Max-Normalized Radon Cumulative Distribution Transform for Limited Data Classification

Jonas Bresch; Matthias Beckmann; Robert Beinert

REVIEW 3 major objections 5 minor 16 references

Max-Normalized Radon Cumulative Distribution Transform for Limited Data Classification

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proposes a max-normalized Radon-CDT and proves that, when two templates have distinct max-normalized profiles, their affine-transformed classes are linearly separable, so classification reduces to separating single points.

desk verdict Clean affine-invariant R-CDT extension with a solid invariance proof, but the linear-separability theorem is a tautology resting on an uncharacterized collision assumption. read the letter →

arxiv 2411.16282 v1 pith:QQXDGXI3 submitted 2024-11-25 math.NA cs.ITcs.NAmath.IT

Matthias Beckmann , Robert Beinert , Jonas Bresch This is my paper · ORCID

classification math.NAcs.ITcs.NAmath.IT MSC 44A1249Q2268T1068U10

keywords RadoncumulativedistributiontransformslicedWassersteindistanceaffineinvariancelinearseparabilitysmalldataclassificationoptimaltransportimagewatermarkrecognition

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 addresses a practical obstacle in small-data image classification: images of the same object can be translated, rotated, scaled, or sheared by the measurement process, and the standard Radon cumulative distribution transform (R-CDT) changes under such affine distortions. The authors propose a two-step normalization—per-direction centering and scaling of the R-CDT projections, followed by taking the maximum over directions—and call the result the max-normalized R-CDT (mNR-CDT). Their main theorem states that if two template measures have different mNR-CDT profiles, then every affine-transformed instance of each template maps to a single feature vector, making the two classes linearly separable. On synthetic affine classes and on affine-transformed MNIST digits, the transform raises nearest-neighbor and linear-SVM accuracy relative to the plain R-CDT, especially when few training samples and few projection angles are used. If the theorem holds, it offers a direct way to make transport-based image features blind to affine nuisance transformations.

What carries the argument

The central object is the max-normalized R-CDT (mNR-CDT): first, each directional R-CDT projection $N_\theta[\mu]$ is standardized to zero mean and unit standard deviation; second, the pointwise supremum over all directions $\theta\in S^1$ is taken. The first step removes the translation and dilation that an affine map induces on each one-dimensional Radon projection, while the second step removes the direction remapping $h_A(\theta)=A^\top\theta/\|A^\top\theta\|$ because the supremum of a family is unchanged when the family is reindexed bijectively. The underlying R-CDT is a collection of one-dimensional optimal transport maps (cumulative distribution transforms) of Radon projections, so the whole construction sits inside the sliced Wasserstein geometry, and the theorem converts affine nuisance variability into exact invariance.

What would settle it

Construct two compactly supported, non-collinear plane measures that are not related by any affine map (for example, an L-shaped polygon and a different polygon found by numerical search) and compute their mNR-CDTs; if $N_{\mathrm{m}}[\mu_0]=N_{\mathrm{m}}[\nu_0]$, then the two affine classes are not linearly separable in mNR-CDT space, showing the theorem's hypothesis can fail in practice.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1. For compactly supported probability measures $\mu_0,\nu_0$ on $\mathbb{R}^2$ whose supports are not contained in a line, define the max-normalized R-CDT by $N_{\mathrm{m}}[\mu](t)=\sup_{\theta\in S^1} N_\theta[\mu](t)$, where $N_\theta[\mu]$ is the R-CDT projection in direction $\theta$ after subtracting its mean and dividing by its standard deviation. If $N_{\mathrm{m}}[\mu_0]\neq N_{\mathrm{m}}[\nu_0]$, then the affine classes $F=\{(A\cdot+y)_\#\mu_0: A\in GL(2), y\in\mathbb{R}^2\}$ and $G=\{(A\cdot+y)_\#\nu_0: A\in GL(2), y\in\mathbb{R}^2\}$ are linearly separable in mNR-CDT space. The proof is short: an earlier proposition shows $N_{\mathrm{m}}$ is invariant under every invertible affine map and translation, so each class collapses to the single feature vector of its template, and distinct vectors in a normed space are always linearly separable. Numerically, the transform achieves near-perfect classification on synthetic affine classes and improves accuracy on affine-transformed MNIST digits relative to plain R-CDT, with the largest gains when few training samples and few angles are available.

Load-bearing premise

The theorem assumes the two templates' max-normalized profiles $N_{\mathrm{m}}[\mu_0]$ and $N_{\mathrm{m}}[\nu_0]$ are different; the paper never characterizes when this holds, and because the maximum over directions discards directional information, distinct visual shapes could plausibly have the same profile and collapse into one inseparable class.

Editorial extensions

If this is right

Each affine class of a template collapses to a single point in mNR-CDT space, so distinguishing classes reduces to separating distinct points, which nearest-neighbor and linear classifiers can do directly.
The transform is invariant to translation, rotation, reflection, anisotropic scaling, and shear, so features no longer need to be aligned or data-augmented to absorb these distortions.
Classification improvements are largest when each class has few members and when few Radon angles are used, exactly the setting of watermark recognition and other limited-data image tasks.
The linear-separability guarantee needs no parameter fitting once the two templates' max-normalized profiles are known to differ; separability is a property of the representation, not of a trained classifier.

Reading between the lines

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

The paper leaves open how often $N_{\mathrm{m}}[\mu_0]\neq N_{\mathrm{m}}[\nu_0]$ holds; because the supremum discards the direction label, two shapes that are not affine-equivalent could share the same upper envelope, and how often this degeneracy occurs in real datasets is an empirical question.
A natural modification is to replace the pure maximum by a smooth or quantile-based aggregation over directions, which would retain some directional information while keeping approximate affine invariance; this could reduce degeneracy without losing the small-data benefit.
The same normalization idea could transfer to other slice-based transforms, such as sliced optimal transport on spheres or parallel slicing, whenever the nuisance group acts only by reindexing the slice parameter.

Share X Bluesky LinkedIn Reddit HN

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

Short version: this is a clean incremental step in the R-CDT program, worth a referee's time, but don't expect deep theorems. The new construction, max-normalized R-CDT, standardizes each directional CDT and then takes a pointwise supremum over directions. That makes the transform invariant under affine transformations, and the proof is straightforward and correct. The experiments, especially on affine-transformed synthetic templates, show a clear accuracy gain over vanilla R-CDT, and the small-data regime is the right setting.

The soft spot is the headline result. Theorem 1's linear separability is basically a tautology: in a normed space, any two distinct points are separable, so the entire content sits in the uncharacterized assumption Nm[μ0]≠Nm[ν0]. The paper never tells you when that holds. The max over directions discards everything below the upper envelope, so two non-affine templates could plausibly collide. The experiments choose templates that don't collide, and LinMNIST isn't an affine-class test, so the failure mode is never probed. This is addressable: characterize the collision condition, measure collision frequency, or add a correction term.

Two smaller issues: the finite-angle discretization has no error analysis, and there is no comparison with the closest prior work, invariance-encoding in sliced-Wasserstein space [15]. The promised code isn't available in the preprint, so reproducibility rests on the experiments alone.

Overall, the invariance construction is sound and useful; the separability theorem is less novel than it looks. I'd send it to peer review and let the reviewers push for a better characterization of the distinctness condition.

Referee Report

3 major / 5 minor

Summary. The paper proposes a two-step normalization of the Radon cumulative distribution transform for probability measures on R^2. First, each directional one-dimensional CDT is normalized to zero mean and unit standard deviation, yielding the NR-CDT; then the pointwise supremum over all directions is taken, yielding the max-normalized R-CDT (mNR-CDT). The authors prove that the mNR-CDT is invariant under affine transformations of the underlying measure (Proposition 7) and claim that affine classes originating from distinct templates are linearly separable in mNR-CDT space (Theorem 1). Numerical experiments on synthetic affine-transformed templates and on an affinely transformed MNIST dataset report improved classification accuracy relative to the plain R-CDT, especially in the small-data regime.

Significance. If the main claims are fully established, the construction is potentially useful for limited-data image classification problems such as watermark recognition, where affine distortions arising from the acquisition process are common. The measure-valued formulation of the Radon transform, the explicit affine action in Proposition 3, and the normalization in Proposition 5 are clean and appear correct. The paper also promises reproducible code and gives a reasonable proof-of-concept comparison. However, the advertised linear-separability theorem is essentially a restatement of the non-collision assumption Nm[μ0] ≠ Nm[ν0], which is neither characterized nor empirically tested. In addition, the numerical experiments use a finite-angle discretization with no error analysis and include a dataset that explicitly violates the affine-class hypothesis. The significance is therefore conditional on the authors either strengthening the theory or clearly repositioning the contribution as an affine-invariant feature representation with heuristic numerical support.

major comments (3)

[Section 3.3, Theorem 1] The linear-separability claim is load-bearing and is almost entirely contained in the uncharacterized assumption Nm[μ0] ≠ Nm[ν0]. By Proposition 7, each affine class F and G collapses to a single element of L∞_ρ(R); once the two images are distinct, Hahn–Banach separation of two points gives the conclusion, so the proof uses no further structure. The authors neither characterize the non-collision condition in terms of μ0 and ν0 nor report an empirical check that it holds for the datasets, e.g., by computing ∥Nm[μ0] − Nm[ν0]∥∞ for the templates. Because the max-envelope operation discards all directional information below the envelope, distinct templates can plausibly share the same mNR-CDT; in that case both classes map to the same feature and no linear classifier can separate them. The authors should prove a checkable sufficient condition for Nm[μ0] ≠ Nm[ν0], give explicit examples or counterexamples, or explicitly weaken and reposition the theorem as an invariance result plus a non-collision assumption.
[Section 3.3, Proposition 6] The proof establishes only pointwise boundedness of sup_{θ∈S1} Nθ[μ](t). To conclude that Nm[μ] belongs to L∞_ρ(R), the function t ↦ sup_{θ∈S1} Nθ[μ](t) must also be measurable; this measurability is not addressed. Since the mNR-CDT space is defined as a subspace of L∞_ρ(R), this is a genuine gap in the definition of the proposed feature. The authors should add a measurability argument or explicitly define Nm via a measurable selection or essential supremum before using it in Theorem 1.
[Section 4, Tables 2–4] The experiments discretize the supremum over S1 using 2–128 equispaced angles, but there is no error analysis relating the finite-angle feature Nm_K[μ] to the continuous Nm[μ]. Consequently, the numerical validation of Theorem 1 is only heuristic; indeed Table 2 shows accuracy varying with the number of angles. Moreover, the LinMNIST experiments explicitly violate the template hypothesis: the classes are not affine orbits of a single template measure, so Theorem 1 does not apply to them and those results should be described as a robustness study rather than as validation of the theorem. The paper would be considerably strengthened by a discretization error estimate of the form ∥Nm_K[μ] − Nm[μ]∥∞ ≤ C Δθ under suitable regularity assumptions, and by reporting the empirical values of ∥Nm[μ0] − Nm[ν0]∥∞ in the academic experiments.

minor comments (5)

[Section 4.1] The name "Chebychev" should be "Chebyshev".
[Section 4.2] The word "angels" should be "angles".
[Table 4] Several entries in Table 4 appear concatenated without separators, for example "1.00000.959±0.005"; the table should be reformatted so that each cell is clearly delimited.
[References] Reference [3] has an incomplete author list: "Bonneel, N., Rabin, J., Peyré, Pfister, H." should supply the missing initial for Peyré.
[Section 2.3 and Table 1] The parameterization θ(ϑ) is used on the half-open interval ϑ∈(−π/2,π/2), while the measure-valued Radon transform is defined on the full circle S1; the relationship between the two parameterizations for the remaining half-circle should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: mNR-CDT invariance and class separability are proved from the definitions; no fitted parameter is renamed as a prediction and no load-bearing self-citation appears.

full rationale

The paper's central claim is a conditional theorem: if the max-normalized R-CDT maps two templates to distinct functions, then their affine classes are linearly separable. Proposition 7 proves affine invariance by direct computation from Proposition 5 and the bijectivity of the direction map h_A; this is a genuine derivation from the definition of Nm as a supremum of normalized projections. Theorem 1 then follows because each affine class collapses to a single point in L^∞_rho, and distinct points in a normed space are always linearly separable. The proof quotes this reduction explicitly, but it is a valid consequence rather than a circular use of the conclusion. The assumption Nm[mu0] != Nm[nu0] is an uncharacterized condition, which is a limitation of the theorem's applicability, not an instance of circularity: no parameter is fitted, and no conclusion is used as a premise. The numerical experiments use synthetic affine classes and the external LinMNIST benchmark, and the authors themselves note that LinMNIST does not satisfy the theoretical assumptions. Self-citations in references [1] and [6] support only the dataset and application motivation, not the main derivation. There is no fitted-input-called-prediction step, no uniqueness imported from the authors' prior work, and no ansatz smuggled in via citation. The paper is therefore self-contained with respect to its claimed mathematical results, and the only substantive concern about the uncharacterized non-collision condition belongs to correctness or strength of the theorem, not to circularity.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central result rests on standard optimal transport and functional analysis, plus domain assumptions about compact, non-collinear measures and invertible affine maps. The only ad hoc assumption is distinctness of the max-normalized templates, which is uncharacterized.

free parameters (2)

Number of equispaced angles in [0, pi) = 2 to 128 in experiments
Computational discretization of the supremum over S1; not fitted from data but controls approximation error and is not analyzed.
Reference measure rho = not specified; uniform on [0,1] is suggested
Choice of non-atomic reference measure affects the embedding but not the affine invariance; not fitted to data.

assumptions (5)

domain assumption Reference measure rho has no atoms
Needed for the CDT to be an optimal Monge map; stated in Section 3.1.
domain assumption Measures lie in P*_c(R2): compact support and support not contained in a line
Ensures the standard deviation of each direction profile is bounded below; Proposition 4.
domain assumption Affine maps are invertible: A in GL(2)
The direction remapping hA is a bijection on S1 only for invertible A; used in Propositions 5 and 7.
ad hoc to paper Distinct templates: Nm[mu0] is not equal to Nm[nu0]
Assumed in Theorem 1 without characterization; without it, classes collapse and are not separable.
standard math Distinct points in a normed space are linearly separable
The proof of Theorem 1 implicitly relies on the fact that two distinct points in a normed space are linearly separable via Hahn-Banach.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Max-Normalized Radon Cumulative Distribution Transform for Limited Data Classification." pith.science (2026). https://pith.science/paper/QQXDGXI3

@misc{pith2026241116282,
  author       = {Pith},
  title        = {Pith review of: Max-Normalized Radon Cumulative Distribution Transform for Limited Data Classification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QQXDGXI3}},
  note         = {Machine review of arXiv:2411.16282}
}

read the original abstract

The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. The aim of this paper is to make the R-CDT and the related sliced Wasserstein distance invariant under affine transformations. For this, we propose a two-step normalization of the R-CDT and prove that our novel transform allows linear separation of affinely transformed image classes. The theoretical results are supported by numerical experiments showing a significant increase of the classification accuracy compared to the original R-CDT.

Figures

Figures reproduced from arXiv: 2411.16282 by the authors.

**Figure 1.** Illustration of the bivariate Radon transform with distance t ∈ R and normal direction θ ∈ S1. The given function is integrated along the lines ℓt,θ. For f ∈ L 1 (R 2 ), we define its Radon transform R[f]: R × S1 → R as the line integral R[f](t, θ) := Z ℓt,θ f(s) ds, (t, θ) ∈ R × S1, where ds denotes the arc length element of ℓt,θ. This defines the Radon operator R: L 1 (R 2 ) → L 1 (R × S1). For fixed θ ∈ S1, we s… view at source ↗

**Figure 2.** Samples of the academic dataset consisting of randomly affine-transformed synthetic template images. class 1 class 5 class 7 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 4.** Visualization of mNR-CDT for the academic dataset and 128 angles in [0, π). Datasets. For our simulations, we rely on two datasets. For academic purposes, the first dataset is based on (up to) three synthetic template symbols, which are randomly translated, rotated, dilated, and sheared, cf [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 1 linked inside Pith

[15]

Pattern Recognition137, 109268 (2023)

Shifat-E-Rabbi, M., Zhuang, Y., Li, S., Rubaiyat, A.H.M., Yin, X., Rohde, G.K.: Invarianceencodinginsliced-Wassersteinspaceforimageclassificationwithlimited training data. Pattern Recognition137, 109268 (2023). https://doi.org/10.1016/ j.patcog.2022.109268

arXiv 2023
[1]

SIAM Journal on Mathematics of Data Science6(3), 579–601 (2024)

Beckmann, M., Heilenkötter, N.: Equivariant neural networks for indirect mea- surements. SIAM Journal on Mathematics of Data Science6(3), 579–601 (2024). https://doi.org/10.1137/23M1582862

doi:10.1137/23m1582862 2024
[2]

IEEE Transactions on Image Processing31, 7292–7305 (2022)

Beier, F., Beinert, R., Steidl, G.: On a linear Gromov–Wasserstein distance. IEEE Transactions on Image Processing31, 7292–7305 (2022). https://doi.org/10.1109/ TIP.2022.3221286

arXiv 2022
[3]

Journal of Mathematical Imaging and Vision51(1), 22–45 (2015)

Bonneel, N., Rabin, J., Peyré, Pfister, H.: Sliced and Radon Wasserstein barycen- ters of measures. Journal of Mathematical Imaging and Vision51(1), 22–45 (2015). https://doi.org/10.1007/s10851-014-0506-3

doi:10.1007/s10851-014-0506-3 2015
[4]

IEEE Signal Processing Magazine 29(6), 141–142 (2012)

Deng, L.: The MNIST database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine 29(6), 141–142 (2012). https://doi. org/10.1109/MSP.2012.2211477

arXiv 2012
[5]

https://doi.org/10.48550/arXiv.2406.15503, arXiv:2406.15503

Diaz Martin, R., Medri, I.V., Rohde, G.K.: Data representation with optimal trans- port (2024). https://doi.org/10.48550/arXiv.2406.15503, arXiv:2406.15503

arXiv doi:10.48550/arxiv.2406.15503 2024
[6]

In: International Conference on Document Analysis and Recognition (ICDAR)

Hauser, D., Beckmann, M., Koliander, G., Stiehl, H.S.: On image processing and pattern recognition for thermograms of watermarks in manuscripts – a first proof- of-concept. In: International Conference on Document Analysis and Recognition (ICDAR). pp. 91–107 (2024). https://doi.org/10.1007/978-3-031-70543-4_6

doi:10.1007/978-3-031-70543-4_6 2024
[7]

IEEE Transactions on Image Processing 25(2), 920–934 (2016)

Kolouri, S., Park, S.R., Rohde, G.K.: The Radon cumulative distribution transform and its application to image classification. IEEE Transactions on Image Processing 25(2), 920–934 (2016). https://doi.org/10.1109/TIP.2015.2509419

arXiv 2016

Show all 16 references

[8]

IEEE Signal Processing Magazine34(4), 43–59 (2017)

Kolouri, S., Park, S.R., Thorpe, M., Slepcev, D., Rohde, G.K.: Optimal mass trans- port. IEEE Signal Processing Magazine34(4), 43–59 (2017). https://doi.org/10. 1109/MSP.2017.2695801

2017
[9]

In- formation and Inference: A Journal of the IMA 12(1), 363–389 (2023)

Moosmüller, C., Cloninger, A.: Linear optimal transport embedding: provable Wasserstein classification for certain rigid transformations and perturbations. In- formation and Inference: A Journal of the IMA 12(1), 363–389 (2023). https: //doi.org/10.1093/imaiai/iaac023

2023 doi
[10]

SIAM, Philadelphia (2001)

Natterer, F.: The Mathematics of Computerized Tomography. SIAM, Philadelphia (2001). https://doi.org/10.1137/1.9780898719284

2001 doi
[11]

In- verse Problems 39(10), 105005 (2023)

Quellmalz, M., Beinert, R., Steidl, G.: Sliced optimal transport on the sphere. In- verse Problems 39(10), 105005 (2023). https://doi.org/10.1088/1361-6420/acf156 Max-Normalized Radon-CDT 13

2023 doi
[12]

Journal of Mathematical Imaging and Vision 66(6), 951–976 (2024)

Quellmalz, M., Buecher, L., Steidl, G.: Parallelly sliced optimal transport on spheres and on the rotation group. Journal of Mathematical Imaging and Vision 66(6), 951–976 (2024). https://doi.org/10.1007/s10851-024-01206-w

2024 doi
[13]

CRC Press (1996)

Ramm, A.G., Katsevich, A.I.: The Radon Transform and Local Tomography. CRC Press (1996). https://doi.org/10.1201/9781003069331

1996 doi
[14]

Journal of Mathematical Imaging and Vision 63, 1185–1203 (2021)

Shifat-E-Rabbi, M., Yin, X., Rubaiyat, A.H.M., Li, S., Kolouri, S., Aldroubi, A., Nichols, J.M., Rohde, G.K.: Radon cumulative distribution transform subspace modeling for image classification. Journal of Mathematical Imaging and Vision 63, 1185–1203 (2021). https://doi.org/10...

2021 doi
[16]

American Mathematical Society (2003)

Villani, C.: Topics in Optimal Transportation. American Mathematical Society (2003). https://doi.org/10.1090/gsm/058

2003 doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [15]

Pattern Recognition137, 109268 (2023)

Shifat-E-Rabbi, M., Zhuang, Y., Li, S., Rubaiyat, A.H.M., Yin, X., Rohde, G.K.: Invarianceencodinginsliced-Wassersteinspaceforimageclassificationwithlimited training data. Pattern Recognition137, 109268 (2023). https://doi.org/10.1016/ j.patcog.2022.109268

arXiv 2023

[2] [1]

SIAM Journal on Mathematics of Data Science6(3), 579–601 (2024)

Beckmann, M., Heilenkötter, N.: Equivariant neural networks for indirect mea- surements. SIAM Journal on Mathematics of Data Science6(3), 579–601 (2024). https://doi.org/10.1137/23M1582862

doi:10.1137/23m1582862 2024

[3] [2]

IEEE Transactions on Image Processing31, 7292–7305 (2022)

Beier, F., Beinert, R., Steidl, G.: On a linear Gromov–Wasserstein distance. IEEE Transactions on Image Processing31, 7292–7305 (2022). https://doi.org/10.1109/ TIP.2022.3221286

arXiv 2022

[4] [3]

Journal of Mathematical Imaging and Vision51(1), 22–45 (2015)

Bonneel, N., Rabin, J., Peyré, Pfister, H.: Sliced and Radon Wasserstein barycen- ters of measures. Journal of Mathematical Imaging and Vision51(1), 22–45 (2015). https://doi.org/10.1007/s10851-014-0506-3

doi:10.1007/s10851-014-0506-3 2015

[5] [4]

IEEE Signal Processing Magazine 29(6), 141–142 (2012)

Deng, L.: The MNIST database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine 29(6), 141–142 (2012). https://doi. org/10.1109/MSP.2012.2211477

arXiv 2012

[6] [5]

https://doi.org/10.48550/arXiv.2406.15503, arXiv:2406.15503

Diaz Martin, R., Medri, I.V., Rohde, G.K.: Data representation with optimal trans- port (2024). https://doi.org/10.48550/arXiv.2406.15503, arXiv:2406.15503

arXiv doi:10.48550/arxiv.2406.15503 2024

[7] [6]

In: International Conference on Document Analysis and Recognition (ICDAR)

Hauser, D., Beckmann, M., Koliander, G., Stiehl, H.S.: On image processing and pattern recognition for thermograms of watermarks in manuscripts – a first proof- of-concept. In: International Conference on Document Analysis and Recognition (ICDAR). pp. 91–107 (2024). https://doi.org/10.1007/978-3-031-70543-4_6

doi:10.1007/978-3-031-70543-4_6 2024

[8] [7]

IEEE Transactions on Image Processing 25(2), 920–934 (2016)

Kolouri, S., Park, S.R., Rohde, G.K.: The Radon cumulative distribution transform and its application to image classification. IEEE Transactions on Image Processing 25(2), 920–934 (2016). https://doi.org/10.1109/TIP.2015.2509419

arXiv 2016