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REVIEW 4 major objections 6 minor 28 references

DeepCormack: Fermi surface tomography using model-based data-driven algorithms

T0 review · 4 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Augmenting the modified Cormack method with neural networks at three stages recovers Fermi-surface momentum densities from twenty times fewer annihilation counts, with an SVD/DMD generator supplying training data from one DFT reference.

desk verdict A well-executed learned reconstruction for a niche but important materials characterization problem; the headline speed-up is real on synthetic data but remains conditional until tested on real ACAR experiments. read the letter →

arxiv 2607.13107 v1 pith:SQC6RYRF submitted 2026-07-14 cond-mat.mtrl-sci cond-mat.str-elcs.LG

Georg F. B. Lovric , Bryn Drury , Carola-Bibiane Schönlieb , Stephen B. Dugdale , Ander Biguri This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.str-elcs.LG PACS 71.18.+y78.70.Bj
keywords Fermisurface2D-ACARtwo-photonmomentumdensitymodifiedCormackmethoddeeplearningreconstructiondynamicmodedecompositionsyntheticdatagenerationpositronannihilation
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

This paper claims that a learned, model-based upgrade to the standard modified Cormack reconstruction can make 2D-ACAR Fermi-surface measurements far faster without sacrificing reconstruction quality. In the synthetic and simulated experiments shown, the full DeepCormack stack at 10M counts per projection reconstructs momentum densities better than plain MCM at 200M counts, implying scan times could drop from months to weeks. Because real experimental training sets are impractical, the paper also introduces a data generator: it decomposes a single DFT copper momentum density into Chebyshev coefficients, samples new central slices with truncated SVD, and evolves them into full 3D volumes using dynamic mode decomposition. The key caveat stated by the authors is distribution dependence—copper-trained models degrade sharply on out-of-distribution ZrZn2 data—so the recommended use is sample-specific training from a DFT calculation of the target material.

What carries the argument

The engine is the modified Cormack method (MCM) itself, written as an operator chain O Z W S^-1 P^-1 C: C interpolates projections onto polar coordinates, P is a symmetry-truncated Fourier transform keeping only harmonics multiple of the crystal's rotational order (4 for FCC), S expands into Chebyshev sine series, W weights the coefficients, Z builds Zernike polynomials, and O recombines radial density functions into the two-photon momentum density. DeepCormack inserts learned components around this operator—a 1D CNN before the MCM, an MLP after the radial-density step, and a UNet on the final Euclidean image, optionally conditioned on the log count level to signal noise. The other load-bear

What would settle it

Run DeepCormack, trained on a copper-derived DMD dataset, on real 10M-count 2D-ACAR projections of a material whose Fermi surface is known from other methods and whose DFT reference was not used in training; if the reconstruction does not match or beat MCM on 200M-count data at the level claimed here, the method's practical value fails. A cheaper falsification: expand the paper's own out-of-distribution test to several materials with multiple Fermi-surface sheets and check whether the degradation seen for ZrZn2 is systematic.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the modified Cormack method can be wrapped in a supervised learning stack—a 1D CNN that denoises the measured projections, an MLP that refines the radial density functions, and a UNet that corrects the reconstructed TPMD image—and that the combined model remains accurate at count levels an order of magnitude below standard practice. The discovery that makes this trainable is the SVD/DMD synthetic-data generator: it turns one DFT-derived reference volume into arbitrarily many paired ground-truth / degraded-measurement training examples. The authors report 40.69 dB PSNR for the best configuration on synthetic test data at 200M counts versus 32.38 dB

Load-bearing premise

The headline gains rest on the assumption that the synthetic training distribution—built from one copper DFT reference plus a hand-calibrated noise model (+60% convolution width, 40% of true counts matched to one ZrZn2 experiment)—is representative enough of the target material's true momentum density and measurement noise that the learned reconstruction transfers to real data.

Editorial extensions

If this is right

  • A 10M-count per projection ACAR scan is sufficient for DeepCormack to reconstruct a momentum density comparable to or better than MCM at 200M counts, so acquisition can be cut by roughly an order of magnitude while keeping or improving quality.
  • Even at standard 200M counts, DeepCormack improves PSNR by about 8.5 dB on in-distribution synthetic data, so Fermi-surface features—especially at high momentum away from the slice centre—should be recovered more faithfully.
  • Because the synthetic training data comes from a DFT reference, the practical workflow becomes: run a DFT calculation of the target material (about a day), generate training data, then collect a shorter experimental measurement; the paper recommends this sample-specific pairing rather than a generic pretrained model.
  • With shorter scans, collecting more than 5 projections at proportionally lower counts becomes viable; the paper argues this richer angular sampling could further improve reconstruction quality, benefiting configurations that exploit multiple input channels.
  • The operator formulation of MCM links ACAR tomography to the standard inverse-problems literature, so improvements from that community (e.g., learned regularisation, uncertainty quantification) can be applied directly to Fermi surface reconstruction.

Reading between the lines

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

  • If the speed-up transfers to real samples, 2D-ACAR could shift from a months-long, one-material measurement to a screening tool for a series of alloys or dopings: the per-sample overhead is reduced to one DFT reference and a shorter scan.
  • The SVD/DMD generator is a general recipe: any tomography problem with a known reference volume and a smooth slice-to-slice evolution could use the same two-step sampling-plus-evolution scheme to generate training data, not just Fermi surface reconstruction.
  • A directly testable extension is to train on multiple DFT references simultaneously and measure out-of-distribution robustness; the paper's own OOD results imply generalization should improve with diversity, but they do not test it.
  • The choice of PSNR/SSIM as evaluation metrics may understate or misstate the real goal (Fermi surface geometry); retraining or at least evaluation with a Fermi-surface-aware metric such as gradient of the LCW occupation could change the relative ranking of configurations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes DeepCormack, a family of learned reconstruction modules (1D CNN, MLP, 2D UNet, optionally FiLM-conditioned) inserted at different stages of the modified Cormack method (MCM) for 2D-ACAR Fermi surface tomography. To obtain training data without months of measurements, the authors generate synthetic 3D TPMD volumes from a single DFT copper reference: SVD sampling of Chebyshev coefficients produces central slices, and DMD/Koopman evolution produces slice-to-slice variation; measurement effects (detector blur, momentum sampling function, Poisson noise) are then simulated. The method is evaluated on synthetic DMD test data, on the copper reference, and on out-of-distribution ZrZn2 data, at count levels from 10M to 200M, using PSNR/SSIM and qualitative Fermi-surface images. The headline result is that the best combined model DCCMU reaches 38.22 dB PSNR at 10M counts on the DMD test set, exceeding MCM at 200M (32.38 dB), a gain of about 8.5 dB at 200M. However, on ZrZn2 the same configuration falls below MCM (32.51–32.82 dB vs. 33.96 dB at 200M), and the paper recommends per-sample DFT training rather than demonstrating transfer to experimental data.

Significance. If the reported gains transferred to real ACAR measurements, DeepCormack could substantially reduce acquisition times (weeks to months) or improve reconstruction quality for Fermi-surface studies. The operator-based reformulation of MCM and the SVD/DMD data-generation pipeline from a single DFT reference are useful and potentially transferable contributions. The paper is also commendably honest: it reports all configurations, including failures on out-of-distribution ZrZn2, and explicitly discusses the distribution-matching limitation. That said, the central practical claim is currently supported only by in-distribution synthetic experiments with a hand-calibrated noise model; the generalization evidence is negative for the flagship configuration on the only truly OOD sample.

major comments (4)
  1. [§3.3.3, Tables 1 and 4] The central speed-up claim is contradicted by the paper's own OOD experiment. In Table 4 (ZrZn2), DCCMU at 10M gives 32.32 dB PSNR, below MCM at 200M (33.96 dB), and at 200M DCCMU gives 32.82 dB vs. 33.96 dB for MCM. The abstract's statements that DeepCormack 'remains stable at reduced counts' and 'enables significantly faster acquisition times' hold for the in-distribution DMD set and for Cu, but not for a sample outside the training distribution. Since real ACAR materials are necessarily outside any training set unless a per-sample DFT is computed—a workflow recommended but not tested here—the practical claim needs to be rephrased as conditional and supported by real-data validation.
  2. [§3.1, noise simulator calibration] The noise simulator is calibrated by hand to a single ZrZn2 experiment using +60% convolution widths and 40% effective counts. No second experiment or independent material is used to check whether these parameters generalize; every training sample and every test measurement (DMD, Cu, ZrZn2) is passed through this same simulator. The learned models may therefore be fitting simulator-specific artifacts (e.g., the particular MSF, blur, and effective count rescaling) rather than the physics of ACAR. A real-data reconstruction, or at minimum a held-out experimental noise calibration, is needed before the reported dB gains can be translated into expected experimental performance.
  3. [§2.3 and §3.2] The training/test protocol makes the headline result in-distribution by construction. Section 2.3 generates all synthetic volumes by SVD/DMD from one DFT copper reference, and the DMD test set is drawn from the same generative pipeline. The copper test is the same reference density used to construct the SVD/DMD models. Hence the +8.5 dB PSNR gain over MCM at 200M (Table 2) demonstrates that the networks reproduce the generative model's manifold, not that they generalize to unseen physics. The ZrZn2 experiment is the only OOD test, and there the flagship DCCMU falls below MCM; this should be stated prominently in the abstract and conclusion.
  4. [§4, evaluation metrics] The quantitative evaluation is in p-space TPMD using PSNR/SSIM, not the downstream Fermi surface in k-space, as acknowledged in §4. Because the LCW step and gradient extraction are nonlinear, a 8.5 dB PSNR gain does not guarantee a proportionally improved Fermi surface. Qualitative Fermi-surface figures are shown, but no quantitative downstream metric is provided. Adding a metric defined on the recovered Fermi surface (e.g., error in high-symmetry plane contours or extracted Fermi-surface features) would materially strengthen the claim that DeepCormack improves Fermiology, not only image quality.
minor comments (6)
  1. [Tables 1 and 4] Tables 1 and 4 report different PSNR values for the same DCCMU/ZrZn2/200M condition (32.51 dB vs. 32.82 dB). Please clarify the evaluation protocol (e.g., slice range, test realization) or explain the discrepancy.
  2. [Figure captions and Discussion] Figure captions 2–4 contain 'Workflow of the The...' typos; the Discussion contains 'wile' (for 'while') and 'desiged' (for 'designed'). A copyedit pass is needed.
  3. [§2.3] The text says 20 ideal projections in [0°,45°] are used to approximate the ground truth copper density, while the experimental pipeline uses 5 projections. Clarify that the 20 projections are used only for SVD/DMD coefficient extraction and not for the learned reconstruction experiments.
  4. [§3.1 and Figure 6] Figure 6 uses 165M counts for the experimental comparison, while Tables 1–4 use 200M. State whether 165M is the experimental total and how the 40% effective-count factor is applied to the simulated data at each count level.
  5. [Tables 3 and 4 captions] The captions for Tables 3 and 4 both refer to 'Figure 9' for the PSNR/SSIM plots; the ZrZn2 results appear to be shown in Figure 11. Check all cross-references.
  6. [Notation, §3.2] The configuration names such as 'CNNPT + MLPPT → UNet' are difficult to parse. Define the arrow notation and the acronyms (e.g., 'PT' = pre-trained) once at first use, and consider a small table mapping configuration names to pipeline stages.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported gains are in-distribution synthetic evaluations, and the paper explicitly conditions real-data generalization on distribution match.

full rationale

The derivation chain is not circular. The MCM is written in operator notation, the DeepCormack components are trained on synthetic paired data generated by SVD/DMD from a copper DFT reference, and the test results are empirical comparisons under the same simulation pipeline. No equation defining the reconstruction is equivalent to the fitted network output by construction, and no fitted parameter is renamed as a prediction. The DMD test set being in the same distribution as the training data is a limitation of external validity, not circularity: the paper states this explicitly ('these results are obtained on synthetic DMD data drawn from the same distribution as the training data, which naturally favours all models evaluated here') and also states that real experimental data will differ and that generalization depends on the training distribution matching the sample. The noise-model calibration to one ZrZn2 experiment (+60% convolution, 40% counts) is a hand-adjusted simulation parameter, but it is used to generate synthetic measurements for comparison, not to claim an experimental prediction; the paper identifies the OOD performance drop of DCCMU, which further shows the evaluation is not forced. Self-citations such as the DFT reference [1] and the ZrZn2 experiment [24] supply reference data and context, but they are not invoked as a load-bearing uniqueness theorem or as evidence for the learned gain. The comparison against MCM under the same simulated conditions is a fair empirical benchmark, and the paper's main claims are appropriately scoped to synthetic test data with an explicit recommendation for sample-specific training. No circular step meeting the quoted-evidence threshold is present.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a data-generation pipeline with several hand-tuned parameters (SVD/DMD truncation levels, noise-simulator calibration, CNN loss weight). The physical forward model and symmetry assumptions are standard for ACAR, but the linear Koopman evolution and the representative-ness of a single copper reference are assumptions introduced for this work.

free parameters (5)
  • SVD truncation K for central slices = K=15
    Section 2.3.1: 'K=15 ... was found to construct reasonable TPMD features, capturing ~88% of the total variance'. Hand-chosen; affects diversity and quality of synthetic training data.
  • DMD truncation K = K=24
    Section 2.3.2: 'K=24 ... was found to give a good trade-off between diversity and stable volume evolution'. Hand-chosen.
  • Noise simulator calibration = +60% convolution widths, 40% of true counts
    Section 3.1: simulated data matched experimental degradation only after increasing convolution by 60% and using 40% of the counts; this calibration directly shapes the training data.
  • CNN loss weighting alpha = alpha=10
    Section 2.2.1: 'We found alpha=10 empirically the best value.' Tuned on validation.
  • SVD latent Gaussian parameters mu_k, Lambda_k = fitted to copper slices
    Section 2.3.1: a multivariate Gaussian is fitted to the latent coordinates of copper Chebyshev coefficients; the generative model depends on this fit.
assumptions (6)
  • domain assumption ACAR forward model is a Radon transform with additive noise and slice-by-slice independence
    Equation (1) and Section 2.1: the entire pipeline assumes the measurement is well described by R̄ρ2γ = f̄ + ē, with detector smearing, Poisson noise, and MSF.
  • domain assumption Crystal symmetry restricts Fourier expansion to harmonics n ≡ 0 mod |G|
    Section 2.1.1, Eq. (3): assumes |G|=4 (FCC) for all samples studied.
  • standard math Zernike/Chebyshev expansion truncated at finite order converges sufficiently for real TPMDs
    Used throughout MCM derivation; no convergence analysis is provided.
  • domain assumption DFT-calculated copper momentum density is a reliable reference ground truth for generating training data
    Section 2.3: all synthetic volumes are derived from a single DFT copper density (Ref [1]).
  • ad hoc to paper DMD/Koopman linear-evolution assumption captures slice-to-slice TPMD variation
    Section 2.3.2: assumes z_{l+1}=K z_l in latent space to extrapolate central slices to full volumes; not derived from physics.
  • ad hoc to paper Calibrated noise model generalizes to future experiments
    Section 3.1: parameters adjusted to match one ZrZn2 dataset; the paper assumes this reproduces real noise for other samples.

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Cite this review

Pith. "Pith review of DeepCormack: Fermi surface tomography using model-based data-driven algorithms." pith.science (2026). https://pith.science/paper/SQC6RYRF

@misc{pith2026260713107,
  author       = {Pith},
  title        = {Pith review of: DeepCormack: Fermi surface tomography using model-based data-driven algorithms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQC6RYRF}},
  note         = {Machine review of arXiv:2607.13107}
}
read the original abstract

The experimental reconstruction of the 3D two-photon momentum density (TPMD) via angular correlation of electron-positron annihilation radiation (ACAR) is a particularly useful method for studying material Fermi surfaces. It does not rely on low temperatures, UHV conditions, or strong magnetic fields, and enables the study of the spin-resolved electronic structure of materials. Yet, it remains a challenging inverse problem. Typically, 10^8 positron annihilation events are measured for 3--6 projections of the TPMD at different angles. The standard reconstruction approach is an ACAR adaptation of Cormack's method (the MCM) that leverages the inherent symmetry in the crystal's structure. However, the poor signal-to-noise ratio means collecting data of sufficient quality for Fermi surface studies can take months per sample. We present DeepCormack, a family of data-driven model-based reconstruction algorithms that augments the MCM by integrating supervised deep-learning models (CNN, MLP, and UNet) at various stages. To overcome the lack of large experimental training sets, we propose a method which leverages singular value decomposition with dynamic mode decomposition to generate realistic synthetic TPMD volumes, requiring only a single reference momentum density computed via density functional theory. On test data, DeepCormack improves reconstruction quality over MCM by about 8.5 dB PSNR at 200M counts and remains stable at reduced counts, enabling significantly faster acquisition times. Generalisation to experimental data depends strongly on how well the training distribution from the reference momentum density matches the sample. We therefore recommend pairing DeepCormack with a DFT calculation of the target material to create sample-specific training data. Our proposed method offers either much higher quality reconstructions, or enables significantly faster ones, on the order of weeks.

Figures

Figures reproduced from arXiv: 2607.13107 by the authors.

Figure 1
Figure 1. Workflow of the DeepCormack Method (10M Counts), illustrating which stages of the Modified Cormack Method (green) neural networks are added (blue). This section describes our proposed data-driven method, enhancing the MCM algorithm with deep neural networks. In general, DeepCormack integrates up to three neural networks at different key stages of the modified Cormack method, as shown in [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 2
Figure 2. Workflow of the The 1D CNN. The first stage, shown in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Workflow of the The MLP. The second stage employs a fully connected multilayer perceptron (MLP) to refine the radial density function representation of the CNN output, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Workflow of the The UNet. The third stage is a UNet architecture ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Step-wise simulation of experimental conditions on projections through the ZrZn2 momentum density. The resulting image degradation from the modified Cormack method is shown for a slice through the centre and 1.1 au outwards. From (i) the ground truth, the procedure inc…
Figure 6
Figure 6. Figure 6: Gradient of the ZrZn2 3D TPMD reconstruction vs counts (165M, 100M, 10M), comparing the experimental measurement with the simulated measurement through the corresponding slice. The simulated measurement data used 60% greater convolution and 40% of the true counts (i.e.…
Figure 7
Figure 7. Figure 7: The steps used for recovering the copper Fermi surface from (i) ACAR projections at 200M counts, (ii) using the modified Cormack or DeepCormack methods to recover ρ 2γ volume, then applying (iii) the LCW theorem to convert the 3D TPMD from p-space to k-space, where (iv…
Figure 8
Figure 8. Figure 8: The PSNR (i) and SSIM (ii) of the DCC, DCM, DCU, and DCCMU models against the modified Cormack method on the synthetic TPMD data at varying counts. DCCMU outperforms all models at every count level on both metrics. Most notably, its reconstruc￾tion at 10M counts (38.22…
Figure 9
Figure 9. Figure 9: The PSNR (i) and SSIM (ii) of the DCC, DCM, DCU, and DCCMU models against the modified Cormack method on the reconstruction from Cu data with simulated measurement conditions. On the Cu data, the DCU is the strongest performer across all count levels, reaching 40.05 PS…
Figure 10
Figure 10. Figure 10: The Cu Fermi surface as reconstructed by the DCU, DCCMU, and MCM compared against the ground truth at (from top to bottom) 200M, 100M, and 10M counts under simulated measurement conditions. Columns showcase slices through different axis of the 3D Fermi surface. 18 [P…
Figure 11
Figure 11. Figure 11: The PSNR (i) and SSIM (ii) of the DCC, DCM, DCU, and DCCMU models against the modified Cormack method on the reconstruction from ZrZn2 data. DCCMUs performance is significantly lower on the ZrZn2 dataset, achieving only 32.82 PSNR at 200M counts, which is below the MC…
Figure 12
Figure 12. Figure 12: The ZrZn2 Fermi surface as reconstructed by the DCU, DCCMU, and MCM compared against the ground truth at 200M, 100M, and 10M counts under simulated measurement conditions [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Comparison of the ZrZn2 Fermi surface reconstruction for DCU with and without FiLM conditioning on the noise. The ZrZn2 simulated measurement data was passed through the models at 200M counts 20 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: The final step of the MCM reconstruction process for copper, which has four-fold symmetry. The radial density functions ρn(p) (here shown normalized) are multiplied by their corresponding cosine modulation term cos(nθ) around a unit circle. The final column shows the …
Figure 15
Figure 15. Figure 15: The difference between the ground truth Fermi surface for copper and its DMD evolved counterpart: taking the Koopman operators computed from the GT copper momentum density to evolve its GT central slice into a 3D TPMD. This is shown through the high-symmetry planes (1…
Figure 16
Figure 16. Figure 16: Comparing synthetic TPMD slices continuously evolved via Dynamic Mode Decomposition. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_16.png]
Figure 17
Figure 17. Figure 17: ZrZn2 3D TPMD reconstruction at varying counts (165M, 100M, 10M), comparing the experimental measurement with the simulated experimental measurement through the corresponding slice. The simulated experimental data used 60% greater convolution and 40% of the true count…

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.