DeepFreqMark: End-To-End Learnable Frequency-Domain Watermarking with Spherical Attack Simulation for Latent Diffusion Models

Chen-Hsiu Huang; Ja-Ling Wu; Mario K\"oppen

REVIEW 3 major objections 5 minor 12 references

DeepFreqMark: End-To-End Learnable Frequency-Domain Watermarking with Spherical Attack Simulation for Latent Diffusion Models

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read DeepFreqMark replaces handcrafted frequency watermarks with a learned encoder-decoder and a Slerp-based attack simulation to lower bit-error rates and scale payloads to 256 bits.

desk verdict DeepFreqMark introduces a genuinely learnable frequency-domain watermark and a Slerp attack surrogate, but the claimed superiority over baselines is not yet supported by a shared-protocol comparison. read the letter →

arxiv 2608.08999 v1 pith:65B7I5WF submitted 2026-08-10 cs.CV cs.AI

Chen-Hsiu Huang , Mario Köppen , Ja-Ling Wu This is my paper · ORCID

classification cs.CVcs.AI

keywords latentdiffusionmodelwatermarkinglearnablefrequency-domainembeddingSlerpattacksimulationDDIMinversionmessagecapacityFFTDCTgenerativeimageprovenance

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

DeepFreqMark aims to establish that watermarking images from latent diffusion models can be done with a learned neural encoder and decoder rather than fixed geometric patterns. It writes a binary message into a low-frequency patch of the initial noise spectrum, and it reads the message back from the DDIM-inverted noise of the generated image. Because running DDIM inversion during every training step is too expensive, the paper introduces a spherical-interpolation (Slerp) attack simulation that nudges the watermarked noise toward a random Gaussian noise while preserving its variance, as a cheap surrogate for real image attacks. The reported experiments support the claim that this approach lowers bit-error rates under attacks such as JPEG, blur, noise, brightness and learned compression, and that it scales the payload to 256 bits, far beyond what the compared ring-based methods achieve.

What carries the argument

The load-bearing mechanism is the Slerp-based attack simulation (Equation 10) applied to the noise latent during training. Spherical linear interpolation moves the watermarked latent $z'_T$ along the great-circle arc toward a random Gaussian vector $z_r$, with strength $s\sim U(0,S)$; because both endpoints lie on the Gaussian annulus, the interpolated latent stays on the high-dimensional sphere and preserves unit variance, avoiding the variance collapse that ordinary linear interpolation would cause. This surrogate stands in for the expensive sequence of image-level attack plus DDIM inversion, letting the encoder and decoder be optimized for robustness in each training step. The second supporting mechanism is the frequency-patch embedding: the encoder outputs a real watermark for DCT and a two-channel complex watermark for FFT, with the FFT scheme enforcing Hermitian symmetry via $F[M-k,N-l]=F[k,l]$ on a free half-region, ensuring the inverse transform is real-valued.

What would settle it

Evaluate a Slerp-trained model and a non-Slerp model under the actual DDIM-inversion pipeline with the attacks in Table I; if the Slerp-trained model does not show clearly lower BER on those real attacks, or if the BER-versus-$S$ trend in Fig. 7 does not match the MSE ranking in Table I, the surrogate-attack assumption is falsified.

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Extended reading notes

Core claim

The paper's central claim is that a learnable end-to-end encoder/decoder operating in the frequency domain of the initial noise latent can outperform handcrafted frequency-domain watermarks for latent diffusion models, and that the key enabler is a Slerp-based surrogate attack. The message encoder maps a binary message to a small $32\times32$ patch in the DCT or FFT spectrum of the last channel of the noise latent; the FFT variant uses a free half-region and Hermitian symmetry to guarantee real-valued latents. During extraction, the attacked image is mapped back by DDIM inversion, cropped in the same frequency patch, and decoded. Against real-world attacks such as JPEG, Gaussian noise/blur, brightness/contrast, and neural re-compression, the FFT model trained with Slerp strength $S=0.6$ reports an average BER of $0.316\%$ versus $1.976\%$ without simulation, and the same setup reports $99.44\%$ bit accuracy at 256-bit payloads. These numbers constitute the paper's evidence that the Slerp proxy transfers to real DDIM-inversion-based decoding.

Load-bearing premise

The robustness gain rests on the unproven premise that moving the watermarked noise partway toward a random Gaussian noise during training mimics what real image attacks do to the latent after DDIM inversion, so if that proxy is wrong the reported low bit-error rates may not appear outside the training setup.

Editorial extensions

If this is right

The training cost of attack-aware watermarking can drop sharply, because the Slerp surrogate replaces the DDIM inversion loop that would otherwise run at every iteration.
Payload capacity for in-generation watermarks jumps from 11–72 bits in the compared handcrafted schemes to 256 bits while keeping reported bit accuracy above 99%, which makes per-user message assignment practical.
The same learnable recipe works in both DCT and FFT domains, with the FFT variant consistently more robust, indicating that the method generalizes across spectral representations.
Average BER below roughly 0.5% leaves room for standard error-correcting codes, so near-zero-error provenance tracking is within reach for the tested attack suite.
Raising the Slerp strength lowers BER but visibly degrades image quality once $S\ge 0.8$, giving a tunable robustness-versus-fidelity dial for deployment.

Reading between the lines

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

Editorial extension: the Slerp-to-real-attack mapping is not directly validated; a natural test is to calibrate the sampled $s$ against Table I's DDIM-inversion MSEs, and if no single $s$ covers all attack severities, training should sample a mixture of strengths rather than a uniform $U(0,S)$.
Editorial extension: if the Slerp proxy transfers faithfully, the same latent-interpolation trick could be applied to other generative models whose inversion is deterministic, not only DDIM-based diffusion.
Editorial extension: the FFT variant's consistent edge suggests that the complex phase carries much of the robust signal; perturbing phase while holding magnitude fixed would reveal whether the decoder truly exploits phase or simply the Hermitian redundancy.

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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

I read DeepFreqMark with interest. The core idea is a real departure from the handcrafted-ring watermarks in this space: a small CNN encodes a binary message into a DCT or FFT patch of the initial noise latent, and a matching decoder recovers the message after DDIM inversion and image attacks. Training the encoder/decoder with a Slerp-based perturbation of the latent, instead of running DDIM inversion at every step, is a clever way to make robustness training feasible. The paper also releases code, which helps reproducibility. On its own ablation, the Slerp training (S=0.6) clearly lowers BER compared with no attack simulation, across both DCT and FFT variants, on the eight attacks tested. That is a meaningful empirical result.

Where the paper falls short is in supporting the abstract's claim that DeepFreqMark "achieves significantly lower Bit Error Rates than baseline methods under real-world attacks." Table II compares only the method's own variants. Table III compares accuracy and capacity values taken directly from the original Tree-Rings, RingID, METR, and HSTR/HSQR papers, which used different attack sets, different image sets, different message lengths, and different metrics (verification accuracy vs. bit accuracy). There is no baseline BER measured on the same 100 prompts, the same eight attacks, and the same Stable Diffusion pipeline. Without that shared-protocol comparison, the superiority claim is not demonstrated.

The Slerp surrogate itself is also under-validated. The paper reports MSE of DDIM-inverted latents under various attacks (Table I), but never shows that a Slerp step of some strength resembles those inverted latents, nor that training with Slerp transfers to real inversion. That is a fixable gap, but it is load-bearing.

One mathematical point needs attention: Eq. (10) is written as the standard Slerp formula for unit-norm vectors, while z'_T and z_r are not unit vectors (z'_T is a watermarked latent that differs slightly from a Gaussian draw, and z_r is sampled from N(0,I)). The claim that Slerp "strictly preserves" Gaussian variance is therefore not obviously true as written. The authors should either normalize the vectors first or clarify why the variance is preserved despite non-unit inputs.

Who is this for? Readers working on generative-model provenance and diffusion watermarking will find the learned embedding and the Slerp idea worth engaging with. The paper deserves a serious referee and revision; I would not reject it outright. But I would send it back with requests for a shared baseline protocol, a direct validation of the Slerp surrogate, and a corrected or clarified Eq. (10).

Referee Report

3 major / 5 minor

Summary. The paper proposes DeepFreqMark, an end-to-end learnable frequency-domain watermarking framework for latent diffusion models. A neural encoder embeds binary messages into the DCT or FFT spectrum of the initial noise latent, and a neural decoder recovers the message after DDIM inversion. To avoid the computational cost of DDIM inversion during training, the authors introduce a Slerp-based attack simulation that perturbs the noise latent directly, with a training objective combining latent reconstruction MSE and binary cross-entropy for message recovery. Experiments with Stable Diffusion v1.5 report BER for DCT and FFT variants with and without Slerp training, and accuracy/capacity comparisons with Tree-Rings, RingID, METR, and HSTR/HSQR.

Significance. If the robustness and capacity claims were substantiated, this would be a useful contribution: it replaces handcrafted frequency patterns with a learned encoder/decoder, extends embedding to DCT as well as FFT, handles Hermitian symmetry properly, and scales to 256-bit payloads. The release of source code is a strength, and the idea of avoiding DDIM inversion during training via a latent-space surrogate is practically motivated. However, the headline BER claim is not currently supported by a shared-protocol comparison, and the Slerp surrogate's validity is not established, so the significance cannot be fully assessed.

major comments (3)

[IV.B (Tables II and III)] The central claim in the abstract and Section I that DeepFreqMark 'achieves significantly lower Bit Error Rates (BER) than baseline methods' is not supported by the experiments as presented. Table II compares only DeepFreqMark's own DCT and FFT variants with S=0 and S=0.6; no baseline method is evaluated under the same protocol (same 100 DiffusionDB prompts, same Stable Diffusion v1.5 pipeline, same eight attacks, same DDIM-inversion extraction). Table III lists accuracy and capacity values taken from the original publications, where attack sets, image sets, message capacities, and metrics (verification accuracy vs. bit accuracy) differ, and no BER row is reported for any baseline. A head-to-head BER comparison on a shared protocol is needed to substantiate the comparative claim.
[III.D and Table I] The Slerp-based attack simulation is a load-bearing ingredient, but the paper does not validate that Slerp on the noise latent approximates the effect of real image-level attacks after DDIM inversion. Table I reports only MSE values of DDIM-inverted latents under several attacks; it does not map a Slerp strength s to any real attack, nor does it show that models trained with S=0.6 transfer to actual DDIM-inversion attacks better than models trained without Slerp. Without such validation, the robustness improvements in Table II may reflect overfitting to the Slerp distribution rather than generalization to real-world attacks. I recommend adding a transfer experiment that compares Slerp-trained and non-Slerp-trained models under the Table I attacks, and ideally a calibration of s against attack MSE.
[III.D, Eq. (10)] Eq. (10) writes Slerp(z'_T, z_r, s) with the standard formula for unit vectors, but z'_T and z_r are Gaussian noise latents that are not normalized in general. Applied to vectors of different norms, this formula does not preserve the norm, and the claimed 'unit variance and underlying Gaussian profile' guarantee is not justified. The authors should either define Slerp on normalized vectors and state the variance-preservation argument explicitly, or provide an alternative formula; as written, this is a technical error in the core simulation.

minor comments (5)

[III.B, Eqs. (1)-(3)] The notation moves between Z'_T (frequency domain) and z'_T (spatial latent) without defining the inverse transform; please clarify which domain each loss operates in.
[Table III] The 'Accuracy' column mixes verification accuracy, bit accuracy, and message accuracy across methods; please define the metric for each row or use a single shared metric for all rows.
[Figures 7-9] No confidence intervals or number of test samples are reported; please add error bars or state the sample size at each point.
[Figure 9] The caption says the gap between |m|=128 and |m|=256 is shortened for better visualization, but the axis appears discontinuous; please make the axis transformation explicit or plot the true scale.
[Section IV.F and Table I] Minor typos: Section IV.F heading has an extra space ('F . Superiority'), and Table I columns are labeled 'MSE Error' though MSE is already an error measure.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: training signal and evaluation protocol are distinct; the baseline-comparison gap is an evidence issue, not a circular derivation.

full rationale

Walked the derivation chain. The training objective (Eqs. 3-5) optimizes the encoder and decoder against a BCE loss on Slerp-perturbed latents (Eq. 10), while the evaluation (Section IV.A-B) applies real image attacks to generated images and performs DDIM inversion (Eqs. 6-9) to compute BER. The reported robustness numbers are therefore not equal by construction to the training loss, and no fitted parameter is renamed as a prediction. The Slerp surrogate may be an imperfect approximation, and Table III's baselines use metrics from their original publications rather than a shared protocol, but that is a validation/evidence gap, not circularity. No load-bearing self-citation appears, and the paper's central comparison is against external prior work. Hence no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The only free parameters are the loss weight alpha and the Slerp strength bound S, both hand-chosen. The key non-standard axiom is the claim that Slerp preserves the Gaussian profile, which is mathematically questionable as stated. No new physical or conceptual entities are invented.

free parameters (2)

alpha
Weights the message loss Lm relative to the reconstruction loss Lr in Eq. (5). No value or tuning procedure is reported, so it is a free parameter that affects the imperceptibility-robustness trade-off.
S (Slerp attack strength bound) = 0.6
Upper bound of the uniform sampling interval for the Slerp attack strength. Set to 0.6 in the main experiments; Section IV.D explores other values (0.7-1.0) but does not describe a held-out validation protocol. This hand-chosen value affects the trained model's robustness.

assumptions (3)

domain assumption DDIM inversion provides a deterministic mapping from a generated image back to the initial noise latent
Invoked in Eq. (6) and throughout the method. This is a property of DDIM taken from prior work (Song et al., 2021), not rederived or tested in this paper.
ad hoc to paper Slerp of Gaussian noise latents preserves the Gaussian variance and profile
Section III.D states this is 'mathematically guaranteed', but the Slerp formula in Eq. (10) is applied to non-unit vectors drawn from N(0,I), whose norms cluster near sqrt(d), not 1. Without normalization, the weighted sum formula does not preserve the Gaussian distribution. This assumption is load-bearing for the attack simulation.
standard math Gaussian Annulus Theorem concentration in high dimension
Used in Section III.D to justify spherical concentration of high-dimensional Gaussian noise. Standard result, but it does not automatically justify Slerp as an attack proxy.

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Cite this review

Pith. "Pith review of DeepFreqMark: End-To-End Learnable Frequency-Domain Watermarking with Spherical Attack Simulation for Latent Diffusion Models." pith.science (2026). https://pith.science/paper/65B7I5WF

@misc{pith2026260808999,
  author       = {Pith},
  title        = {Pith review of: DeepFreqMark: End-To-End Learnable Frequency-Domain Watermarking with Spherical Attack Simulation for Latent Diffusion Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/65B7I5WF}},
  note         = {Machine review of arXiv:2608.08999}
}

read the original abstract

The proliferation of AI-generated images produced by Latent Diffusion Models (LDMs) has raised critical concerns regarding copyright infringement and misinformation. Although existing frequency-domain watermarking methods embed handcrafted geometric patterns into the initial latent noise prior to generation, they suffer from limited capacity and rigid pattern designs. We propose DeepFreqMark, an end-to-end learnable frequency-domain watermarking framework that replaces manual pattern engineering with a neural message encoder and decoder. To circumvent the computational bottleneck caused by Denoising Diffusion Implicit Model (DDIM) inversion during training, we introduce a Spherical Linear Interpolation (Slerp)-based attack simulation. This approach operates directly on the noise latent while strictly preserving the Gaussian variance. Extensive experiments demonstrate that DeepFreqMark achieves significantly lower Bit Error Rates (BER) than baseline methods under real-world attacks and scales to 256 bits message capacity. Our source code is available at https://github.com/chenhsiu48/DeepFreqMark.

Figures

Figures reproduced from arXiv: 2608.08999 by the authors.

**Figure 2.** DeepFreqMark embedding process in the FFT domain. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Architectural overview of the message encoder and decoder for DCT [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Architectural overview of the message encoder and decoder for [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

**Figure 7.** Figure 7: Average BER for DCT and FFT across attack strength [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

**Figure 5.** Figure 5: Qualitative comparison of images generated via DeepFreqMark. The [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 8.** Figure 8: Visual quality degradation of watermarked images under elevated [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]

**Figure 9.** Figure 9: Average BER vs. message length (at S = 0.6). The gap between |m| = 128 and |m| = 256 is shortened for better visualization. F. Superiority of FFT over DCT Comparing the DCT and FFT embedding methods in Figures 7 and 9, we observe that the FFT-based DeepFreqMark varian…

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Reference graph

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