{"id":"a3d127dc-d3f1-42a0-9ff9-61c9052dab3b","arxiv_id":"2506.00564","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Image restoration can be learned from noisy targets in the Fourier domain, where noise becomes approximately Gaussian and clean targets are not needed.","lead":"This paper shows that image restoration networks can be trained with noisy images as targets instead of clean ones, as long as the loss is computed in the Fourier domain. This matters because clean high-resolution images are often impossible to collect in low light, remote sensing, and high-speed imaging.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central Theorem 3.3 is sound; the independence claim in Theorems 3.1/3.2 is overstated due to Hermitian symmetry of real-valued noise, but Theorem 3.3 does not require independence.","rationale":"The reader's verdict of CONDITIONAL is reasonable, but their identified weakest assumption (the single-target loss in Eq. 11 replacing the expectation in Eq. 10) is not actually load-bearing. In supervised learning, the empirical loss over i.i.d. training pairs is an unbiased estimator of the expected loss; the theorem's pointwise equivalence E_n[L(f(x), z+n)] = L_phi(f(x), z) then implies that the expected empirical loss matches clean-target training with the blurred penalty. No self-similarity is needed, so the Remark 7 heuristic is unnecessary but the practice is standard. The more concrete issue is the false independence claim in Theorems 3.1 and 3.2, which results from overlooking Hermitian symmetry of real-valued noise. Nevertheless, Theorem 3.3's proof does not use independence, so the central claim survives. The experiments are broad and internally consistent: IR-NSF closely tracks IR-CSF across tasks and datasets, and the reported gains over N2N in the periodic-noise denoising case are credible. The lack of error bars and the small SR/DB margins are minor and do not change the conclusion. Because the central theoretical claim is sound and the empirical evaluation supports the framework, the reader's CONDITIONAL verdict remains appropriate; no adjustment is needed.","tokens_in":23205,"tokens_out":32144,"duration_ms":308418,"concrete_test":"Re-derive Eq. (10) while allowing arbitrary dependence among the Fourier coefficients of n, keeping only the marginal Gaussianity assumption. If the equality still holds term-by-term, the false independence claim is not load-bearing for Theorem 3.3. As a direct computational check, generate i.i.d. Gaussian noise of size 255x255, compute the empirical covariance between a(n)[k,l] and a(n)[-k,-l] for a non-zero frequency, and confirm it equals the variance of a(n)[k,l] rather than zero.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central mathematical claim, Theorem 3.3, is correct under its stated assumptions, and the practical loss in Eq. (11) is justified by standard empirical risk minimization rather than the self-similarity heuristic in Remark 7. The most concrete technical flaw is in Theorems 3.1(ii) and 3.2(ii), which state that Fourier coefficients of real-valued noise are independent of each other. For real-valued noise, the 2D-DFT obeys Hermitian symmetry, F(n)[-k,-l] = conj(F(n)[k,l]), so the coefficients at (k,l) and (U-k,V-l) are deterministically related. The proof's orthogonality argument overlooks this: Cov(a(n)[k,l], a(n)[-k,-l]) equals Var(a(n)[k,l]), not zero. Thus the independence statement is false as written. However, this does not undermine the central equivalence: the proof of Theorem 3.3 in Appendix C only requires the marginal distribution of each Fourier coefficient to be Gaussian, and the expectation of a sum is the sum of expectations regardless of dependence. The overstated independence weakens the paper's motivating narrative but does not invalidate the main result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23384,"tokens_out":15484,"duration_ms":142016,"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":[{"comment":"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.","section":"Theorems 3.1(ii)/3.2(ii) and Appendix A"},{"comment":"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.","section":"Theorem 3.3, Eq. (10), Appendix C"},{"comment":"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.","section":"Remark 7 and Eq. (11)"}],"minor_comments":[{"comment":"The name 'Neighbor2Neighrbor' is a typo and should be 'Neighbor2Neighbor'.","section":"Section 4.4.1"},{"comment":"'denosing' should be 'denoising'.","section":"Figure 11 caption"},{"comment":"'CHIRS' should be 'CHRIS' (the dataset is the Compact High Resolution Imaging Spectrometer dataset).","section":"Section 5.2"},{"comment":"'T arget' contains an unwanted space and should read 'Target'.","section":"Figure 12 caption"},{"comment":"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":"Appendix A"},{"comment":"The experiments use a Huber loss for denoising, but the Huber parameter is never specified; please provide the numerical value for reproducibility.","section":"Section 4.1"}],"recommendation":"major_revision","confidential_remarks":"This is a solid TPAMI submission with a correct core theorem and extensive experiments. The false independence claim in Theorems 3.1(ii)/3.2(ii) will be raised by any expert reviewer; the authors should fix it before acceptance, but the modification is localized. The ERM justification for Eq. (11) is an easy replacement. The per-frequency variance issue in Theorem 3.3 is more subtle and should be handled carefully in revision. The public code and the broad experimental coverage are strengths that weigh in favor of eventual acceptance after the stated corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat'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.\n\nSoft spots, in proportion:\n\n1. 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.\n\n2. 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.\n\n3. 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.\n\n4. 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.\n\nWho 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.\n\nRecommendation: 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.","headline":"The Fourier-domain noisy-supervision theorem is sound and useful; the independence claim is false but not load-bearing.","tokens_in":23938,"tokens_out":3759,"would_cite":true,"duration_ms":33246,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A08","60F05","62E20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["noisy supervision","image restoration","Fourier domain","self-supervised denoising","spatially correlated noise","stripe removal","statistical equivalence","weakly supervised learning"],"falsifier":"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.","tokens_in":22939,"feed_emoji":"📷","tokens_out":4688,"duration_ms":43203,"temperature":0.7,"pith_summary":"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.","feed_headline":"Fourier loss lets noisy targets match clean-target training","feed_subtitle":"By averaging in the Fourier domain, one noisy photo per scene rivals clean supervision, even for stripe and periodic noise.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the pixel-domain statistical equivalence that this paper extends to the Fourier domain and serves as the main comparison baseline.","marker":"[13]"},{"why":"Introduces the convolution noise model n = h * η and variational stationary-noise removal, providing the correlated-noise model used in the analysis.","marker":"[25]"},{"why":"Shows that |F(h)| is sparse for diverse correlation kernels, supporting the sparsity claim for spatially correlated noise in the Fourier domain.","marker":"[26]"},{"why":"Applies the central limit theorem to transform coefficient distributions for images, serving as the template for the Gaussian-convergence proof.","marker":"[31]"},{"why":"Supplies the heteroscedastic Gaussian model for Poisson-Gaussian noise, justifying the noise models covered by the unified framework.","marker":"[33]"},{"why":"Documents periodic and row noise in extremely low-light images, motivating the stripe and periodic noise experiments.","marker":"[20]"}],"fun_headline_variants":["Fourier domain makes noisy supervision match clean","Noisy targets equal clean targets via Fourier loss","Fourier transform turns noisy supervision into clean","Proof: Fourier noisy supervision equals clean training","Fourier trick: noisy targets rival clean targets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Fourier domain makes noisy supervision match clean","Noisy targets equal clean targets via Fourier loss","Fourier transform turns noisy supervision into clean","Proof: Fourier noisy supervision equals clean training","Fourier trick: noisy targets rival clean targets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1814,"prompt_tokens":911,"completion_tokens":903,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":832}},"tokens_in":527,"tokens_out":903,"duration_ms":9218,"temperature":1.0,"reasoning_tokens":832,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:03:13.249199+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Noise2Noise: Learning image restoration with- out clean data,","cited_arxiv_id":null,"evidence_quote":"Establishes the pixel-domain statistical equivalence that this paper extends to the Fourier domain and serves as the main comparison baseline."},{"cited_title":"Variational algorithms to remove stationary noise: applications to microscopy imaging,","cited_arxiv_id":null,"evidence_quote":"Introduces the convolution noise model n = h * η and variational stationary-noise removal, providing the correlated-noise model used in the analysis."},{"cited_title":"Collaborative filtering of correlated noise: Exact transform-domain variance for improved shrinkage and patch matching,","cited_arxiv_id":null,"evidence_quote":"Shows that |F(h)| is sparse for diverse correlation kernels, supporting the sparsity claim for spatially correlated noise in the Fourier domain."},{"cited_title":"A mathematical analysis of the DCT coefficient distributions for images,","cited_arxiv_id":null,"evidence_quote":"Applies the central limit theorem to transform coefficient distributions for images, serving as the template for the Gaussian-convergence proof."},{"cited_title":"Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data,","cited_arxiv_id":null,"evidence_quote":"Supplies the heteroscedastic Gaussian model for Poisson-Gaussian noise, justifying the noise models covered by the unified framework."},{"cited_title":"Dancing under the stars: video denoising in starlight,","cited_arxiv_id":null,"evidence_quote":"Documents periodic and row noise in extremely low-light images, motivating the stripe and periodic noise experiments."}],"review_version":1}