Neural Vector Tomography for Reconstructing a Magnetization Vector Field

Giorgi Butbaia; Jiadong Zang

REVIEW 5 major objections 5 minor 31 references

Neural Vector Tomography for Reconstructing a Magnetization Vector Field

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper argues that representing a magnetization vector field as a smooth neural field, rather than as a discretized voxel array, gives tomographic reconstructions that remain accurate under measurement noise, and that encoding a known…

desk verdict Strong idea, but the quantitative evidence is compromised by the gradient ambiguity and an impossible SSIM; the method warrants peer review after major fixes. read the letter →

arxiv 2412.09927 v1 pith:YBMXE26T submitted 2024-12-13 cond-mat.dis-nn cs.CV

Giorgi Butbaia , Jiadong Zang This is my paper · ORCID

classification cond-mat.dis-nncs.CV

keywords vectortomographymagnetizationneuralfieldsraytransformSO(2)-equivariancetomographicreconstructionnoiserobustness

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

Reconstructing a three-dimensional magnetization vector field from X-ray or electron projections is partially ill-posed: with a single probe, fields that differ by the gradient of a scalar function produce identical measurements. The paper proposes to represent the unknown field as a neural network with smooth activations and to train it against ray-integrated projections, adding a gradient penalty to suppress noise. On a simulated Bloch-point phantom, this neural-field method keeps reconstruction error low and image similarity high as Gaussian noise grows, while a voxel-discretized iterative baseline degrades quickly. For a Hopfion sample with rotational symmetry, a network built to be rotation-equivariant reconstructs more accurately than a non-equivariant network. If the approach carries to experiments, it offers a route to noise-robust three-dimensional magnetic imaging without high-resolution memory costs.

What carries the argument

The load-bearing object is the neural field $F_\theta$, a multilayer perceptron with smooth activations mapping $(x,y,z)$ to a vector in $\mathbb{R}^3$, trained with the loss $L_\alpha = L_{\mathrm{rec}} + \alpha L_{\mathrm{reg}}$. Here $L_{\mathrm{rec}}$ approximates the probed ray transform by stratified Monte Carlo sampling along each ray, and $L_{\mathrm{reg}}$ penalizes the field's gradient at uniformly sampled points to suppress noise-induced irregularities. For symmetric samples, the architecture enforces $SO(2)$-equivariance by rotating the network output back to the local frame, so the symmetry is exact by construction rather than learned. The theory framing the problem is Theorem 1: single-probe measurements cannot distinguish fields differing by a gradient.

What would settle it

Compute the MSE and SSIM after Helmholtz-decomposing both the reconstruction and the ground truth into solenoidal and irrotational parts on the Bloch-point phantom, and also add a large smooth gradient field to the ground truth, which leaves all single-probe projections unchanged; if the neural method's advantage disappears when only the solenoidal components are compared, the central claim of improved physical reconstruction is refuted.

Watch

Extended reading notes

Core claim

The central claim is that modeling the field as a continuous neural function $F_\theta$ and solving $\min_\theta \frac{1}{N}\sum_i |R_p(F_\theta)(L_i)-R_p(F)(L_i)|^2 + \alpha \sum_j |\nabla F_\theta(x_j)|^2$ removes the artifacts that voxel discretization introduces, and that the error advantage grows with noise. The paper reports lower mean-squared error and higher structural similarity for the neural method than for the discretized baseline at all tested noise levels, and shows that stitching the field's $SO(2)$ symmetry into the architecture reduces MSE from 0.39 to 0.14 for a Hopfion. Because $\ker R_p \simeq \operatorname{im}\nabla$, only the solenoidal part is determined by the measurements; the reconstruction is therefore understood as recovering the field up to a gradient.

Load-bearing premise

The reported MSE and SSIM gains assume that the neural network does not introduce a large spurious gradient component, even though the single-probe measurements used here cannot constrain that component at all.

Editorial extensions

If this is right

At every tested noise level from $\sigma=0$ to $\sigma=7.7$, the neural-field reconstruction achieves lower MSE and higher SSIM than the discretized baseline, with the gap growing as noise increases.
Because the network maps coordinates directly to field values, reconstructions can be evaluated at arbitrary spatial resolution without increasing memory usage, only training time.
For $SO(2)$-symmetric samples, an equivariant network gives MSE 0.14 versus 0.39 for a non-equivariant network and higher SSIM.
Smooth activations together with the gradient regularization term suppress noise-induced artifacts, so the method remains stable even when the projections are strongly corrupted.

Reading between the lines

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

If the unrecoverable gradient component stays small in practice, the same smoothness prior should carry over to experimental XMCD or Lorentz TEM data, where noise and missing-wedge artifacts dominate; this could be tested by running the same loss on experimental projections.
A fairer benchmark would compare only the solenoidal parts of reconstruction and ground truth, because single-probe measurements cannot determine the irrotational part; until that is reported, the quantitative gains over the baseline may partly reflect how each method handles an invisible component.
The equivariance trick is not limited to $SO(2)$: discrete symmetries can be enforced by group averaging, and more realistic sample symmetries such as mirror planes or dihedral rotations could be encoded the same way.
The same neural-field formulation applies to any inner-product-probe vector tomography problem, including electron holography and neutron imaging, whenever the unknown field is smooth.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Colleague,

The paper applies neural fields to vector tomography of magnetization, which is a reasonable transfer of the NeRF toolkit to a real problem. The SO(2)-equivariant architecture for the Hopfion case is a nice touch, and the qualitative noise robustness in Fig. 2 is genuinely suggestive.

But the quantitative claims do not hold up. The authors state Theorem 1: only the solenoidal component is recoverable from these probed ray transforms. Then they evaluate full-field MSE and SSIM against the ground truth. That measures an unidentifiable component, and the neural network's gradient penalty (Eq. 13) will suppress the irrotational part that any method cannot recover. So the numerical advantage they report could just reflect the network's bias on an invisible subspace, not better estimation of what is actually identifiable. The stress-test note is correct. They need to report errors on the solenoidal component (projected onto the kernel) or at least use projection-domain residuals.

Second, Table I reports SSIM values of 2.38 and 2.79. SSIM is capped at 1. That is not a debatable choice; it is a sign of a computational bug. This alone should have been caught before submission.

Third, the experimental section is under-specified. No hyperparameters (alpha, network depth/width, learning rate, sampling count), no error bars, no code release, and a single baseline that is modified in footnote 1 (the scalar part replaced by the ground-truth mask). That baseline modification might make the comparison more favorable to the proposed method, but the lack of details and the single dataset make the robustness claim unsupported.

The method itself is plausible; the idea of a smooth neural field for vector tomography is not new in structure (NEAT, NeRF), but the probed ray-transform formulation and the equivariant variant are sensible. The paper is worth engaging with—if the authors fix the evaluation, correct the SSIM, and release code and settings, it could become a useful tool for 3D nanomagnetism.

For now, I would not cite it in this form. The intended audience is the computational imaging / 3D magnetism community, and they would get value from the architectural ideas, but only after the evidence is rebuilt.

Regarding peer review: send it out. The core idea is not dead, and the flaws are fixable in revision, not fatal. But the referee should hold it to a high bar on the evaluation metrics.

Referee Report

5 major / 5 minor

Summary. The paper proposes a neural-field-based method for vector tomography of magnetization vector fields, using a network F_theta to represent the field and a reconstruction loss (Eq. 12) plus a gradient-smoothness regularizer (Eq. 13). The authors argue that smooth neural representations avoid discretization artifacts and remain robust under measurement noise, and they introduce an SO(2)-equivariant architecture for fields with continuous symmetry. Numerical experiments compare the method against a discretized iterative baseline on simulated XMCD-type projections of two Bloch-point and Hopfion fields, reporting MSE and SSIM over a range of noise levels. The central quantitative claim is that the neural method substantially improves reconstruction accuracy, especially under symmetry.

Significance. If the claims are supported, the paper would offer a useful application of implicit neural representations to vector-field tomography, with a plausible mechanism (smoothness bias) for noise robustness and a well-motivated equivariant construction. The use of neural fields to avoid explicit voxelization and the deterministic ray-sampling strategy are sensible and align with recent NeRF-style tomographic methods. However, the experimental evidence as presented is not yet sufficient to support the central claim: the evaluation metric is mismatched to the identifiability limitation established by the paper's own Theorem 1, and the reported SSIM values are internally inconsistent. The paper also lacks the hyperparameter and implementation details needed for reproducibility. The core idea is viable, but the quantitative comparison must be redone.

major comments (5)

[Section IV, Theorem 1] The quantitative evaluation is mismatched to the identifiability result. Theorem 1 states that the single-probe ray transform Rp has kernel im ∇, so measurements do not distinguish F from F + ∇φ. Section IV nevertheless computes MSE and SSIM against the full ground-truth vector field (Fig. 5, Table I). Any reconstruction will incur an unavoidable error on the irrotational component, and the relative performance of the neural method and the discretized baseline on that unidentifiable subspace is driven by their priors: the gradient penalty in Eq. (13) explicitly suppresses ∇φ for the neural model, while the discretized baseline has no such bias. The reported superiority may therefore reflect behavior on the kernel, not improved recovery of the solenoidal component that the data actually determine. The authors should quantify the irrotational content of their phantoms and report errors after projecting both the reconstruction and ground truth onto the solenoidal subspace (for example, comparing curl-field errors) or, at minimum, report projection-domain residuals. Without this, the abstract's claim of 'substantially improves the accuracy' is not supported as stated.
[Table I] The SSIM values 2.38 and 2.79 in Table I exceed the maximum possible value of 1 for the structural similarity index, which indicates a computational or reporting error in the Hopfion comparison. Since this table is the only quantitative evidence for the SO(2)-equivariant advantage, the result cannot be assessed until the metric is corrected and recomputed.
[Section IV, footnote 1] The baseline comparison is not a faithful implementation of the method in [2]: the scalar part is replaced with the ground-truth mask, so only the vector part is optimized, while no equivalent ground-truth information is given to the neural method. The reported metrics are computed on the full vector field, so it is unclear what advantage or disadvantage this hybrid baseline confers. The authors should either run the complete method from [2] on the same data, or explicitly state and justify the modified protocol and its effect on the comparison.
[Section III-IV] The manuscript provides no network architecture details, training hyperparameters (optimizer, learning rate, number of iterations, batch size), the regularization weight α in Eq. (14), the stratified sample count M in Eq. (16), or the number of random seeds. Without these, the numerical results cannot be reproduced, and the claimed robustness under noise cannot be independently verified. This is a load-bearing omission for a methods paper.
[Section IV, Fig. 5] Figure 5 appears to show a single noise realization per σ level, with no error bars or multiple trials. The visual difference between methods at high noise could be due to a single favorable initialization of the neural network. Multiple runs with different seeds and noise realizations are needed to establish that the observed stability is a property of the method rather than a random fluctuation.

minor comments (5)

[Section III-C] In the symmetry paragraph, 'g · F = g' should read 'g · F = F' (typo).
[Section III-C] The sentence 'It is easy to see that a neural network constructed using (19) indeed satisfies is indeed SO(2)-equivariant' has a duplicated verb and should be corrected.
[Section II, Theorem 1] The statement 'ker Rp ≃ im ∇' would benefit from a precise domain specification, since the kernel of the probed ray transform on compactly supported vector fields consists of gradients of compactly supported potentials; the current wording '≃' leaves this implicit.
[Section IV, Fig. 5] The y-axis label '100' in both subplots is ambiguous; presumably it means 10^0, but the SSIM panel must also be checked because SSIM is bounded above by 1. The axis should be labeled clearly and the plotted values verified.
[References] Several references lack complete bibliographic information (e.g., [5], [6], [8], [10], [11], [12], [16] have missing volume/page or journal fields). The citation of [16] as a bare arXiv preprint should be updated if a published version exists.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; neural reconstruction is benchmarked against an external discretized baseline. Only minor, non-load-bearing self-citations appear.

full rationale

The central method solves the optimization problem (11)-(14): a neural field F_theta is trained to fit probed ray-transform measurements R_P(F)(L_i), with a gradient penalty. The resulting reconstruction is compared against a discretized baseline from Donnelly et al. [2] using MSE/SSIM on the ground-truth field (Fig. 5). This is an external benchmark, not a self-referential definition: the target quantity is not constructed from the fitted parameters, and no parameter is renamed as a prediction. Theorem 1 (kernel = im grad, cited from Ilmavirta and Mönkkönen [26]) is an external mathematical result, not imported from the authors' prior work. The paper explicitly acknowledges that only the solenoidal component is recoverable, yet still reports full-field errors; this is an identifiability/evaluation limitation, not circularity, because the ground-truth field is independent of the reconstruction algorithm and the error metric is not used to define the loss. The self-citations to MagNet [16] and the physical-constraint reconstruction paper [13] are related-work or phantom references and do not carry any load-bearing assumption of the derivation chain. The SO(2)-equivariant construction (19) is an architectural constraint, and its reported advantage (Table I) is an empirical comparison between two networks, not a proof that relies on its own conclusion.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. The free parameters are all experimental or training choices that are not reported; the axioms are standard tomographic results plus the physical forward-model and capacity assumptions of the neural field.

free parameters (4)

Regularization weight alpha = not reported
Controls the strength of the smoothness penalty in Eq. (14); chosen by hand, no sensitivity analysis or selection procedure described.
Network architecture (depth, width, activation) = not reported
The neural network is described only as 'neural network'; no layer count, hidden width, or activation is given, so reproducibility depends on undisclosed choices.
Stratified sampling count M = not reported
Number of point samples per ray in Eq. (16); affects integral accuracy but is not specified.
Training hyperparameters (optimizer, learning rate, iterations) = not reported
Not specified; training time and convergence are mentioned but no values are provided.

assumptions (5)

standard math The scalar ray transform is invertible on C^infinity_c(R^n).
Used in Section II to justify vector ray transform invertibility when enough probes are available; cited to [24].
standard math For a single probe along the ray direction, ker Rp = im grad (Theorem 1).
Invoked in Section II and IV to state that only the solenoidal component can be reconstructed; cited to [26].
domain assumption XMCD projections are modeled by the probed ray transform with probe p(L) = tau_L (Eq. 8).
Section IV states that X-ray projections correspond to the probed measurements with this probe; this physical modeling assumption connects the math to experiment.
ad hoc to paper The neural network with smooth activations can approximate the true magnetization field well enough, and the gradient-penalty regularization biases the solution toward the true field rather than toward a spurious gradient component.
Section III introduces Lreg to suppress noise-induced irregularities; the paper does not prove that this bias does not distort the solenoidal component, nor does it analyze the unrecoverable gradient part.
domain assumption The simulated measurements are exactly the probed ray transform of a known ground-truth field plus i.i.d. Gaussian noise with variance sigma^2.
Section IV uses Eq. (21) to generate noisy measurements; all conclusions about robustness depend on this synthetic noise model.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Neural Vector Tomography for Reconstructing a Magnetization Vector Field." pith.science (2026). https://pith.science/paper/YBMXE26T

@misc{pith2026241209927,
  author       = {Pith},
  title        = {Pith review of: Neural Vector Tomography for Reconstructing a Magnetization Vector Field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBMXE26T}},
  note         = {Machine review of arXiv:2412.09927}
}

read the original abstract

Discretized techniques for vector tomographic reconstructions are prone to producing artifacts in the reconstructions. The quality of these reconstructions may further deteriorate as the amount of noise increases. In this work, we instead model the underlying vector fields using smooth neural fields. Owing to the fact that the activation functions in the neural network may be chosen to be smooth and the domain is no longer pixelated, the model results in high-quality reconstructions, even under presence of noise. In the case where we have underlying global continuous symmetry, we find that the neural network substantially improves the accuracy of the reconstruction over the existing techniques.

Figures

Figures reproduced from arXiv: 2412.09927 by the authors.

**Figure 2.** Slices of reconstructions at different levels of noise [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Comparison of the reconstructed and ground truth [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Comparison of (a) isosurfaces Mz = 0 and (b) slices of reconstructions using different reconstruction techniques. based reconstructions. This can be understood by following observation: in discretized domain, there are no additional conditions favoring a continuous app…

**Figure 5.** Figure 5: SSIM and MSE metrics of the reconstructions using [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

31 extracted references · 27 canonical work pages

[2]

Tomographic reconstruction of a three-dimensional magnetization vector field,

C. Donnelly, S. Gliga, V . Scagnoli, M. Holler, J. Raabe, L. J. Heyderman, and M. Guizar-Sicairos, “Tomographic reconstruction of a three-dimensional magnetization vector field,” New Journal of Physics, vol. 20, no. 8, p. 083009, aug 2018. [Online]. Available: https://dx.doi.org/10.1088/1367-2630/aad35a

doi:10.1088/1367-2630/aad35a 2018
[1]

Three-dimensional nanomagnetism,

A. Fern ´andez-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, and R. P. Cowburn, “Three-dimensional nanomagnetism,” Nature Communications, vol. 8, no. 1, p. 15756, Jun. 2017, publisher: Nature Publishing Group. [Online]. Available: https://www.nature.com/articles/ ncomms15756

work page 2017
[3]

Imaging three-dimensional magnetic systems with x-rays,

C. Donnelly and V . Scagnoli, “Imaging three-dimensional magnetic systems with x-rays,” Journal of Physics: Condensed Matter , vol. 32, no. 21, p. 213001, Feb. 2020, publisher: IOP Publishing. [Online]. Available: https://dx.doi.org/10.1088/1361-648X/ab5e3c

work page doi:10.1088/1361-648x/ab5e3c 2020
[4]

Revealing 3D magnetization of thin films with soft X-ray tomography: magnetic singularities and topological charges,

A. Hierro-Rodriguez, C. Quir ´os, A. Sorrentino, L. M. Alvarez- Prado, J. I. Mart ´ın, J. M. Alameda, S. McVitie, E. Pereiro, M. V ´elez, and S. Ferrer, “Revealing 3D magnetization of thin films with soft X-ray tomography: magnetic singularities and topological charges,” Nature Communications , vol. 11, no. 1, p. 6382, Dec. 2020, publisher: Nature Publish...

work page 2020
[5]

Three-Dimensional Study of the Vector Potential of Magnetic Structures,

C. Phatak, A. K. Petford-Long, and M. De Graef, “Three-Dimensional Study of the Vector Potential of Magnetic Structures,” vol. 104, no. 25, p. 253901. [Online]. Available: https://link.aps.org/doi/10.1103/ PhysRevLett.104.253901

work page
[6]

Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,

T. Tanigaki, Y . Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” vol. 15, no. 2, pp. 1309–1314. [Online]. Available: https://doi.org/10.1021/nl504473a

work page doi:10.1021/nl504473a
[7]

Holographic vector field electron tomography of three- dimensional nanomagnets,

D. Wolf, N. Biziere, S. Sturm, D. Reyes, T. Wade, T. Niermann, J. Krehl, B. Warot-Fonrose, B. B ¨uchner, E. Snoeck, C. Gatel, and A. Lubk, “Holographic vector field electron tomography of three- dimensional nanomagnets,” vol. 2, no. 1, pp. 1–9. [Online]. Available: https://www.nature.com/articles/s42005-019-0187-8

work page
[8]

Unveiling the three-dimensional magnetic texture of skyrmion tubes,

A. Lubk, “Unveiling the three-dimensional magnetic texture of skyrmion tubes,” vol. 17, no. 3, pp. 250–255. [Online]. Available: https://www.nature.com/articles/s41565-021-01031-x

work page

Show all 31 references

[9]

Three-dimensional reconstruction of magnetic vector fields using electron-holographic interferometry,

G. Lai, T. Hirayama, A. Fukuhara, K. Ishizuka, T. Tanji, and A. Tonomura, “Three-dimensional reconstruction of magnetic vector fields using electron-holographic interferometry,” vol. 75, no. 9, pp. 4593–4598. [Online]. Available: https://doi.org/10.1063/1.355955

doi
[10]

Vector field electron tomography of magnetic materials: Theoretical development,

C. Phatak, M. Beleggia, and M. De Graef, “Vector field electron tomography of magnetic materials: Theoretical development,” vol. 108, no. 6, pp. 503–513. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/S0304399107001957
[11]

Iterative reconstruction of magnetic induction using Lorentz transmission electron tomography,

C. Phatak and D. G ¨ursoy, “Iterative reconstruction of magnetic induction using Lorentz transmission electron tomography,” vol. 150, pp. 54–64. [Online]. Available: https://www.sciencedirect.com/science/ article/pii/S0304399114002496
[12]

3D reconstruction of the magnetic vector potential using model based iterative reconstruction,

K. C. Prabhat, K. Aditya Mohan, C. Phatak, C. Bouman, and M. De Graef, “3D reconstruction of the magnetic vector potential using model based iterative reconstruction,” vol. 182, pp. 131–144. [Online]. Available: https://www.sciencedirect.com/science/article/pii/ S0304399117301948
[13]

Three-dimensional magnetization reconstruction from electron optical phase images with physical constraints,

B. Lyu, S. Zhao, Y . Zhang, W. Wang, F. Zheng, R. E. Dunin-Borkowski, J. Zang, and H. Du, “Three-dimensional magnetization reconstruction from electron optical phase images with physical constraints,” Science China Physics, Mechanics & Astronomy , vol. 67, no. 11, Sep. 2024. [...

2024 doi
[14]

Neat: Neural adaptive tomography,

D. R ¨uckert, Y . Wang, R. Li, R. Idoughi, and W. Heidrich, “Neat: Neural adaptive tomography,” ACM Transactions on Graphics (TOG) , vol. 41, no. 4, pp. 1–13, 2022

2022
[15]

Nerf: Representing scenes as neural radiance fields for view synthesis,

B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoorthi, and R. Ng, “Nerf: Representing scenes as neural radiance fields for view synthesis,” in ECCV, 2020

2020
[16]

B. Lyu, S. Zhao, Y . Zhang, W. Wang, H. Du, and J. Zang. MagNet: Machine learning enhanced three-dimensional magnetic reconstruction. [Online]. Available: http://arxiv.org/abs/2210.03066

arXiv
[17]

Inverse radon transform with deep learning: an application in cardiac motion correction,

H. Chang, V . Kobzarenko, and D. Mitra, “Inverse radon transform with deep learning: an application in cardiac motion correction,” Physics in Medicine & Biology , vol. 69, no. 3, p. 035010, Jan. 2024. [Online]. Available: http://dx.doi.org/10.1088/1361-6560/ad0eb5

2024 doi
[18]

Radon inversion via deep learning,

J. He, Y . Wang, and J. Ma, “Radon inversion via deep learning,” IEEE transactions on medical imaging , vol. 39, no. 6, pp. 2076–2087, 2020

2020
[19]

Strong Lensing Source Reconstruction Using Continuous Neural Fields,

S. Mishra-Sharma and G. Yang, “Strong Lensing Source Reconstruction Using Continuous Neural Fields,” in 39th International Conference on Machine Learning Conference , 6 2022

2022
[20]

Grav- itationally lensed black hole emission tomography,

A. Levis, P. P. Srinivasan, A. A. Chael, R. Ng, and K. L. Bouman, “Grav- itationally lensed black hole emission tomography,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 19 841–19 850

2022
[21]

V . A. Sharafutdinov, Integral Geometry of Tensor Fields . Berlin, New York: De Gruyter, 1994. [Online]. Available: https://doi.org/10.1515/ 9783110900095

1994
[22]

Vector field tomography, an overview,

G. Sparr, K. Str, and A. En, “Vector field tomography, an overview,”
[23]

Tensor tomography: progress and challenges,

G. P. Paternain, M. Salo, and G. Uhlmann, “Tensor tomography: progress and challenges,” arXiv preprint arXiv:1303.6114 , 2013

2013 arXiv
[24]

Deans, The Radon Transform and Some of Its Applications, ser

S. Deans, The Radon Transform and Some of Its Applications, ser. Dover Books on Mathematics Series. Dover Publications, 2007. [Online]. Available: https://books.google.com/books?id=xSCc0KGi0u0C

2007
[25]

Tomographic reconstruction of 3-d vector fields using inner product probes,

J. Prince, “Tomographic reconstruction of 3-d vector fields using inner product probes,” IEEE Transactions on Image Processing , vol. 3, no. 2, p. 216–219, Mar. 1994. [Online]. Available: http: //dx.doi.org/10.1109/83.277903

1994 doi
[26]

X-ray tomography of one- forms with partial data,

J. Ilmavirta and K. M ¨onkk¨onen, “X-ray tomography of one- forms with partial data,” SIAM Journal on Mathematical Analysis , vol. 53, no. 3, pp. 3002–3015, 2021. [Online]. Available: https: //doi.org/10.1137/20M1344779

2021 doi
[27]

Theoretical aspects of group equivariant neural networks,

C. Esteves, “Theoretical aspects of group equivariant neural networks,” arXiv preprint arXiv:2004.05154 , 2020

2004 arXiv
[28]

Group equivariant convolutional networks,

T. Cohen and M. Welling, “Group equivariant convolutional networks,” in International conference on machine learning . PMLR, 2016, pp. 2990–2999

2016
[29]

Scalars are universal: Equivariant machine learning, structured like classical physics,

S. Villar, D. W. Hogg, K. Storey-Fisher, W. Yao, and B. Blum-Smith, “Scalars are universal: Equivariant machine learning, structured like classical physics,” Advances in Neural Information Processing Systems , vol. 34, pp. 28 848–28 863, 2021

2021
[30]

Hopfion rings in a cubic chiral magnet,

F. Zheng, N. S. Kiselev, F. N. Rybakov, L. Yang, W. Shi, S. Bl ¨ugel, and R. E. Dunin-Borkowski, “Hopfion rings in a cubic chiral magnet,” Nature, vol. 623, no. 7988, p. 718–723, Nov. 2023. [Online]. Available: http://dx.doi.org/10.1038/s41586-023-06658-5

2023 doi
[1998]

Available: https://api.semanticscholar.org/CorpusID: 16235452

[Online]. Available: https://api.semanticscholar.org/CorpusID: 16235452

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [2]

Tomographic reconstruction of a three-dimensional magnetization vector field,

C. Donnelly, S. Gliga, V . Scagnoli, M. Holler, J. Raabe, L. J. Heyderman, and M. Guizar-Sicairos, “Tomographic reconstruction of a three-dimensional magnetization vector field,” New Journal of Physics, vol. 20, no. 8, p. 083009, aug 2018. [Online]. Available: https://dx.doi.org/10.1088/1367-2630/aad35a

doi:10.1088/1367-2630/aad35a 2018

[2] [1]

Three-dimensional nanomagnetism,

A. Fern ´andez-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, and R. P. Cowburn, “Three-dimensional nanomagnetism,” Nature Communications, vol. 8, no. 1, p. 15756, Jun. 2017, publisher: Nature Publishing Group. [Online]. Available: https://www.nature.com/articles/ ncomms15756

work page 2017

[3] [3]

Imaging three-dimensional magnetic systems with x-rays,

C. Donnelly and V . Scagnoli, “Imaging three-dimensional magnetic systems with x-rays,” Journal of Physics: Condensed Matter , vol. 32, no. 21, p. 213001, Feb. 2020, publisher: IOP Publishing. [Online]. Available: https://dx.doi.org/10.1088/1361-648X/ab5e3c

work page doi:10.1088/1361-648x/ab5e3c 2020

[4] [4]

Revealing 3D magnetization of thin films with soft X-ray tomography: magnetic singularities and topological charges,

A. Hierro-Rodriguez, C. Quir ´os, A. Sorrentino, L. M. Alvarez- Prado, J. I. Mart ´ın, J. M. Alameda, S. McVitie, E. Pereiro, M. V ´elez, and S. Ferrer, “Revealing 3D magnetization of thin films with soft X-ray tomography: magnetic singularities and topological charges,” Nature Communications , vol. 11, no. 1, p. 6382, Dec. 2020, publisher: Nature Publish...

work page 2020

[5] [5]

Three-Dimensional Study of the Vector Potential of Magnetic Structures,

C. Phatak, A. K. Petford-Long, and M. De Graef, “Three-Dimensional Study of the Vector Potential of Magnetic Structures,” vol. 104, no. 25, p. 253901. [Online]. Available: https://link.aps.org/doi/10.1103/ PhysRevLett.104.253901

work page

[6] [6]

Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,

T. Tanigaki, Y . Takahashi, T. Shimakura, T. Akashi, R. Tsuneta, A. Sugawara, and D. Shindo, “Three-dimensional observation of magnetic vortex cores in stacked ferromagnetic discs,” vol. 15, no. 2, pp. 1309–1314. [Online]. Available: https://doi.org/10.1021/nl504473a

work page doi:10.1021/nl504473a

[7] [7]

Holographic vector field electron tomography of three- dimensional nanomagnets,

D. Wolf, N. Biziere, S. Sturm, D. Reyes, T. Wade, T. Niermann, J. Krehl, B. Warot-Fonrose, B. B ¨uchner, E. Snoeck, C. Gatel, and A. Lubk, “Holographic vector field electron tomography of three- dimensional nanomagnets,” vol. 2, no. 1, pp. 1–9. [Online]. Available: https://www.nature.com/articles/s42005-019-0187-8

work page

[8] [8]

Unveiling the three-dimensional magnetic texture of skyrmion tubes,

A. Lubk, “Unveiling the three-dimensional magnetic texture of skyrmion tubes,” vol. 17, no. 3, pp. 250–255. [Online]. Available: https://www.nature.com/articles/s41565-021-01031-x

work page