{"id":"77c3e28a-f74d-4329-b7a6-f144034949eb","arxiv_id":"2509.07647","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Enforcing Hermitian symmetry when embedding watermarks in the Fourier domain of latent diffusion models improves detection robustness and image fidelity.","lead":"This paper improves invisible watermarks for AI-generated images by fixing a frequency-domain flaw in earlier methods. The new embedding strategy boosts detection accuracy and keeps image quality high, making AI-made pictures easier to trace.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Random-crop results at scale 0.2 are hard to reconcile with center-confined embedding; the crop protocol must be verified.","rationale":"The reader's weakest assumption and my independent read converge on the same load-bearing point: the random-crop robustness numbers are unexplained and implausible given the center-confined embedding. The center-aware strategy is one of the paper's two main contributions and is the basis for the cropping robustness claims; if the R.C. protocol is non-standard, that contribution is not established. This is a concrete, testable issue rather than a stylistic objection, and it is more central than the misprinted Hermitian equation (Eq. 1), which is likely a typographical error that does not affect the actual implementation, and more significant than the verification overclaim (HSQR and RingID are statistically tied in several verification columns). Because the concern is addressable by inspecting the code and re-running one attack configuration, the reader's CONDITIONAL verdict remains appropriate; no verdict change is needed.","tokens_in":32354,"tokens_out":6907,"duration_ms":60967,"concrete_test":"Inspect the released repository's random-crop attack implementation and log the actual crop bounding boxes for the seed used to produce Tab. 15. If every box is centered (or centers are strongly biased), the R.C. results are explained by a non-uniform crop protocol. Then run a strict uniformly random crop-and-resize at area scale 0.2 (e.g., torchvision.transforms.RandomResizedCrop with scale=(0.2, 0.2), ratio=(1,1)) and measure HSQR identification accuracy; also force a corner crop that excludes the original central 44x44 region and re-measure. If accuracy drops to near chance under these conditions, the claimed R.C. robustness does not hold for the standard protocol.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's center-aware embedding (Sec. 4.2) embeds the watermark only in the central 44x44 of the 64x64 latent, and detection also uses the central 44x44 region. For a truly random crop at scale 0.2, the retained linear dimension is about 44.7% of the original, and the crop can be offset so that the original center (and with it the watermark) lies partly or wholly outside the central detection region after resize. The paper never analyzes this geometric mapping. The reported numbers are internally suspicious: in Tab. 15, HSQR at scale 0.2 achieves 0.955 random-crop accuracy but only 0.555 center-crop accuracy, while HSTR shows the opposite ordering (0.489 vs 0.818). A random crop should be at least as destructive as a center crop, because it can miss the watermark center; HSQR's 40-point inversion suggests the implemented random-crop protocol is not the standard uniformly random crop-and-resize. If the R.C. results are invalid, the central claim that center-aware embedding significantly enhances cropping robustness (Sec. 5.3.2, Fig. 5, ablation Case C in Tab. 8) is unsupported, which weakens the overall robustness-quality balance claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Hermitian Symmetric Fourier Watermarking (SFW), a training-free modification for semantic watermarks embedded in the latent Fourier domain of diffusion models. SFW enforces Hermitian symmetry when constructing Fourier-domain patterns, avoiding the need to discard imaginary components during inverse transformation. The authors further propose a center-aware embedding strategy that embeds and detects watermarks only in the central 44×44 region of the 64×64 latent space. These ideas are instantiated in two methods: HSTR, an Hermitian-symmetric Tree-Ring variant, and HSQR, an Hermitian-symmetric QR-code watermark. The methods are evaluated on MS-COCO, SD-Prompts, and DiffusionDB with Stable Diffusion v2-1, under signal-processing, regeneration, and cropping attacks, with verification, identification, FID, and CLIP metrics. The central claim is that SFW simultaneously improves detection robustness and generation quality, and that center-aware embedding substantially improves cropping robustness.","tokens_in":32616,"tokens_out":10192,"duration_ms":95635,"significance":"If the results hold, this is a simple and practically useful contribution: it converts a known mathematical constraint (real-valued inverse DFT) into a concrete embedding rule, and it is applicable to existing semantic watermarking schemes without retraining. The paper has several genuine strengths. The four-case ablation in Tab. 4 isolates the effect of SFW and shows a clear detection gain (Case D vs Case A: average verification 0.971 vs 0.653). The HSQR amplitude Lambda is set from the Fourier-domain noise level rather than tuned on the headline metrics. The attack suite is broad, including learned compression, diffusion-based regeneration, and cropping, and the code is promised publicly. The main claims are falsifiable. The weaknesses that need attention are a mathematical typo in the central symmetry equation, an under-specified and internally suspicious random-crop protocol, and some overstatement in the verification comparison.","major_comments":[{"comment":"The stated Hermitian symmetry condition omits complex conjugation. It should read F[M-k,N-l] = conj(F[k,l]) (with the appropriate periodic-index convention, or F[-k,-l] = conj(F[k,l]) in shifted coordinates). As printed, the condition is not Hermitian symmetry and does not guarantee a real-valued inverse DFT. Because this equation is the mathematical foundation of the proposed method, it must be corrected.","section":"Sec. 3.2, Eq. (1)"},{"comment":"The random-crop protocol is not defined precisely enough to support the cropping-robustness claims. The text states only that random crop (R.C.) is applied with a crop scale of 0.7, where the scale is the ratio of cropped area to original, but Fig. 5 and Tab. 15 report scales from 0.2 to 0.8 without specifying the offset distribution, whether crops are resized back to 512x512, how many random crops are used per image, or whether the reported scale is an area ratio or a linear ratio. More seriously, the reported numbers are internally difficult to reconcile: at scale 0.2, HSQR achieves 0.955 random-crop identification accuracy but only 0.555 center-crop accuracy, while HSTR shows the opposite ordering (0.489 vs 0.818). A uniformly random crop can displace the watermark outside the central detection region and should not be systematically easier than a center crop for a center-embedded watermark. Because the center-aware cropping claim in Sec. 5.3.2 and Tab. 8 depends on these comparisons, the authors must either specify the exact random-crop implementation, release the attack code, and explain the ordering, or remove and re-run the R.C. results.","section":"Sec. 5.1, Sec. 5.3.2, Tab. 15, Fig. 5"},{"comment":"The embedding and detection rules for HSQR are not fully reconciled. Eq. (3) writes signed magnitudes plus or minus |F(x,c)| of the original Fourier coefficient, while Eq. (4) defines the detection reference as a fixed-amplitude pattern plus or minus Lambda. The L1 distance between a query key region and a constant-amplitude template therefore includes the full magnitude fluctuation of the query, and the paper does not explain why this distance remains discriminative. In addition, replacing F_c by a signed half-normal variable changes the marginal distribution of the modified coefficient, so the Gaussianity-preservation discussion in Sec. 3.2 should be qualified for HSQR. A short derivation of the expected Delta-L1 statistic, or a statement that detection is performed on sign-normalized values, would remove this ambiguity.","section":"Sec. 4.3.2, Eqs. (3)-(4)"}],"minor_comments":[{"comment":"The claim that the methods achieve state-of-the-art verification performance and surpass previous approaches is stronger than the reported numbers support. On average verification (TPR@1%FPR), HSQR ties RingID (0.997 vs 0.997 on MS-COCO) and HSTR is below RingID (0.971 vs 0.997). The paper should either soften the verification claim to competitive, with state-of-the-art identification, or provide a statistical test demonstrating superiority.","section":"Abstract, Sec. 5.2, Tab. 1"},{"comment":"The definition of crop scale is ambiguous across the paper. The main text says the scale is the ratio of cropped image area to original, but Fig. 5 and Tab. 15 use scales from 0.2 to 0.8 without restating this convention. Please state explicitly whether scale is an area ratio or a linear dimension ratio in every table and figure.","section":"Sec. 5.1"},{"comment":"The tables mix Bit Accuracy (for bitstream methods) with TPR@1%FPR (for semantic methods) in the same column. A footnote already explains this, but the cross-family comparisons should be explicitly flagged as not directly comparable so that readers do not interpret the numbers as apples-to-apples measurements.","section":"Tab. 1 and Tab. 2"},{"comment":"The figures contain unresolved LaTeX artifacts: Fig. 3 has the labels 'fig:wm-fft-a (lab4.py)' and 'fig:wm-fft-b (lab4.py)' above the caption, and Fig. 4 has 'fig:concept ???'. These should be cleaned before publication.","section":"Fig. 3 and Fig. 4"},{"comment":"The text says accuracy declines as capacity increases for all methods, but RingID in Fig. 6 stays near 1.0 up to 2,048 keys and only degrades afterward. Please rephrase to describe the plateauing behavior accurately.","section":"Sec. 5.3.3, Fig. 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is within scope and the core SFW mechanism appears sound; the four-case ablation gives real evidence for the detection gain. The main risk is the random-crop protocol. If the reported R.C. numbers cannot be reproduced with a standard random-offset crop-and-resize, the cropping-robustness claim would be substantially weakened, although the center-crop evidence would remain. The missing conjugation in Eq. (1) is an easily fixable but important mathematical typo. I would ask the authors to provide the exact attack code or a precise textual protocol and to reconcile the Tab. 15 ordering before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the good news: the paper's central idea—enforcing Hermitian symmetry when embedding Fourier-domain semantic watermarks in latent diffusion models—is real, and the ablations support it. They show that Tree-Ring without SFW drops to 0.653 verification and 0.114 identification, while the same pattern with SFW reaches 0.971 and 0.889. The supplementary ablation cleanly separates the SFW gain from the center-aware gain. HSQR matches RingID verification while improving FID from about 27 to 24.9. That is a practical win, not just a tweak. Code is provided.\n\nNow the soft spots. Eq. (1) writes Hermitian symmetry as F[M−k,N−l] = F[k,l], which omits the complex conjugation. It's a typo, but it's in the definition of the central mechanism, so fix it. More serious: the random-crop numbers at scale 0.2 don't add up. In Tab. 15, HSQR gets 0.955 under random crop and 0.555 under center crop. A center-confined watermark should survive a center crop better than a random crop that may miss the center entirely. HSTR shows the expected ordering (0.818 center, 0.489 random), so the HSQR result is anomalous. Either the random-crop protocol is not uniformly random, or detection does not actually use the central 44x44 after cropping, or there is a bug. The paper does not say. This matters because Fig. 5 and the 'center-aware embedding enhances cropping robustness' claim lean on these numbers. The main attack tables use R.C. at scale 0.7, which is mild and consistent, so the core SFW result survives, but the cropping story needs to be re-run and spelled out.\n\nAlso, 'state-of-the-art verification' is a bit strong: HSQR ties RingID on verification averages, it wins on identification and on fidelity. That's the real claim.\n\nBottom line: the SFW mechanism is a legit contribution, the ablations are the right kind of evidence, and the crop issue is a fixable protocol problem rather than a fatal flaw. Worth a serious referee; the referee should ask for the random-crop definition, the resizing method, and a corrected Eq. (1).","headline":"SFW is a genuine fix for Fourier-domain semantic watermarks, but the random-crop numbers at scale 0.2 are internally inconsistent and need a second look before I'd trust the cropping claims.","tokens_in":33121,"tokens_out":5849,"would_cite":false,"duration_ms":49282,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that enforcing Hermitian symmetry when embedding semantic watermarks in the latent Fourier domain raises detection robustness and image quality together, and that the HSQR variant beats RingID on identification accuracy…","keywords":["semantic watermarking","latent diffusion models","Fourier domain","Hermitian symmetry","cropping robustness","QR code watermark","frequency integrity","generation quality"],"falsifier":"Watermark an image with HSQR, crop out a corner region that excludes the original image center (for example the top-left 70% of the area), resize it back to $512 \\times 512$, run DDIM inversion, and measure identification accuracy within the central $44 \\times 44$ latent region; if accuracy drops well below the reported 0.955 at random-crop scale 0.2, the center-aware robustness claim holds only for centered crops. A second check: verify that the inverse-transformed watermarked latent has a vanishingly small imaginary component for SFW; if it does not, the frequency-integrity mechanism is not actually enforced.","tokens_in":32144,"feed_emoji":"🏷️","tokens_out":6035,"duration_ms":49967,"temperature":0.7,"pith_summary":"Existing semantic watermarks for latent diffusion models embed a pattern in the Fourier domain of the latent noise and then discard the imaginary part of the inverse transform, which the paper identifies as the cause of both weak detection and degraded image quality. The paper proposes Hermitian Symmetric Fourier Watermarking (SFW), which enforces the mirror symmetry a real latent requires so the full complex Fourier pattern survives, together with a center-aware embedding that applies the Fourier transform to only the central 44×44 of the 64×64 latent. Applied to the tree-ring pattern (HSTR) and to a new QR-code-based pattern (HSQR), the method claims state-of-the-art verification and identification accuracy across signal-processing, regeneration, and cropping attacks while improving FID and CLIP scores over prior semantic methods. If right, this means the robustness/fidelity trade-off in latent-domain watermarking is largely an artifact of broken frequency structure.","feed_headline":"Fourier fix boosts watermark detection and image quality","feed_subtitle":"Enforcing mirror symmetry in the latent Fourier domain beats prior semantic watermarks on accuracy and FID.","key_machinery":"The load-bearing object is the Hermitian symmetry condition on the discrete Fourier transform of the real latent, written as $F[M-k, N-l] = F[k,l]$, which guarantees that the inverse transform is real-valued so no imaginary information must be thrown away; the paper positions the pattern one pixel off the DC axis and zeroes the imaginary parts at DC and Nyquist frequencies. The second mechanism is center-aware embedding: the Fourier transform is applied only to the central $44 \\times 44$ region of the $64 \\times 64$ latent, so a crop-and-resize attack that keeps the image center retains the pattern. For HSQR, the QR code's error correction and a $2 \\times 2$-pixel cell layout add redundancy that allows detection across nearly the full frequency spectrum.","core_discovery":"The central claim is that the frequency loss caused by discarding the imaginary component after inverse Fourier transform—not the embedding pattern itself—limits both detection and generation quality in latent Fourier watermarking. Enforcing Hermitian symmetry, so that the modified spectrum still corresponds to a real-valued latent, preserves the information in both real and imaginary Fourier components and keeps the latent noise statistically close to a real Gaussian. With this constraint, the paper constructs HSTR (a symmetric tree-ring watermark) and HSQR (a QR code split across the real and imaginary parts of the free half-region) and reports that they outperform the strongest baseline RingID on identification accuracy (HSQR 0.985 vs 0.964 average on MS-COCO) and on FID (24.895 vs 27.052) while matching or exceeding its verification performance.","pith_inferences":["If frequency integrity is the real bottleneck, the same Hermitian-symmetry enforcement could be applied to other Fourier-domain watermarking schemes, including post-hoc ones like Zodiac, and should transfer to other generative models that operate on real-valued latents.","The result that full-spectrum QR encoding outperforms low-frequency ring patterns suggests that statistical redundancy (error-correcting codes, multi-pixel cells) matters more than frequency band choice in latent space; a direct extension would test this with other error-correcting structures.","The crop-robustness claims in the paper implicitly assume the watermark stays geometrically aligned in the central detection region after crop-and-resize; a natural extension is to estimate the crop offset from the detected pattern shift and realign before comparison, which could extend robustness to arbitrary off-center crops."],"forward_implications":["HSTR and HSQR achieve state-of-the-art verification (TPR@1%FPR) and identification (Perfect Match Rate) across signal-processing, regeneration, and cropping attacks on MS-COCO, SD-Prompts, and DiffusionDB.","HSQR improves average identification accuracy to 0.985 versus RingID's 0.964 on MS-COCO, and improves FID from 27.052 to 24.895, so the previous trade-off between robustness and image quality is shifted.","Because SFW keeps the latent close to a real Gaussian, generation quality improves (better FID and CLIP, and higher MOS in the paper's human study) without any extra processing time since the watermark is merged in generation.","The finding that near-full-spectrum embedding works in latent space implies that the conventional low–mid frequency constraint, inherited from pixel-domain watermarking, does not apply to latent-domain semantic watermarking."],"supporting_citations":[{"why":"The tree-ring baseline whose discarded imaginary component causes the frequency loss SFW restores; HSTR directly refines this method.","marker":"[48]"},{"why":"The strongest prior semantic watermark (RingID), which supplies the verification/identification evaluation protocol and the main baseline on accuracy and image quality.","marker":"[11]"},{"why":"The post-hoc Zodiac semantic watermark, included as a comparison baseline and an example of frequency-integrity loss the paper challenges.","marker":"[53]"},{"why":"Stable Diffusion, the latent diffusion model that defines the real-valued latent space in which the watermark is embedded and detected.","marker":"[37]"},{"why":"DDIM, the scheduler used for both generation and inversion, so the detection pipeline depends on DDIM inversion to recover the latent key region.","marker":"[41]"},{"why":"The diffusion-based regeneration attack used to evaluate robustness, establishing the threat model that semantic watermarks must survive.","marker":"[56]"}],"fun_headline_variants":["Hermitian symmetry preserves Fourier integrity for robust watermarks","Center-aware embedding resists cropping attacks on latent watermarks","Fourier symmetry fix boosts watermark detection and image fidelity","Symmetric Fourier watermarking tops detection and FID scores","Frequency integrity unlocks better semantic watermarking"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The cropping-robustness results depend on the assumption that after a crop-and-resize attack the embedded pattern still sits in the same central $44 \\times 44$ region of the latent with its alignment unchanged, so the $L^1$ distance comparison finds it.","fun_headline_variants_meta":{"raw":{"variants":["Hermitian symmetry preserves Fourier integrity for robust watermarks","Center-aware embedding resists cropping attacks on latent watermarks","Fourier symmetry fix boosts watermark detection and image fidelity","Symmetric Fourier watermarking tops detection and FID scores","Frequency integrity unlocks better semantic watermarking"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000212,"raw_usage":{"total_tokens":1412,"prompt_tokens":931,"completion_tokens":481,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":547,"completion_tokens_details":{"reasoning_tokens":405}},"tokens_in":547,"tokens_out":481,"duration_ms":4608,"temperature":1.0,"reasoning_tokens":405,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:12:27.162487+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Watermark an image with HSQR, crop out a corner region that excludes the original image center (for example the top-left 70% of the area), resize it back to $512 \\times 512$, run DDIM inversion, and measure identification accuracy within the central $44 \\times 44$ latent region; if accuracy drops well below the reported 0.955 at random-crop scale 0.2, the center-aware robustness claim holds only for centered crops. A second check: verify that the inverse-transformed watermarked latent has a vanishingly small imaginary component for SFW; if it does not, the frequency-integrity mechanism is not actually enforced.","supporting_citations":[{"cited_title":"Tree-rings watermarks: Invisible fingerprints for diffusion images.Advances in Neural Information Process- ing Systems, 36, 2024","cited_arxiv_id":null,"evidence_quote":"The tree-ring baseline whose discarded imaginary component causes the frequency loss SFW restores; HSTR directly refines this method."},{"cited_title":"Attack-resilient image water- marking using stable diffusion.Advances in Neural Infor- mation Processing Systems, 37:38480–38507, 2025","cited_arxiv_id":null,"evidence_quote":"The post-hoc Zodiac semantic watermark, included as a comparison baseline and an example of frequency-integrity loss the paper challenges."},{"cited_title":"High-resolution image synthesis with latent diffusion models","cited_arxiv_id":null,"evidence_quote":"Stable Diffusion, the latent diffusion model that defines the real-valued latent space in which the watermark is embedded and detected."},{"cited_title":"Invisible image watermarks are provably removable using generative ai.Advances in Neural Information Processing Systems, 37:8643–8672, 2025","cited_arxiv_id":null,"evidence_quote":"The diffusion-based regeneration attack used to evaluate robustness, establishing the threat model that semantic watermarks must survive."}],"review_version":1}