independence follows from orthogonality of the DFT basis.

What would settle it

Train IR-NSF on a dataset where every image patch is unique (no repeated or similar patches) and the target contains spatially correlated noise; if the restored PSNR then falls clearly below clean-target training while Noise2Noise also fails, the self-similarity replacement in Remark 7 is the broken link. A more direct check: compare the per-coefficient gradient variance under single-target and expected losses; if they differ substantially, the practical equivalence can fail even where Theorem 3.3 holds.

Watch

Extended reading notes

Core claim

Training a restoration network on noisy targets is statistically equivalent to training on clean targets, provided the loss is computed on Fourier coefficients and the target noise has zero-mean Gaussian Fourier coefficients. The key identity is E[L_φ(fθ(x), y)] = L_ϕ(fθ(x), z), where ϕ = φ * p is the penalty blurred by the noise coefficient distribution p; the minimizer of both is the clean image z. The paper also proves that Fourier coefficients of i.i.d. noise and of correlated noise n = h * η converge in distribution to independent Gaussian variables as the image grows, so the assumption covers Poisson-Gaussian noise, uniform noise, stripe noise, and periodic noise. This extends noisy supervision to spatially correlated noise, which Noise2Noise cannot handle, and lets the loss be restricted to corrupted coefficients in cases like stripe removal.

Load-bearing premise

The theorem requires averaging the loss over many noise realizations, but the implemented training uses a single noisy target per input, relying on repeated similar image patches to supply that average; there is no proof or error bound that this replacement works.

Editorial extensions

If this is right

A practitioner can skip the denoising step when collecting training pairs and use noisy targets directly, obtaining sharper results than pseudo-clean targets generated by BM3D.
Unsupervised stripe removal can be built by applying the Fourier loss only to the corrupted coefficients (k=0 for row noise), leaving image structure and pixel-wise noise intact.
The framework works across three restoration tasks (denoising, super-resolution, deblurring) and six network architectures, so the equivalence is not tied to a specific model.
When targets carry periodic or stripe noise, IR-NSF needs less training data than Noise2Noise, because the noise occupies few Fourier coefficients and averages out faster.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the single-target replacement in Remark 7 is a real statistical effect, it should be measurable: one could estimate the effective number of independent noise samples per Fourier coefficient and test whether the gradient noise variance matches the theorem's prediction.
The blurred-penalty view suggests an unexplored link to robust losses: because φ * p is a smoothed version of φ, noisy-target Fourier training may be inherently more tolerant to outliers in the target than pixel-space training.
The same equivalence might extend to other global transforms (wavelets, DCT, learned spectral transforms) wherever coefficients of structured noise are sparse and approximately Gaussian, offering a recipe for noisy supervision beyond the Fourier basis.
A testable practical claim: for scenes with unique textures and few self-similar patches, the gap between IR-NSF and clean-target training should widen, which could provide a data-collection criterion for when two noisy captures are worth collecting.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the central result—Theorem 3.3—is sound, and the framework is worth taking seriously. The stated independence of Fourier coefficients is overstated and technically false for real-valued noise, but that flaw is not load-bearing. The paper deserves serious peer review.

What's new: Noise2Noise only covers spatial L2. Here they extend noisy-supervision equivalence to Fourier-domain losses with any increasing penalty, and the 'blurred penalty' ϕ=φ*p is a clean way to see the effect of noise. The asymptotic Gaussianity of Fourier coefficients for correlated noise (via the convolution model) unifies several noise types and is genuinely useful, especially for stripe noise. The experiments are broad: three tasks, six networks, synthetic and real datasets (SIDD, ReLoBlur, CHRIS), and they cut the data burden from 150 noisy images per scene to 2. The code and trained models are public, which is real evidence.

Soft spots, in proportion:

1. The independence claim in Thm 3.1(ii) and 3.2(ii) is wrong. For real-valued noise, F[-k,-l] = conj(F[k,l]), so coefficients at symmetric positions are deterministically related. The orthogonality argument misses that. But the proof of Thm 3.3 only needs marginal Gaussianity of each coefficient plus linearity of expectation; dependence is irrelevant. So the main equivalence survives. The paper should weaken that claim (say 'uncorrelated' with caveats, or state it only for the limiting infinite-size case where the symmetric pairs degenerate). This is a fixable overstatement, not a fatal flaw.

2. The practical loss (11) is justified in Remark 7 by a hand-wavy 'similar patches act as repeated samples.' That's not the right justification. The theorem is per-input; the empirical average over i.i.d. training pairs approximates the expected loss by standard ERM. The self-similarity argument is unnecessary and weak, but the practice is fine.

3. The gains over N2N in SR and DB are small (≤0.2 dB) and partly come from the Fourier loss itself even with clean targets (IR-CSF vs N2C). The paper acknowledges this in Sec. 4.3.2. The headline advantage is for correlated noise, and that's where the story is.

4. No error bars or significance tests anywhere. For a paper claiming small but consistent improvements, that matters. The tables are dense but we don't know if the 0.1 dB gaps are noise.

Who is it for: anyone working on noisy supervision, self-supervised denoising, or destriping. The theorem is a useful new tool, and the experiments are broad enough that the practical claim is credible.

Recommendation: send it to peer review, not desk reject. The referee should ask for the independence claim to be corrected, an ERM-based justification of (11), and ideally error bars. The core idea is solid; I'd cite the theorem in my own work.

Referee Report

3 major / 6 minor

Summary. The paper proposes IR-NSF, a framework for training image restoration networks from noisy targets by defining the training loss in the 2D-DFT domain. The central theoretical result (Theorem 3.3) states that if the target noise is zero-mean and its Fourier coefficients are Gaussian, then the expected Fourier-domain loss with noisy targets equals a blurred-penalty loss with clean targets, and the minimizer is exactly the clean image. The paper also analyzes the asymptotic distribution of Fourier coefficients for i.i.d. and spatially correlated noise, and supports the theory with extensive experiments on grayscale denoising, color super-resolution, and deblurring, including real-world SIDD and ReLoBlur datasets, as well as an unsupervised stripe removal application.

Significance. The core idea is original and timely: it extends Noise2Noise-style noisy supervision to spatially correlated noise by moving the loss to the Fourier domain, where the noise has a simpler distribution and the loss encodes global structure. As I read it, the main intended conclusion of Theorem 3.3 is correct under the stated assumptions, and the experimental validation is broad (two network architectures per task, six test sets, real-world datasets) and strongly supports the claim that Fourier-domain noisy supervision can match clean-target performance. The paper also ships public code and trained models, which is a substantial reproducibility asset. The main caveats are a false independence statement in Theorems 3.1(ii)/3.2(ii), a notational imprecision in the equivalence theorem regarding per-frequency noise variances, and an unsupported heuristic bridging the theorem to the implemented single-sample loss. These issues are fixable without changing the central result.

major comments (3)

[Theorems 3.1(ii)/3.2(ii) and Appendix A] Theorems 3.1(ii) and 3.2(ii) assert that the Fourier coefficients of real-valued noise are mutually independent. This is false as stated. For a real-valued n, the 2D-DFT satisfies F(n)[U-k,V-l] = conj(F(n)[k,l]), so a(n)[U-k,V-l] = a(n)[k,l] and b(n)[U-k,V-l] = -b(n)[k,l]; such coefficients are deterministically related, not independent. The proof in Appendix A uses the orthogonality of the DFT basis to conclude that coefficients are uncorrelated, but the diagonal complex covariance E[F(n)[k,l] conj(F(n)[k',l'])] does not imply zero covariance between the real components of conjugate-symmetric pairs. The authors should correct the theorem statements (e.g., restrict independence to coefficients not linked by conjugate symmetry) and adjust the abstract/contributions, which currently claim 'independence' as a key motivation. This does not affect Theorem 3.3, whose proof only uses the marginal Gaussianity of each coefficient.
[Theorem 3.3, Eq. (10), Appendix C] Equation (10) states the equivalence using a single blurred loss L_phi with one penalty phi = phi * p. However, the Fourier coefficients of noise have different variances across frequencies: for i.i.d. noise the DC coefficient has variance sigma_n^2/(UV) while other coefficients have variance sigma_n^2/(2UV) (Remark 2), and for correlated noise the variance is |F(h)|^2 times this (Eq. (8)). The correct equivalence is per-coefficient: E[L_phi(f(x),y)] = sum_{k,l} [phi_{k,l}(a(f)-a(z)) + phi_{k,l}(b(f)-b(z))] with phi_{k,l} = phi * p_{k,l}, where p_{k,l} is the zero-mean Gaussian density with the variance of that coefficient. The proof in Appendix C derives the single-component identity correctly, but the extension to a single phi in (10) does not follow. The minimizer statement (ii) remains true because each phi_{k,l} is minimized at 0, so the central conclusion is intact, but the theorem statement needs to be rewritten to use per-frequency blurring kernels.
[Remark 7 and Eq. (11)] Remark 7 justifies the implemented single-sample loss in Eq. (11) by asserting that similar patches across the image and training set act as repeated samples, providing an implicit expectation over noise. This is a heuristic with no proof or error bound. The standard and correct justification is empirical risk minimization: if the training pairs are i.i.d. draws from the data distribution, then the empirical loss (11) is an unbiased estimator of the population risk, and its gradient is an unbiased estimator of the risk gradient, so SGD with one noisy target per input minimizes the expected loss of Theorem 3.3. The authors should replace the self-similarity argument with this reasoning, or provide a concrete bound; the current remark suggests a mechanism that is neither necessary nor established.

minor comments (6)

[Section 4.4.1] The name 'Neighbor2Neighrbor' is a typo and should be 'Neighbor2Neighbor'.
[Figure 11 caption] 'denosing' should be 'denoising'.
[Section 5.2] 'CHIRS' should be 'CHRIS' (the dataset is the Compact High Resolution Imaging Spectrometer dataset).
[Figure 12 caption] 'T arget' contains an unwanted space and should read 'Target'.
[Appendix A] In the proof of Theorem 3.1(i), the condition 'kl̸=0' should be phrased as 'except k=l=0', since the same argument applies to k=0,l≠0 and to k≠0,l=0.
[Section 4.1] The experiments use a Huber loss for denoising, but the Huber parameter is never specified; please provide the numerical value for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.3 is derived from explicit zero-mean Gaussian Fourier-coefficient assumptions and a monotonicity condition on the penalty, with no fitted parameter or self-citation chain carrying the result.

full rationale

I examined the claimed derivation chain. The central result (Theorem 3.3, Eq. (10)) is a direct consequence of the stated assumptions: if each Fourier coefficient of the noise has density p(t) and is independent of the input, then E[phi(a_f - a_y)] = phi * p (a_f - a_z) by the convolution identity, and the Appendix C derivative argument proves that phi * p has its global minimum at zero when phi increases with |t| and p is zero-mean Gaussian. Thus no predicted value is secretly a fitted value, and no parameter is calibrated to the clean image. Theorems 3.1 and 3.2 are CLT-style supporting analysis; although the pairwise-independence assertion in Theorems 3.1(ii) and 3.2(ii) is overstated because real-valued 2D-DFT coefficients obey Hermitian symmetry, that flaw is a correctness issue, not a circular reduction, and Theorem 3.3 does not use independence across coefficients. The practical loss in Eq. (11) (Section 3.3, Remark 7) replaces the expectation in Eq. (10) with a single noisy target per input; Remark 7 justifies this only heuristically, but that is an approximation gap, not a circular definition. Self-citations (e.g., [31] for DCT coefficient analysis) serve as background analogies and do not carry the equivalence. Comparison against external benchmarks (DIV2K, SIDD, ReLoBlur) makes the experimental claims independently checkable. Overall, the derivation is self-contained under its explicit assumptions, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central theorem is self-contained once the Gaussian-coefficient assumption is granted. The main unpaid inputs are the convolution noise model for correlated noise, the finite-image Gaussian approximation, and the self-similarity heuristic that turns a theoretical expectation into a single-sample loss. No fitted constants are used in the derivation; the loss penalty and USR amplification factor are hand-selected.

free parameters (2)

Loss penalty φ(t) and Huber parameter = Huber for denoising; L1 (q=1) for super-resolution and deblurring; Huber δ not reported
Theorem 3.3 holds for any even increasing φ, so the specific choice is not fitted to make the derivation work, but it is hand-selected and affects the reported metrics.
USR noise amplification factor ε = not reported (stated as ε>1)
Introduced in Section 5.2 to compensate under-estimated stripe noise in the unsupervised stripe removal; its value is not given and affects USR results.

assumptions (6)

standard math Central limit theorem / Lyapunov conditions for weighted sums of i.i.d. noise with bounded cumulants
Used in Theorem 3.1 to show Fourier coefficients of i.i.d. noise converge to Gaussian.
domain assumption Correlated noise is modeled as n = h * η with zero-mean i.i.d. η
Theorem 3.2 relies on this model from [25],[26]; not all spatially correlated noise, e.g. non-stationary noise, fits it.
domain assumption Finite-image Fourier coefficients are approximately Gaussian and independent
Theorems 3.1 and 3.2 are asymptotic in UV; the practical loss (9)-(12) treats the approximation as exact for images around 480x480.
domain assumption Target noise is zero-mean and independent of the input
Assumed in Theorem 3.3 and satisfied by two independent captures of the same scene; may fail for signal-dependent or input-correlated noise.
ad hoc to paper Self-similar patches make the single-sample loss (11) approximate the expectation in (10)
Remark 7 gives this heuristic without proof; it is load-bearing for the implemented training procedure.
domain assumption The correlation kernel h has sparse |F(h)| for diverse correlated noise
Section 3.2.2 cites [26] for sparsity; used to explain the advantage over N2N and to restrict the USR loss to k=0.

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Cite this review

Pith. "Pith review of Image Restoration Learning via Noisy Supervision in the Fourier Domain." pith.science (2026). https://pith.science/paper/LNNJMTXC

@misc{pith2026250600564,
  author       = {Pith},
  title        = {Pith review of: Image Restoration Learning via Noisy Supervision in the Fourier Domain},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LNNJMTXC}},
  note         = {Machine review of arXiv:2506.00564}
}

read the original abstract

Noisy supervision refers to supervising image restoration learning with noisy targets. It can alleviate the data collection burden and enhance the practical applicability of deep learning techniques. However, existing methods suffer from two key drawbacks. Firstly, they are ineffective in handling spatially correlated noise commonly observed in practical applications such as low-light imaging and remote sensing. Secondly, they rely on pixel-wise loss functions that only provide limited supervision information. This work addresses these challenges by leveraging the Fourier domain. We highlight that the Fourier coefficients of spatially correlated noise exhibit sparsity and independence, making them easier to handle. Additionally, Fourier coefficients contain global information, enabling more significant supervision. Motivated by these insights, we propose to establish noisy supervision in the Fourier domain. We first prove that Fourier coefficients of a wide range of noise converge in distribution to the Gaussian distribution. Exploiting this statistical property, we establish the equivalence between using noisy targets and clean targets in the Fourier domain. This leads to a unified learning framework applicable to various image restoration tasks, diverse network architectures, and different noise models. Extensive experiments validate the outstanding performance of this framework in terms of both quantitative indices and perceptual quality.

Figures

Figures reproduced from arXiv: 2506.00564 by the authors.

**Figure 4.** Statistical equivalence. Utilizing φ(t) with noisy targets is statistically equivalent to utilizing ϕ(t) = φ(t) ∗ p(t) with clean targets. The p(t) corresponds to the zero-mean Gaussian distribution with σ = 0.2. maintain both the independence and the sparsity in the Fourier domain. These findings emphasize the advantages of utilizing the Fourier domain for noisy supervision and provide insights into the developmen… view at source ↗

**Figure 5.** Illustration of training pairs for DN, SR, and DB. These images [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

Figures from the paper (10 more)

**Figure 6.** Figure 6: (b), the convergence curves for IR-NSF and IR-CSF are almost consistent everywhere. As for the DN experiment shown in [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: Visual results of denoising experiments. The image contrast is adjusted for better comparison. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Visual results of super-resolution experiments. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Visual results of deblurring experiments. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: Visual results of deblurring experiments on ReLoBlur. The [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: Illustration of the data collection pipeline adopted by existing [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

**Figure 14.** Figure 14: PSNR (dB) on Urban100 as a function of training step. For (a), the noise type is the Poisson-Gaussian noise and the training set is the DIV2K. For (b), the noise type is the mixture of Poisson-Gaussian and periodic noise. The training set is the combination of DIV2K a…

**Figure 15.** Figure 15: Visual results of denoising experiments under limited train [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]

**Figure 16.** Figure 16: Visual comparison among VSNR [25], SEID [42], and USR. [PITH_FULL_IMAGE:figures/full_fig_p012_16.png]

**Figure 17.** Figure 17: Visual results on real-world remote sensing dataset CHRIS. [PITH_FULL_IMAGE:figures/full_fig_p013_17.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 55 canonical work pages

[1]

Nonlinear total variation based noise removal algorithms,

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,”Physica D: Nonlinear Phenomena, vol. 60, no. 1-4, pp. 259–268, 1992

1992
[2]

K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Transactions on Signal Processing, vol. 54, no. 11, pp. 4311– 4322, 2006

work page 2006
[3]

Image denois- ing by sparse 3-D transform-domain collaborative filtering,

K. Dabov, A. Foi, V . Katkovnik, and K. Egiazarian, “Image denois- ing by sparse 3-D transform-domain collaborative filtering,”IEEE Transactions on Image Processing, vol. 16, no. 8, pp. 2080–2095, 2007

work page 2007
[4]

Nonlocal image restoration with bilat- eral variance estimation: A low-rank approach,

W. Dong, G. Shi, and X. Li, “Nonlocal image restoration with bilat- eral variance estimation: A low-rank approach,”IEEE Transactions on Image Processing, vol. 22, no. 2, pp. 700–711, 2012

work page 2012
[5]

Weighted nuclear norm minimization with application to image denoising,

S. Gu, L. Zhang, W. Zuo, and X. Feng, “Weighted nuclear norm minimization with application to image denoising,” inIEEE Con- ference on Computer Vision and Pattern Recognition, 2014, pp. 2862– 2869

work page 2014
[6]

Beyond a Gaussian denoiser: Residual learning of deep cnn for image denoising,

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian denoiser: Residual learning of deep cnn for image denoising,”IEEE Transactions on Image Processing, vol. 26, no. 7, pp. 3142–3155, 2017

work page 2017
[7]

Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections,

X. Mao, C. Shen, and Y. Yang, “Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections,”Advances in Neural Information Processing Systems, vol. 29, 2016

work page 2016
[8]

Learning a deep convolutional network for image super-resolution,

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” inEuropean Conference on Computer Vision. Springer, 2014, pp. 184–199

work page 2014

Show all 56 references

[9]

From local to global: Efficient dual attention mechanism for single image super-resolution,

P . Zhang and E. Y. Lam, “From local to global: Efficient dual attention mechanism for single image super-resolution,”IEEE Access, vol. 9, pp. 114 957–114 964, 2021. SUBMITTED TO IEEE TRANSACTIONS ON PATTERN ANAL YSIS AND MACHINE INTELLIGENCE 16

2021
[10]

Deep multi-scale convolutional neural network for dynamic scene deblurring,

S. Nah, T. H. Kim, and K. M. Lee, “Deep multi-scale convolutional neural network for dynamic scene deblurring,” inIEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 3883–3891

2017
[11]

A high-quality de- noising dataset for smartphone cameras,

A. Abdelhamed, S. Lin, and M. S. Brown, “A high-quality de- noising dataset for smartphone cameras,” inIEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 1692–1700

2018
[12]

Real-world blur dataset for learning and benchmarking deblurring algorithms,

J. Rim, H. Lee, J. Won, and S. Cho, “Real-world blur dataset for learning and benchmarking deblurring algorithms,” inEuropean Conference on Computer Vision. Springer, 2020, pp. 184–201

2020
[13]

Noise2Noise: Learning image restoration with- out clean data,

J. Lehtinen, J. Munkberg, J. Hasselgren, S. Laine, T. Karras, M. Ait- tala, and T. Aila, “Noise2Noise: Learning image restoration with- out clean data,” inInternational Conference on Machine Learning. PMLR, 2018, pp. 2965–2974

2018
[14]

Noise2Void-learning denois- ing from single noisy images,

A. Krull, T.-O. Buchholz, and F. Jug, “Noise2Void-learning denois- ing from single noisy images,” inIEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp. 2129–2137

2019
[15]

Noise2Self: Blind denoising by self- supervision,

J. Batson and L. Royer, “Noise2Self: Blind denoising by self- supervision,” inInternational Conference on Machine Learning. PMLR, 2019, pp. 524–533

2019
[16]

Self2Self with dropout: Learning self-supervised denoising from single image,

Y. Quan, M. Chen, T. Pang, and H. Ji, “Self2Self with dropout: Learning self-supervised denoising from single image,” in IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 1890–1898

2020
[17]

Neighbor2neighbor: Self-supervised denoising from single noisy images,

T. Huang, S. Li, X. Jia, H. Lu, and J. Liu, “Neighbor2neighbor: Self-supervised denoising from single noisy images,” inIEEE Conference on Computer Vision and Pattern Recognition, 2021, pp. 14 781–14 790

2021
[18]

Noisier2Noise: Learning to denoise from unpaired noisy data,

N. Moran, D. Schmidt, Y. Zhong, and P . Coady, “Noisier2Noise: Learning to denoise from unpaired noisy data,” inIEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 12 064–12 072

2020
[19]

Recorrupted-to- Recorrupted: Unsupervised deep learning for image denoising,

T. Pang, H. Zheng, Y. Quan, and H. Ji, “Recorrupted-to- Recorrupted: Unsupervised deep learning for image denoising,” inIEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 2043–2052

2021
[20]

Dancing under the stars: video denoising in starlight,

K. Monakhova, S. R. Richter, L. Waller, and V . Koltun, “Dancing under the stars: video denoising in starlight,” inIEEE/CVF Confer- ence on Computer Vision and Pattern Recognition, 2022, pp. 16 241– 16 251

2022
[21]

Toward universal stripe removal via wavelet-based deep con- volutional neural network,

Y. Chang, M. Chen, L. Yan, X.-L. Zhao, Y. Li, and S. Zhong, “Toward universal stripe removal via wavelet-based deep con- volutional neural network,”IEEE Transactions on Geoscience and Remote Sensing, vol. 58, no. 4, pp. 2880–2897, 2019

2019
[22]

Single- image-based nonuniformity correction of uncooled long-wave in- frared detectors: A deep-learning approach,

Z. He, Y. Cao, Y. Dong, J. Yang, Y. Cao, and C.-L. Tisse, “Single- image-based nonuniformity correction of uncooled long-wave in- frared detectors: A deep-learning approach,”Applied optics, vol. 57, no. 18, pp. D155–D164, 2018

2018
[23]

Optical sectioning deep inside live embryos by selective plane illumination microscopy,

J. Huisken, J. Swoger, F. Del Bene, J. Wittbrodt, and E. H. Stelzer, “Optical sectioning deep inside live embryos by selective plane illumination microscopy,”Science, vol. 305, no. 5686, pp. 1007– 1009, 2004

2004
[24]

Detector blur and corre- lated noise modeling for digital breast tomosynthesis reconstruc- tion,

J. Zheng, J. A. Fessler, and H.-P . Chan, “Detector blur and corre- lated noise modeling for digital breast tomosynthesis reconstruc- tion,”IEEE Transactions on Medical Imaging, vol. 37, no. 1, pp. 116– 127, 2017

2017
[25]

Variational algorithms to remove stationary noise: applications to microscopy imaging,

J. Fehrenbach, P . Weiss, and C. Lorenzo, “Variational algorithms to remove stationary noise: applications to microscopy imaging,” IEEE Transactions on Image Processing, vol. 21, no. 10, pp. 4420–4430, 2012

2012
[26]

Collaborative filtering of correlated noise: Exact transform-domain variance for improved shrinkage and patch matching,

Y. M ¨akinen, L. Azzari, and A. Foi, “Collaborative filtering of correlated noise: Exact transform-domain variance for improved shrinkage and patch matching,”IEEE Transactions on Image Pro- cessing, vol. 29, pp. 8339–8354, 2020

2020
[27]

Fourier space losses for efficient perceptual image super-resolution,

D. Fuoli, L. V . Gool, and R. Timofte, “Fourier space losses for efficient perceptual image super-resolution,” inIEEE/CVF Inter- national Conference on Computer Vision, 2021, pp. 2360–2369

2021
[28]

Spectral bayesian uncertainty for im- age super-resolution,

T. Liu, J. Cheng, and S. Tan, “Spectral bayesian uncertainty for im- age super-resolution,” inIEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023, pp. 18 166–18 175

2023
[29]

Rethinking coarse-to-fine approach in single image deblurring,

S.-J. Cho, S.-W. Ji, J.-P . Hong, S.-W. Jung, and S.-J. Ko, “Rethinking coarse-to-fine approach in single image deblurring,” inIEEE/CVF International Conference on Computer Vision, 2021, pp. 4641–4650

2021
[30]

Focal frequency loss for image reconstruction and synthesis,

L. Jiang, B. Dai, W. Wu, and C. C. Loy, “Focal frequency loss for image reconstruction and synthesis,” inIEEE/CVF International Conference on Computer Vision, 2021, pp. 13 919–13 929

2021
[31]

A mathematical analysis of the DCT coefficient distributions for images,

E. Y. Lam and J. W. Goodman, “A mathematical analysis of the DCT coefficient distributions for images,”IEEE Transactions on Image Processing, vol. 9, no. 10, pp. 1661–1666, 2000

2000
[32]

An efficient statistical method for image noise level estimation,

G. Chen, F. Zhu, and P . A. Heng, “An efficient statistical method for image noise level estimation,” inIEEE International Conference on Computer Vision, 2015, pp. 477–485

2015
[33]

Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,

A. Foi, M. Trimeche, V . Katkovnik, and K. Egiazarian, “Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,”IEEE Transactions on Image Processing, vol. 17, no. 10, pp. 1737–1754, 2008

2008
[34]

Fast and accurate sCMOS noise correction for fluorescence mi- croscopy,

B. Mandracchia, X. Hua, C. Guo, J. Son, T. Urner, and S. Jia, “Fast and accurate sCMOS noise correction for fluorescence mi- croscopy,”Nature Communications, vol. 11, no. 1, p. 94, 2020

2020
[35]

Nakamura,Image Sensors and Signal Processing for Digital Still Cameras

J. Nakamura,Image Sensors and Signal Processing for Digital Still Cameras. CRC Press, 2006

2006
[36]

Training deep learning based de- noisers without ground truth data,

S. Soltanayev and S. Y. Chun, “Training deep learning based de- noisers without ground truth data,”Advances in Neural Information Processing Systems, vol. 31, 2018

2018
[37]

AmbientGAN: Generative models from lossy measurements,

A. Bora, E. Price, and A. G. Dimakis, “AmbientGAN: Generative models from lossy measurements,” inInternational Conference on Learning Representations, 2018

2018
[38]

Noise robust generative adversarial networks,

T. Kaneko and T. Harada, “Noise robust generative adversarial networks,” inIEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 8404–8414

2020
[39]

Self-supervised image prior learning with GMM from a single noisy image,

H. Liu, X. Liu, J. Lu, and S. Tan, “Self-supervised image prior learning with GMM from a single noisy image,” inIEEE/CVF International Conference on Computer Vision, 2021, pp. 2845–2854

2021
[40]

Deep image prior,

D. Ulyanov, A. Vedaldi, and V . Lempitsky, “Deep image prior,” in IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 9446–9454

2018
[41]

Amplitude- phase recombination: Rethinking robustness of convolutional neu- ral networks in frequency domain,

G. Chen, P . Peng, L. Ma, J. Li, L. Du, and Y. Tian, “Amplitude- phase recombination: Rethinking robustness of convolutional neu- ral networks in frequency domain,” inIEEE/CVF International Conference on Computer Vision, 2021, pp. 458–467

2021
[42]

Simultaneous destriping and image denoising using a nonparametric model with the EM algorithm,

L. Song and H. Huang, “Simultaneous destriping and image denoising using a nonparametric model with the EM algorithm,” IEEE Transactions on Image Processing, vol. 32, pp. 1065–1077, 2023

2023
[43]

Robust estimation of a location parameter,

P . J. Huber, “Robust estimation of a location parameter,”The Annals of Mathematical Statistics, vol. 35, no. 1, pp. 73–101, 1964

1964
[44]

Residual dense network for image restoration,

Y. Zhang, Y. Tian, Y. Kong, B. Zhong, and Y. Fu, “Residual dense network for image restoration,”IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 43, no. 7, pp. 2480–2495, 2020

2020
[45]

Understanding and evaluating blind deconvolution algorithms,

A. Levin, Y. Weiss, F. Durand, and W. T. Freeman, “Understanding and evaluating blind deconvolution algorithms,” inIEEE Confer- ence on Computer Vision and Pattern Recognition. IEEE, 2009, pp. 1964–1971

2009
[46]

NTIRE 2017 challenge on single image super-resolution: Dataset and study,

E. Agustsson and R. Timofte, “NTIRE 2017 challenge on single image super-resolution: Dataset and study,” inIEEE Conference on Computer Vision and Pattern Recognition Workshops, 2017, pp. 126– 135

2017
[47]

Low-complexity single-image super-resolution based on nonneg- ative neighbor embedding,

M. Bevilacqua, A. Roumy, C. Guillemot, and M.-L. A. Morel, “Low-complexity single-image super-resolution based on nonneg- ative neighbor embedding,” inThe British Machine Vision Confer- ence, 2012

2012
[48]

On single image scale-up using sparse-representations,

R. Zeyde, M. Elad, and M. Protter, “On single image scale-up using sparse-representations,” inInternational Conference on Curves and Surfaces. Springer, 2010, pp. 711–730

2010
[49]

Color demosaicking by lo- cal directional interpolation and nonlocal adaptive thresholding,

L. Zhang, X. Wu, A. Buades, and X. Li, “Color demosaicking by lo- cal directional interpolation and nonlocal adaptive thresholding,” Journal of Electronic Imaging, vol. 20, no. 2, pp. 023 016–023 016, 2011

2011
[50]

Single image super- resolution from transformed self-exemplars,

J.-B. Huang, A. Singh, and N. Ahuja, “Single image super- resolution from transformed self-exemplars,” inIEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 5197–5206

2015
[51]

Sketch-based manga retrieval using manga109 dataset,

Y. Matsui, K. Ito, Y. Aramaki, A. Fujimoto, T. Ogawa, T. Yamasaki, and K. Aizawa, “Sketch-based manga retrieval using manga109 dataset,”Multimedia Tools and Applications, vol. 76, pp. 21 811– 21 838, 2017

2017
[52]

Enhanced deep residual networks for single image super-resolution,

B. Lim, S. Son, H. Kim, S. Nah, and K. M. Lee, “Enhanced deep residual networks for single image super-resolution,” inIEEE Conference on Computer Vision and Pattern Recognition Workshops, 2017, pp. 136–144

2017
[53]

Image super- resolution using very deep residual channel attention networks,

Y. Zhang, K. Li, K. Li, L. Wang, B. Zhong, and Y. Fu, “Image super- resolution using very deep residual channel attention networks,” inEuropean Conference on Computer Vision, 2018, pp. 286–301. SUBMITTED TO IEEE TRANSACTIONS ON PATTERN ANAL YSIS AND MACHINE INTELLIGENCE 17

2018
[54]

Real-world deep local motion deblurring,

H. Li, Z. Zhang, T. Jiang, P . Luo, H. Feng, and Z. Xu, “Real-world deep local motion deblurring,” inAAAI Conference on Artificial Intelligence, vol. 37, no. 1, 2023, pp. 1314–1322

2023
[55]

NTIRE 2017 challenge on single image super-resolution: Methods and results,

R. Timofte, E. Agustsson, L. Van Gool, M.-H. Yang, and L. Zhang, “NTIRE 2017 challenge on single image super-resolution: Methods and results,” inIEEE Conference on Computer Vision and Pattern Recognition Workshops, 2017, pp. 114–125

2017
[56]

J. K. Blitzstein and J. Hwang,Introduction to probability. CRC Press, 2019

2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Nonlinear total variation based noise removal algorithms,

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,”Physica D: Nonlinear Phenomena, vol. 60, no. 1-4, pp. 259–268, 1992

1992

[2] [2]

K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,

M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Transactions on Signal Processing, vol. 54, no. 11, pp. 4311– 4322, 2006

work page 2006

[3] [3]

Image denois- ing by sparse 3-D transform-domain collaborative filtering,

K. Dabov, A. Foi, V . Katkovnik, and K. Egiazarian, “Image denois- ing by sparse 3-D transform-domain collaborative filtering,”IEEE Transactions on Image Processing, vol. 16, no. 8, pp. 2080–2095, 2007

work page 2007

[4] [4]

Nonlocal image restoration with bilat- eral variance estimation: A low-rank approach,

W. Dong, G. Shi, and X. Li, “Nonlocal image restoration with bilat- eral variance estimation: A low-rank approach,”IEEE Transactions on Image Processing, vol. 22, no. 2, pp. 700–711, 2012

work page 2012

[5] [5]

Weighted nuclear norm minimization with application to image denoising,

S. Gu, L. Zhang, W. Zuo, and X. Feng, “Weighted nuclear norm minimization with application to image denoising,” inIEEE Con- ference on Computer Vision and Pattern Recognition, 2014, pp. 2862– 2869

work page 2014

[6] [6]

Beyond a Gaussian denoiser: Residual learning of deep cnn for image denoising,

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian denoiser: Residual learning of deep cnn for image denoising,”IEEE Transactions on Image Processing, vol. 26, no. 7, pp. 3142–3155, 2017

work page 2017

[7] [7]

Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections,

X. Mao, C. Shen, and Y. Yang, “Image restoration using very deep convolutional encoder-decoder networks with symmetric skip connections,”Advances in Neural Information Processing Systems, vol. 29, 2016

work page 2016

[8] [8]

Learning a deep convolutional network for image super-resolution,

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” inEuropean Conference on Computer Vision. Springer, 2014, pp. 184–199

work page 2014