Gaussian beam Radon transform for tensor fields in $\mathbb{R}^2$

Indrani Zamindar; Rohit Kumar Mishra; Souvik Roy

REVIEW 4 major objections 4 minor 31 references

Gaussian beam Radon transform for tensor fields in $\mathbb{R}^2$

T0 review · 4 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read The paper establishes that a compactly supported smooth vector field in the unit disk is uniquely recoverable from its longitudinal and transverse Gaussian beam Radon transforms, and a symmetric 2-tensor field from longitudinal…

desk verdict Genuine extension of Gaussian-beam Radon tomography to tensor fields, but every recovery theorem divides by unproved Mellin transform factors and unproved determinants; the central results are not closed as written. read the letter →

arxiv 2608.21966 v1 pith:ASHW7LLG submitted 2026-08-22 math.CA

Rohit Kumar Mishra , Souvik Roy , Indrani Zamindar This is my paper · ORCID

classification math.CA MSC 44A1244A3544A3042B05

keywords GaussianbeamRadontransformtensortomographyvectorfieldreconstructionsymmetric2-tensorfieldslongitudinalandtransversetransformsmixedMellinFourier-Mellininversion

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

The paper extends the Gaussian beam Radon transform—an integral model in which the measurement beam has a Gaussian transverse intensity profile rather than being a thin line—from scalar functions to vector fields and symmetric 2-tensor fields in the plane. It proves explicit reconstruction theorems: every compactly supported smooth vector field in the unit disk is determined by its longitudinal and transverse Gaussian beam Radon transforms, and every compactly supported smooth symmetric 2-tensor field is determined by the triple of longitudinal, transverse, and mixed transforms, or by one transform together with its first two moments. The proofs give Fourier-Mellin inversion formulas that deconvolve the beam profile frequency-by-frequency. The motivation is optical tomography, where the measured quantities are electromagnetic fields and real beams are closer to Gaussian than to idealized lines.

What carries the argument

The mechanism that carries the proof is the Fourier-Mellin decomposition of the Gaussian beam kernel. The kernel $K(x_1,x_2)=w_0/w(x_2)\exp(-x_1^2/w^2(x_2))$ has the Fourier symbol $\widehat K(\sigma,\theta^\perp\cdot x)=w_0\sqrt{\pi}\exp(-w_0^2(1+((\theta^\perp\cdot x)/z_R)^2)\sigma^2/4)$. After Fourier transform in the offset $s$ and Fourier series in the beam angle $\theta$, the remaining radial integrals have the form $\int_0^\infty h_n(r)g_n(r^{-1}\sigma^{-1})\,rdr$, a Mellin convolution. The Mellin transform turns this into the product $Nh_n(\rho+2)Ng_n(\rho)$. The problem of inverting the GbRt therefore reduces to dividing by $Ng_n(\rho-2)$ or, for moment data, to solving small linear systems in those Mellin products. This is the deconvolution identity at the center of all four theorems.

What would settle it

Evaluate $Ng_n(\rho-2)$ numerically for small $n$ and for the strip of $\rho$ where the inverse Mellin transform runs, and evaluate the determinants of the systems in (17) and (33)–(34). A zero in any divisor, or a nonzero compactly supported field whose longitudinal and transverse GbRt data vanish identically, would disprove the corresponding recovery theorem.

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Extended reading notes

Core claim

The paper's central claim is that the generalized Gaussian beam Radon transforms defined in Definition 3, $$M_k^\ell f(s,\$\theta$)=\int_{\mathbb{R}^2}(\$\theta$^\perp\cdot x)^k K(s-\$\theta$\cdot x,\$\theta$^\perp\cdot x)\langle \$\theta$^\ell(\$\theta$^\perp)^{m-\ell},f(x)\rangle\,dx$$, can be inverted in the following combinations. Theorem 1: every $f\in C_c^\infty(S^1;D)$ is uniquely recovered from $(Lf,Tf)$; after a Fourier transform in $s$ and a Fourier series in $\theta$, the Mellin transform of the angular coefficient $(f_1)_n+i(f_2)_n$ (and similarly $(f_1)_n-i(f_2)_n$) equals a data term divided by $Ng_n(\rho-2)$, and inverse Mellin transform recovers the field. Theorem 2 recovers a vector field from $(Lf,L_1f)$ or from $(Tf,T_1f)$. For symmetric 2-tensor fields, Theorem 3 recovers $f$ from $(Lf,Tf,Mf)$, and Theorem 4 recovers $f$ from $(Lf,L_1f,L_2f)$ or $(Tf,T_1f,T_2f)$ by solving a $2\times2$ linear system whose coefficients are products of Mellin transforms of the kernel coefficients $g_n$; Remark 3.1 shows $(Mf,M_1f,M_2f)$ is insufficient.

Load-bearing premise

The load-bearing premise is that none of the Mellin-transform divisors the inversion divides by—the kernel coefficients $Ng_n(\rho-2)$ and the determinants of the $2\times2$ systems in (17) and (33)–(34)—ever vanish on the range of integration; the paper assumes this instead of proving it.

Editorial extensions

If this is right

For vector fields, longitudinal and transverse Gaussian beam data determine the field completely, so two measurement scans contain the full information and no third transform is needed.
For symmetric 2-tensor fields, either three transform types $(Lf,Tf,Mf)$ or three moments of one type $(Lf,L_1f,L_2f)$ determine the field, while the mixed triple $(Mf,M_1f,M_2f)$ does not.
The Fourier-Mellin formulas give a concrete reconstruction procedure for each proven pair: Fourier transform the data in $s$, expand in Fourier series in $\theta$, divide by kernel-coefficient Mellin transforms, and apply inverse Mellin transforms.
The authors expect the same divide-by-$Ng_n$ structure to extend to tensor fields of any order, with the calculations growing cumbersome rather than changing in principle.

Reading between the lines

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

Because the inversion divides by $Ng_n(\rho-2)$, the numerical conditioning of any implementation depends on how close these divisors come to zero; testing that numerically is a concrete next step.
The failure of mixed moments to give full recovery in Remark 3.1 mirrors the classical kernel of mixed ray transforms, suggesting the minimal data sets for Gaussian-beam tensor tomography likely match the classical minimal data sets.
If a zero of $Ng_n(\rho-2)$ is found, the affected transform pair would have non-injective behavior at specific frequencies, and a different extra datum such as one mixed moment might restore uniqueness.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know before you spend time on this. First, it is genuinely new: nobody has written down Gaussian-beam Radon transforms for symmetric tensor fields before, and the authors give explicit inversion formulas for vector fields and symmetric 2-tensors from longitudinal, transverse, and mixed moments. Second, the proofs as written are not complete. Every reconstruction divides by the Mellin transform N g_n(ρ−2) of the Gaussian kernel coefficient, and the 2x2 systems in Theorems 2 and 4 have determinants that are never shown to be nonzero. This is not a technicality. The g_n are oscillatory (in the zero-beam-waist limit they reduce to Bessel-type functions with infinitely many zeros), so the division steps may genuinely fail, and even if the zeros are isolated, the paper needs an analytic-continuation or removable-singularity argument that it does not provide.

What the paper does well: the definitions of longitudinal, transverse, and mixed GbRts and their moments are natural; the Fourier-Mellin strategy is coherent; and the algebra is detailed enough to check. The citation pattern is honest—the main self-citation [23] is the scalar predecessor, and using that technique is not circular. There are no fitted parameters.

The soft spots are real but not all equal. First, the missing invertibility proofs sit at the final step of each theorem, so the claimed explicit recovery is not established. Second, the paper applies the inverse Mellin formula and the Mellin transform to the kernel coefficients without proving convergence or holomorphy in the needed half-plane. That may be a routine exercise, but it is omitted. Third, equation (24) in Theorem 3 contains a sign/index error: the Fourier coefficient of cLf−cTf−2iMf is written with +2iMf and with e^{-i(n+2)θ}; this looks correctable but as printed the formula is inconsistent. Fourth, there are no numerical experiments, which would have helped indicate whether the division troubles actually appear.

Overall, the core idea is sound and likely salvageable, but the central theorems are not established as written. I would not cite the recovery theorems in their current form, but I would be interested to see the fix.

If you are asked to referee this, take it. The novelty is real and the problem is well-motivated by optical tomography. A serious referee should demand proofs that N g_n(ρ−2) is nonzero on the relevant strip (or that the zeros are removable), and that the determinants in (17), (33), and (34) do not vanish identically. If those gaps close, the paper is publishable; without them, the inversion claims are conditional.

Referee Report

4 major / 4 minor

Summary. The paper defines generalized Gaussian beam Radon transforms (longitudinal, transverse, mixed, and their integral moments) for vector fields and symmetric 2-tensor fields in R^2, extending the scalar Gaussian beam Radon transform of Roy, Jeon, and Moon. The main results are four recovery theorems: Theorem 1 recovers a compactly supported smooth vector field from its longitudinal and transverse GbRts; Theorem 2 from either (Lf, L1f) or (Tf, T1f); Theorem 3 recovers a symmetric 2-tensor field from (Lf, Tf, Mf); and Theorem 4 from either (Lf, L1f, L2f) or (Tf, T1f, T2f). The proofs proceed by Fourier transforming in the beam coordinate, expanding in angular Fourier series, and using Mellin transforms to deconvolve the Gaussian kernel, yielding explicit inversion formulas.

Significance. If the stated inversion formulas are valid, this is a useful and nontrivial extension of scalar Gaussian-beam Radon inversion to tensor fields, with potential applications to optical tomography of vectorial and tensor quantities. The derivation is parameter-free, the angular algebra in Theorems 1 and 3 is checked in detail, and the paper is explicit about the kernel's structure. However, the central deconvolution step repeatedly divides by Mellin transforms of the kernel coefficient g_n and by determinants of 2x2 systems, and no proof that these quantities are nonzero is provided. Since g_n(t) is oscillatory (e.g., proportional to J_n(1/t) for zero beam width), zeros are a real possibility, so the recovery formulas are not established as written. The paper also applies Mellin inversion to quantities whose convergence and meromorphy are not justified in the required half-plane. These are load-bearing gaps, but they appear fixable by adding analyticity and nonvanishing arguments or by restricting to an open dense set of parameters.

major comments (4)

[§3.1, Eqs. (12) and (13)] The recovery of (f1)n and (f2)n divides by Ng_n(ρ−2) on both sides. The paper never proves that Ng_n(ρ−2) is nonzero on the vertical line used for the inverse Mellin transform (8). Since g_n(t) is oscillatory—for A=0, g_n(t) = (−i)^n J_n(1/t)—its Mellin transform can have zeros, so the pointwise division is not justified. An isolated zero would make the formula invalid as written, and an open set of zeros would destroy the recovery step entirely. The same issue affects Eqs. (19), (22), and (25) in Theorem 3.
[§3.2, Eq. (17)] The reconstruction in Theorem 2 solves a 2x2 linear system in the Mellin domain and then divides by the determinant N g_{n−2}(ρ−2) N(g_{n−1}−g_{n+1})(ρ−3) − N g_n(ρ−2) N(g_{n−3}−g_{n−1})(ρ−3). No argument is given that this determinant is nonzero on the inversion line, nor that the individual Mellin factors are all nonzero. Similar unproved determinants appear in Eqs. (33)–(34) for Theorem 4, which are essential to recovering N(f22−f11+2if12)_{n−2} and N(f22−f11−2if12)_{n+2}.
[§3.1–§3.4, Eq. (8) and surrounding text] The inverse Mellin formula (8) is quoted for functions in C_c^∞(D), but the paper applies it to Mellin transforms of products involving the Fourier-transformed data, the exponential factor exp(w0^2 σ^2/4), and kernel coefficients g_n(σ/r). The convergence of the Mellin transform, the existence of a common half-plane of holomorphy, and the integrability of the data side on the inversion line are not established. Moreover, when division by Ng_n(ρ−2) is performed, any zeros of Ng_n create poles; without a removable-singularity or meromorphic-continuation argument, the inverse Mellin integral is not even well defined.
[§3.3, Eq. (24)] Equation (24) contains a sign/index inconsistency: the text says it computes the (n−2)-th Fourier coefficient, but the displayed integral contains e^{-i(n+2)θ}, and the left side is written as (cLf−cTf−2i dMf)_{n−2}. This makes Eq. (25) suspect as written. The error appears localized, but because Eq. (25) is later combined with Eqs. (19) and (22) in the reconstruction, it must be corrected and the subsequent formulas re-checked.

minor comments (4)

[Throughout] The manuscript has numerous typographical and formatting issues: 'R 2', 'θ θθ', 'fourier' with lowercase, and inconsistent use of bold and non-bold symbols. These should be cleaned up for publication.
[§3.1, Eq. (11)] After the convolution notation, the equation 'r^2{(f1)n(r) + i(f2)n(r)}×g_n(σ)' is ambiguous; the convolution variable and the constant factors should be displayed more clearly to avoid confusion about where the Mellin transform is applied.
[§3.4, Eq. (30)] In the displayed formula for d L2f_n(σ), the kernel is written as bK(σ,−rcψ) in one place and bK(σ,−rsψ) in another; check whether the second argument should be −r s_ψ consistently.
[§3.5] The operator A is introduced via [2] but its action on symmetric 2-tensors is only partially defined in the text; the symmetry of f is used implicitly, so it would help to state explicitly that A is applied to the tensor as a whole and then symmetrized.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity: the tensor-field inversions are derived by explicit Fourier/Mellin algebra from the definitions, and the cited scalar technique is a borrowed method, not a premise that forces the result.

full rationale

The claimed derivations are self-contained in the sense relevant to circularity. Theorems 1-4 do not fit parameters and do not rename known data as predictions: each proof starts from the definitions (2)-(4) with kernel (1), takes Fourier transforms in s, expands in angular Fourier series, passes to polar coordinates, and applies Mellin inversion. The unknowns N{(f1)_n ± i(f2)_n}, N{(f11+f22)_n}, etc., are expressed in terms of Mellin transforms of the measured data divided by N g_n(ρ-2) or by determinants of such factors (e.g., Sections 3.1, 3.3, and equations (17), (33)-(34)). These are inversion formulas, not restatements of the inputs. The citation [23] is a prior scalar GbRt paper by co-author Roy; the text says 'we adopted the techniques introduced by the authors in [23]', but the present proofs re-derive the needed Fourier-Mellin manipulations and do not assume Theorems 1-4 from [23]. Thus the self-citation is a methodological reference, not a load-bearing premise. The operator A in Remark 3.1 is attributed to [2], which has no author overlap with this paper, and is used only to relate Mf to -L tilde f; that relation is a symmetry calculation, not an import of the conclusion. The genuine weaknesses here are correctness gaps rather than circularity: the paper never proves N g_n(ρ-2) is nonvanishing in the Mellin-inversion strip, nor that the 2x2 determinants in (17) and (33)-(34) are nonzero, and equation (24) contains an apparent sign/index slip. Those omissions make the pointwise division formulas conditional, but they do not make any output equal to an input by construction. Accordingly, no circular step can be exhibited, and the appropriate score is 0.

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

The central claim rests on the physical Gaussian beam model (with parameters w0 and z_R as inputs, not fitted), standard Mellin/Fourier analysis, and two unproved technical assumptions: nonzero Mellin denominators and invertibility of the systems in Theorems 2 and 4. No new entities or fitted constants are introduced.

assumptions (4)

domain assumption The Gaussian kernel model (1) and the beam waist formula w(u) = w0 sqrt(1+(u/z_R)^2) accurately represent the imaging physics.
The entire forward model is based on this kernel, chosen from laser optics literature; if the physical beam deviates, the transforms and reconstructions change.
ad hoc to paper The Mellin transform N g_n(ρ-2) is nonzero on the inversion line, and the linear systems in equations (17) and (33)-(34) have nonzero determinants.
The reconstruction divisions by N g_n and the 'Solving the equations' steps in Theorems 2 and 4 require this, but the paper does not verify it.
standard math The inverse Mellin formula (8) applies to the reconstructed Fourier coefficients.
This is a classical inversion formula, but its use here requires the relevant Mellin transforms to converge in a suitable strip, which is not discussed.
standard math Interchange of Fourier, Fourier series, and Mellin integrals is valid for compactly supported smooth tensor fields.
Standard in this context; f in C_c^∞ makes most integrations legitimate.

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Pith. "Pith review of Gaussian beam Radon transform for tensor fields in $\mathbb{R}^2$." pith.science (2026). https://pith.science/paper/ASHW7LLG

@misc{pith2026260821966,
  author       = {Pith},
  title        = {Pith review of: Gaussian beam Radon transform for tensor fields in $\mathbbR^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ASHW7LLG}},
  note         = {Machine review of arXiv:2608.21966}
}

abstract

In this article, we introduce and study a set of generalized Gaussian beam Radon transforms (GbRt) acting on tensor fields in $\mathbb{R}^2$. The operators considered include longitudinal, transverse, mixed GbRts, along with their integral moments. These operators extend the corresponding notions of the classical generalized Radon transforms for tensor fields. We establish reconstruction results for vector and symmetric 2-tensor fields using appropriate combinations of the defined transforms. This work extends a recent study on the recovery of scalar functions from their GbRt to the recovery of vector and tensor fields from analogously defined generalized GbRts.

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Reference graph

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [23]

Radon transform with gaussian beam: Theoretical and numerical reconstruction scheme.Applied Mathematics and Computation, 452:128024, 2023

Souvik Roy, Gihyeon Jeon, and Sunghwan Moon. Radon transform with gaussian beam: Theoretical and numerical reconstruction scheme.Applied Mathematics and Computation, 452:128024, 2023

work page 2023

[2] [1]

Generalized Radon transforms over symmetric m-tensor fields inR n.Analysis and Mathematical Physics, 16:11, 2026

Anuj Abhishek, Rohit Kumar Mishra, and Chandni Thakkar. Generalized Radon transforms over symmetric m-tensor fields inR n.Analysis and Mathematical Physics, 16:11, 2026

work page 2026

[3] [2]

Generic uniqueness and stability for the mixed ray transform.Transactions of the American Mathematical Society, 374(9):6085–6144, 2021

Maarten de Hoop, Teemu Saksala, Gunther Uhlmann, and Jian Zhai. Generic uniqueness and stability for the mixed ray transform.Transactions of the American Mathematical Society, 374(9):6085–6144, 2021

work page 2021

[4] [3]

Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve.Inverse Problems, 22(2):399, 2006

Alexander Denisjuk. Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve.Inverse Problems, 22(2):399, 2006

work page 2006

[5] [4]

Derevtsov and Ivan E

Evgeny Yu. Derevtsov and Ivan E. Svetov. Tomography of tensor fields in the plane.Eurasian Journal of Mathematical and Computer Applications, 3(2):24–68, 2015. 15

work page 2015

[6] [5]

Birkh¨ auser, 1999

Sigurdur Helgason.The Radon Transform. Birkh¨ auser, 1999

work page 1999

[7] [6]

Gaussian light model in brightfield optical projection tomography.Scientific reports, 9(1):13934, 2019

Olli Koskela, Toni Montonen, Birhanu Belay, Edite Figueiras, Sampsa Pursiainen, and Jari Hyttinen. Gaussian light model in brightfield optical projection tomography.Scientific reports, 9(1):13934, 2019

work page 2019

[8] [7]

Krishnan, Ramesh Manna, Suman K

Venkateswaran P. Krishnan, Ramesh Manna, Suman K. Sahoo, and Vladimir A. Sharafutdinov. Momentum ray transforms.Inverse Problems & Imaging, 13(3):679–701, 2019

work page 2019