{"id":"d8d01d93-9204-4505-97a3-a132b7fdf691","arxiv_id":"2501.03030","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A training-free method (DDRM-PR) that combines pretrained diffusion priors with alternating projection improves Fourier phase retrieval over HIO alone, but with only modest simulated gains and a heuristic theoretical basis.","lead":"This paper adapts a diffusion-based inverse problem solver, DDRM, to phase retrieval, where only the brightness of a Fourier transform is measured. It replaces the linear pseudo-inverse step with the classic HIO alternating projection algorithm and shows modest reconstruction gains on simulated and experimental images.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central assumption that HIO can stand in for the linear pseudo-inverse in Eq. (13) is unverified; if the projection analogy fails, DDRM-PR is not DDRM and the reported gains may come from the diffusion prior or averaging.","rationale":"I agree with the reader's weakest_assumption: the load-bearing premise is the HIO-as-pseudo-inverse analogy in Section 4.1. The proof of Theorem 4.1 also contains an unstated approximation, sqrt(1-η²)≈1-η, which is inaccurate for typical η values (e.g., η=0.25 gives LHS 0.968 vs RHS 0.75), and Lemma A.4's matrix manipulation is not generally valid; however, these issues affect the simplified DDRM form inherited from prior work, whereas the novel step is the HIO substitution. The empirical results are positive but narrow, and the authors disclose limitations. Thus the concern is real but does not overturn the verdict: the paper should be accepted only conditionally, with the diagnostic/ablation required to substantiate the projection analogy. No change to the reader's CONDITIONAL verdict.","tokens_in":14497,"tokens_out":7936,"duration_ms":71827,"concrete_test":"Run a diagnostic on the actual diffusion trajectory: for a held-out set of CelebA-HQ images and all noise levels α=0.5,1,2,3, collect pairs (x_{t+1}, xθ,t) at the sampled timesteps and apply the exact HIO subroutine used in Eq. (13). Measure (i) the relative Fourier-magnitude residual || |F HIO(|F xθ,t|)| - y ||₂ / ||y||₂ before and after the HIO step, and (ii) the distance to the ground truth (up to trivial ambiguities) before and after. If HIO does not consistently decrease both quantities (e.g., in at least 90% of steps and on average at every noise level), the pseudo-inverse analogy in Section 4.1 fails and Eq. (13) is not a data-consistency update.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.1 replaces H† in the simplified DDRM update, Eq. (12), with the HIO algorithm, yielding Eq. (13). The justification is two bullet-point observations about the noiseless case: HIO applied to a Fourier magnitude recovers the image up to trivial ambiguities, and re-taking the magnitude reproduces the measurement. But the actual operator used in Eq. (13) is not a function of the denoiser output alone: HIO(|F fθ(x_{t+1})|) is initialized from x_{t+1}, is run for only k=100 iterations, and is known to stall in local minima with residual errors. Unlike H†H (an orthogonal projection) and H†y (a fixed least-squares point), HIO is nonlinear, initialization-dependent, and does not exactly enforce |F·|=y. Therefore x'_t = xθ - HIO(|F xθ|) + RandomInit(y) is not a data-consistency correction; the added RandomInit(y) term can dominate and push the iterate away from the denoiser estimate. If HIO is not effectively a projection, the sampler is not sampling a conditioned diffusion posterior, and the empirical improvement over HIO could be due to the unconditional diffusion prior, to averaging over N=8 samples, or to the random init, rather than to the claimed DDRM mechanism. The paper does not provide any direct evidence for the two bullet properties along the diffusion trajectory, nor does it compare against an ablation that removes the HIO correction. This is the load-bearing gap in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes DDRM-PR, a training-free method for Fourier phase retrieval that combines the simplified noiseless DDRM update with the classical HIO alternating-projection algorithm as a stand-in for the pseudo-inverse of the nonlinear Fourier-magnitude operator. The method uses a pretrained unconditional diffusion model to refine HIO reconstructions, with a random-init procedure and averaging over multiple samples. Experiments on simulated CelebA-HQ data and on experimental scattering-medium data report consistent improvements over HIO in PSNR, SSIM, and LPIPS. The authors also provide a proof of the simplified linear DDRM update in an appendix and discuss extensions to other phase retrieval settings.","tokens_in":14895,"tokens_out":2169,"duration_ms":22392,"significance":"If the central heuristic were rigorously validated, DDRM-PR would be a useful addition to the phase retrieval toolbox, demonstrating a plug-and-play use of pretrained diffusion priors for a nonlinear inverse problem without task-specific training. The paper is commendable for providing source code, testing on experimental transmission-matrix data, and being transparent about the heuristic nature of the HIO substitution and about the method's remaining artifacts. However, the significance currently rests on an unproven analogy between HIO and the linear pseudo-inverse, and the empirical evaluation is too thin to establish that the proposed mechanism, rather than the unconditional diffusion prior or sample averaging, is responsible for the reported gains.","major_comments":[{"comment":"The load-bearing assumption that HIO can replace the pseudo-inverse H† in the simplified DDRM update is not justified. The two bullet-point observations about the noiseless case establish only that HIO, when initialized appropriately and run to convergence, can recover the image up to trivial ambiguities and reproduce the Fourier magnitude; they do not imply that the concrete operator used in Eq. (13), namely HIO(|F fθ(x_{t+1})|) initialized from x_{t+1} and run for k=100 iterations, behaves like the orthogonal projection H†H along the entire diffusion trajectory, nor that it preserves data consistency under the noisy measurements used in the experiments. The paper provides no direct evidence for these properties and no ablation that removes the HIO correction or replaces it with an exact projection, so the reader cannot tell whether the improvement over HIO arises from the claimed DDRM mechanism or from the diffusion prior and averaging alone.","section":"§4.1, Eq. (13)"},{"comment":"The derivation of the simplified DDRM form uses the approximation sqrt(1-η²) ≈ 1-η without any error bound or justification. For intermediate values such as η=0.5, the two expressions differ by about 0.37 in absolute value, and this error is multiplied by the noise term in Eq. (32) and propagated into the final update. Since Theorem 4.1 is stated as a formal result, the approximation should either be removed by an exact derivation or stated as an explicit approximation whose validity range is quantified; as written, the proof does not establish the claimed equality.","section":"Appendix A, Eq. (32)"},{"comment":"The evaluation protocol is insufficiently specified for the reported numbers to be fully credible. The grid search over η, η_b, t, and T_init is described as 'linear grid search' but no validation split is mentioned; if the hyperparameters were selected on the test set, the reported PSNR/SSIM/LPIPS values are optimistic. In addition, Table 1 reports averages without error bars or the number of test images, and the experimental section reports a single configuration. The paper should include a clear train/validation/test split for hyperparameter selection, report variability across the test set, and state the number of images used for each entry.","section":"§5.1 and §5.2"},{"comment":"The comparison is limited to a single baseline (HIO), and that baseline is also the initialization stage of the proposed method. This makes it impossible to attribute the improvement to the diffusion prior, the HIO correction, the random-init procedure, or the averaging over N=8 samples. At minimum, the authors should include an ablation with the diffusion prior but without the HIO correction term, an ablation with the HIO correction but without the diffusion prior, and a comparison with a stronger classical or learning-based PR baseline such as prDeep or oversampling smoothness; without such controls the central claim of 'improving alternating-projection methods' is not convincingly supported.","section":"§5.1, Table 1"}],"minor_comments":[{"comment":"The sentence 'RandomInitrepresentstotheHIOinitializationprocedureproposedintheprDeeppaper' contains a typographical error ('representstothe') and would benefit from rewording for clarity.","section":"§4.1"},{"comment":"The figure captions list hyperparameter values such as η=0.15 and t=15, but the main text states that a grid search was performed; it is unclear whether the values shown in the captions are the selected optimal values for each noise level or just illustrative examples. Please clarify.","section":"Figures 1–4"},{"comment":"The use of cyan color to denote scaled variables in Definition A.2 is not visible in a black-and-white print version and is not self-explanatory; please use an explicit notation such as a tilde or superscript instead of or in addition to color.","section":"Appendix A, Definition A.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a reasonable heuristic contribution, but the gap between the formal-sounding Theorem 4.1 and the nonlinear extension in Section 4.1 is wide, and the empirical section currently lacks the controls needed to support the paper's central attribution of improvement to the DDRM mechanism. The authors may need to add ablations and a clearer evaluation protocol before this is suitable for publication in a serious journal. The scope is appropriate for an applied optics or computational imaging venue, but the current level of validation is on the borderline."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look, but the central claim is shakier than the writing lets on. What's new: Eq. (13) is a real algorithmic proposal, DDRM-PR, substituting HIO for the pseudo-inverse in the simplified DDRM update, and it has no direct precedent in the literature. The proof of the simplified DDRM form (Theorem 4.1) is also new, though it leans on the approximation sqrt(1-eta^2) ~ 1-eta, which is rough for eta near 1 and not justified beyond convenience. The scattering-media experiment is a genuine extra, and the code is public. Credit where due: the authors are transparent that the derivation is noiseless, that HIO artifacts persist, and that the whole thing is a heuristic extension.\n\nThe soft spots are real but proportionate. The load-bearing assumption is in Section 4.1: HIO behaves like H-dagger for the nonlinear Fourier magnitude operator. The two bullet-point justifications describe the noiseless, idealized behavior, but the actual algorithm runs HIO for only 100 iterations from the current diffusion iterate, which is initialization-dependent and does not exactly enforce the magnitude constraint. So the correction term in Eq. (13) is not a clean data-consistency projection. The stress-test is right that there's no ablation removing the HIO correction or isolating the effect of averaging N=8 samples versus the diffusion prior. That's the gap a referee should probe. On top of that, the hyperparameters (eta, eta_b, t, T_init) are grid-searched per noise level, and I don't see a described validation split, so the reported PSNR gains of 0.4–1.7 dB over HIO are likely optimistic. Only HIO as baseline, no error bars, and the noisy simulations are run against a noiseless-derived update. The experimental scattering results (7.85 to 13.12 dB) look striking, but that's partly because HIO is almost useless there; the diffusion prior does heavy lifting.\n\nWho is this for? Anyone working on phase retrieval with pretrained diffusion priors, or on nonlinear extensions of DDRM. It's a solid methods paper with an honest limitations section and a testable hypothesis, even if the mechanism is not yet proven. It deserves a serious referee, and a good referee would push for an ablation study, a validation split, and comparison with at least one other diffusion-based PR method. If the ablation shows the improvement mostly comes from the prior plus averaging, the paper is still useful but the framing should change.","headline":"A genuinely new training-free combination of DDRM and HIO for phase retrieval, with an honest limitations section, but the central analogy that HIO approximates the pseudo-inverse is a load-bearing heuristic that the paper does not verify.","tokens_in":15402,"tokens_out":1384,"would_cite":false,"duration_ms":15118,"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":"Phase retrieval improves when a diffusion prior refines alternating-projection reconstructions.","keywords":["phase retrieval","Fourier magnitude","diffusion models","denoising diffusion restoration models","hybrid input-output","alternating projections","inverse problems","pretrained priors"],"falsifier":"Take a noiseless natural image $x$, compute the HIO output from its Fourier magnitude with the paper's parameters, and compare both the reconstruction error and the Fourier-magnitude residual of that output; if for a nontrivial fraction of images the update $x - \\text{HIO}(|Fx|)$ is not closer to $x$ than before, or the magnitude residual is not near zero, the data-consistency correction in Eq. (13) is not doing what the pseudo-inverse analogy claims.","tokens_in":14286,"feed_emoji":"🔬","tokens_out":6463,"duration_ms":130758,"temperature":0.7,"pith_summary":"This paper tries to establish that Fourier phase retrieval—recovering an image from noisy measurements of only the magnitude of its Fourier transform—can be improved by wrapping the classical Hybrid Input-Output (HIO) alternating-projection algorithm inside a Denoising Diffusion Restoration Model (DDRM). The difficulty is that DDRM was derived for linear inverse problems; the paper's move is to let HIO play the role of the pseudo-inverse $H^{\\dagger}$ in the noiseless DDRM update, producing a training-free update that uses a pretrained unconditional diffusion model as the image prior. On simulated face images the method reports higher PSNR and SSIM and lower LPIPS than HIO alone at every tested noise level, and on experimental imaging-through-scattering data it reports a large quantitative improvement. The paper also argues the same recipe extends to other phase retrieval settings, such as coded diffraction patterns and complex-valued images, provided a matching pretrained denoiser is available.","feed_headline":"Diffusion prior lifts phase-retrieval quality at every noise level","feed_subtitle":"A pretrained denoiser refines classical alternating-projection reconstructions with no retraining, in simulations and scattering…","key_machinery":"The load-bearing device is the analogy between the Moore–Penrose pseudo-inverse $H^{\\dagger}$ in the linear DDRM update and the HIO algorithm—an alternating-projection method that alternates between Fourier magnitude constraints and spatial constraints—applied to Fourier magnitudes. In the noise-free case, HIO is expected to return the original image from its Fourier magnitude, mirroring $H^{\\dagger}H x \\approx x$, and to leave the noiseless measurement unchanged, mirroring $H H^{\\dagger}H = H$. The authors use this analogy to rewrite the correction term as $\\text{HIO}(|F f_\\theta^{(t+1)}(x_{t+1})|)$, with a random-initialization HIO step supplying the data-consistency component. Everything else in the pipeline—the pretrained denoiser, the noise schedule, and the averaging over multiple outputs—is standard DDRM machinery.","core_discovery":"The central claim is that the simplified noiseless DDRM update $x'_t = x_{\\theta,t} - H^{\\dagger}H x_{\\theta,t} + H^{\\dagger} y$ remains valid for the nonlinear Fourier phase retrieval operator if HIO, run on the Fourier magnitude of the denoiser output, is substituted for the pseudo-inverse terms. The resulting update is Eq. (13): $x'_t = f_\\theta^{(t+1)}(x_{t+1}) - \\text{HIO}(|F f_\\theta^{(t+1)}(x_{t+1})|) + \\text{RandomInit}(y)$, followed by the standard diffusion interpolation with noise. The paper treats this as a regularized extension of alternating projection: the diffusion prior cleans HIO artifacts, while the HIO term keeps the iterate consistent with the measured Fourier magnitude. It reports that this combination outperforms the HIO initialization on all evaluated metrics in simulations and substantially improves PSNR, SSIM, and LPIPS on experimental scattering data, with the caveat that HIO artifacts can persist and the noisy-measurement theory is not developed.","pith_inferences":["If the HIO $\\approx H^{\\dagger}$ analogy is quantitatively tested on natural images, a large gap between HIO$(|Fx|)$ and $x$ would predict where Eq. (13) injects the wrong target; this is a testable diagnostic the paper does not report.","The method's per-step cost includes a 100-iteration HIO run inside every diffusion step, so timings on a standard GPU would clarify whether the gains are practical for real-time imaging; the paper does not report runtime.","The claimed extension to complex-valued images likely requires a pretrained diffusion model whose training distribution matches the amplitude and phase statistics of the target, which may be harder to obtain than the natural-image prior used here.","One could test the prior's role directly by replacing the pretrained diffusion denoiser with a classical denoiser of similar strength; if the gain mostly disappears, the diffusion sampling procedure rather than the prior is what matters."],"forward_implications":["The same recipe can be applied to any phase retrieval problem with an alternating-projection solver, so coded diffraction patterns, ptychography, and scattering measurements become candidates for diffusion-prior refinement without retraining.","Using a pretrained unconditional diffusion model means a new measurement setup does not require collecting paired training data; only the alternating-projection component needs to be matched to the measurement.","Simulation results imply that even when the baseline HIO reconstruction is noisy, the diffusion update improves both distortion metrics (PSNR, SSIM) and perceptual quality (LPIPS) across the tested noise range.","The experimental scattering results suggest the prior generalizes beyond its natural-image training distribution, since a black-and-white target is recovered with a model trained on color natural images.","Because the method still inherits some HIO artifacts, its ceiling is tied to the quality of the alternating-projection initialization; better initializations would translate directly into better final reconstructions."],"supporting_citations":[{"why":"Defines DDRM as an SVD-based posterior sampler for linear inverse problems, the framework this paper adapts to nonlinear phase retrieval.","marker":"[17]"},{"why":"Provides the simplified noiseless DDRM update that becomes Theorem 4.1; the paper supplies the proof and builds Eq. (13) on it.","marker":"[19]"},{"why":"Supplies the random-initialization HIO procedure (RandomInit) used in every DDRM-PR update to obtain a measurement-consistent starting point.","marker":"[12]"},{"why":"Defines the HIO algorithm whose update is substituted for the pseudo-inverse in the Fourier phase retrieval setting.","marker":"[28]"},{"why":"Provides the experimental scattering-media dataset and the estimated transmission matrix used in Section 5.2 to test the extension beyond classical Fourier PR.","marker":"[35]"},{"why":"Supplies the simulated face-image test set used to compare HIO and DDRM-PR under four measurement noise levels.","marker":"[41]"}],"fun_headline_variants":["Diffusion prior sharpens phase retrieval without retraining","Pretrained denoiser improves phase retrieval across noise levels","Alternating projection plus diffusion: a phase retrieval boost","Diffusion model refines Fourier phase retrieval","Phase retrieval enhanced by diffusion priors, no retraining needed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole update hinges on the unproven assumption that HIO acts like the linear pseudo-inverse $H^{\\dagger}$: that running HIO on the Fourier magnitude of a denoised image brings it closer to the true image and preserves the measured magnitude, rather than getting trapped in a local minimum with large residual error.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion prior sharpens phase retrieval without retraining","Pretrained denoiser improves phase retrieval across noise levels","Alternating projection plus diffusion: a phase retrieval boost","Diffusion model refines Fourier phase retrieval","Phase retrieval enhanced by diffusion priors, no retraining needed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000256,"raw_usage":{"total_tokens":1546,"prompt_tokens":889,"completion_tokens":657,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":590}},"tokens_in":505,"tokens_out":657,"duration_ms":6462,"temperature":1.0,"reasoning_tokens":590,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:58:05.614342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a noiseless natural image $x$, compute the HIO output from its Fourier magnitude with the paper's parameters, and compare both the reconstruction error and the Fourier-magnitude residual of that output; if for a nontrivial fraction of images the update $x - \\text{HIO}(|Fx|)$ is not closer to $x$ than before, or the magnitude residual is not near zero, the data-consistency correction in Eq. (13) is not doing what the pseudo-inverse analogy claims.","supporting_citations":[{"cited_title":"Denoising diffusion restoration models,","cited_arxiv_id":null,"evidence_quote":"Defines DDRM as an SVD-based posterior sampler for linear inverse problems, the framework this paper adapts to nonlinear phase retrieval."},{"cited_title":"JPEG artifact correction using denoising diffusion restoration models,","cited_arxiv_id":null,"evidence_quote":"Provides the simplified noiseless DDRM update that becomes Theorem 4.1; the paper supplies the proof and builds Eq. (13) on it."},{"cited_title":"prDeep: Robust phase retrieval with a flexible deep network,","cited_arxiv_id":null,"evidence_quote":"Supplies the random-initialization HIO procedure (RandomInit) used in every DDRM-PR update to obtain a measurement-consistent starting point."},{"cited_title":"Phase retrieval algorithms: a comparison,","cited_arxiv_id":null,"evidence_quote":"Defines the HIO algorithm whose update is substituted for the pseudo-inverse in the Fourier phase retrieval setting."},{"cited_title":"Coherent inverse scattering via transmission matrices: Efficient phase retrieval algorithms and a public dataset,","cited_arxiv_id":null,"evidence_quote":"Provides the experimental scattering-media dataset and the estimated transmission matrix used in Section 5.2 to test the extension beyond classical Fourier PR."},{"cited_title":"Progressive growing of GANs for improved quality, stability, and variation,","cited_arxiv_id":null,"evidence_quote":"Supplies the simulated face-image test set used to compare HIO and DDRM-PR under four measurement noise levels."}],"review_version":1}