An Inversion Formula for Horizontal Conical Radon Transform

Duy N. Nguyen; Linh V. Nguyen

REVIEW 3 major objections 4 minor 23 references

An Inversion Formula for Horizontal Conical Radon Transform

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A function in 3D can be reconstructed exactly from integrals over all cones with horizontal axes and vertices on a line, for every integer weight.

desk verdict A useful new inversion formula for a practical cone transform, but the proof of Lemma 2.5 drops boundary terms that appear to break the identity. read the letter →

arxiv 1908.08255 v1 pith:AWHKEZRT submitted 2019-08-22 math.CA

Duy N. Nguyen , Linh V. Nguyen This is my paper · ORCID

classification math.CA MSC 44A1253C65

keywords conicalRadontransforminversionformulaComptoncameraimagingverticalsliceV-lineweightedX-raysphericalintegralgeometrycone-beamreconstruction

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 establishes an exact, explicit inversion formula for a weighted conical Radon transform in $\mathbb{R}^3$, defined by integrating a compactly supported smooth function over all one-sided circular cones whose vertices lie on a straight line and whose axes are horizontal. The formula recovers the original function from the transform values and their derivatives by a finite combination of integrals, for every integer weight parameter $k$. This matters because this three-parameter family of cones models Compton-camera detection with detectors arranged along a line: each detected photon gives a cone of possible source directions, and an explicit inversion makes direct analytic reconstruction possible rather than iterative approximation.

What carries the argument

The proof is carried by three transforms and one connecting identity. The vertical slice transform $\Gamma g(\phi,t)$ integrates a function on the sphere over the circle where $e_\phi\cdot\omega=t$, and its spherical inversion (Theorem 2.3) is the first reconstruction step. The weighted X-ray transform $\chi_k f(z,\omega)$ integrates $f$ along the ray from the vertex $b(z)$ in direction $\omega$ with weight $r^k$. Lemma 2.4 connects them: $\Gamma(\chi_k f)(z,\beta,\cos\psi)=T_k(f)(z,\beta,\psi)/(2\pi\sin^2\psi)$, so cone data give slice data of the weighted X-ray transform. Because slice circles are symmetric about the horizontal plane, only the even part $(\chi_k f)_e$ survives, and an identity expresses the unweighted even X-ray data $(\chi_0 f)_e$ as an integral of $z$-derivatives of $(\chi_k f)_e$, reducing any weight $k$ to the unweighted case. The even X-ray data are then read as V-line integrals in each vertical half-plane, and two-dimensional Radon inversion finishes the reconstruction.

What would settle it

Take a compactly supported smooth $f$ whose conical transform $T_k$ can be computed to high accuracy, evaluate the right-hand side of Theorem 2.6 at a point outside the support of $f$, and check that the principal-value integral is zero; a nonzero value, or divergence as the regularization is removed, would falsify the claimed inversion.

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Extended reading notes

Core claim

The central claim, Theorem 2.6, is that for every integer $k\ge 1$ and every compactly supported smooth $f$, the value $f(x)$ at $x=(r\,e_\varphi,x_3)$ is reproduced by the displayed integral of the conical data $T_k(f)(p/\sin\eta,\beta,\psi)$ and its derivatives, with the singular kernels $1/(\cos\psi-\sin\gamma\cos(\beta-\varphi))$ and $1/(r\cos\eta+x_3\sin\eta-p)$. The integrations run over the opening angle, the detector direction, and the cone-vertex position. Thus the three-dimensional family of conical projections with line vertices and horizontal axes determines $f$ exactly, not merely approximately. The proof converts cone integrals into vertical-slice data on the sphere, recovers the even weighted X-ray transform, and inverts the resulting V-line data by the two-dimensional Radon transform.

Load-bearing premise

The derivation assumes that the singular integrals in the inversion formula are interpreted as principal values and that data for obtuse cone opening angles are known through the symmetry relation; if either assumption fails, the equality in Theorem 2.6 need not hold.

Editorial extensions

If this is right

For every positive integer $k$, a compactly supported smooth function in $\mathbb{R}^3$ is determined exactly by its horizontal conical Radon transform on the three-parameter family of cones with vertices on a line.
Compton-camera setups with detectors on a line acquire a direct analytic reconstruction route: recover even weighted X-ray data, then invert a planar Radon transform slice by slice.
The inversion separates over the azimuth $\varphi$, so the reconstruction can be carried out independently for each vertical half-plane.
The formula handles every integer weight $k$ with the same structure, so stronger radial weighting needs no new inversion theory.

Reading between the lines

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

The $\psi$-integral in Theorem 2.6 runs to $\pi$ although $T_k$ is defined only for $\psi\in(0,\pi/2)$; the paper leaves implicit that obtuse-angle data must come from the symmetry $T_k(z,\beta,\pi-\psi)=T_k(z,\beta+\pi,\psi)$, so a detector recording only acute cones would need additional measurements or an extrapolation step.
The two kernel denominators are singular inside the integration domain, and the paper states no principal-value prescription; any numerical implementation must choose and validate a regularization before the formula can handle measured data.
Because the proof factors through the vertical slice transform, the same strategy could plausibly invert other three-parameter cone manifolds with horizontal axes, such as vertices lying on a closed curve; the paper does not pursue those geometries.
The inversion differentiates the data $k+1$ times in $p$ and once in $\psi$, so noise sensitivity is a likely practical obstacle; simulated-phantom tests of the formula's stability would be a natural next step.

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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

Short version: the paper has a novel setup—an inversion formula for the conical Radon transform over cones with horizontal axes and vertices on a line—but there is a gap in the proof of Lemma 2.5 that looks serious. I'd send it to a referee, but the referee should focus on that lemma.

What the paper does well: the problem is well-motivated from Compton cameras with line detectors. The even/odd decomposition of the weighted X-ray transform is clean, and Lemma 2.4 is correct. The idea to compose Gindikin's vertical slice inversion with V-line inversion is a reasonable strategy, and the final formula is explicit.

The soft spot is in Lemma 2.5. When applying Theorem 2.3, the function Γ(χ_k f)_e is extended by zero for |t|>1, but it has nonzero values at t=±1. The distributional derivative therefore contains delta functions at those endpoints. In the change of variables to ψ∈[0,π], they keep only the interior derivative and discard the deltas. The omitted terms involve χ_k f(z, ±e_β), i.e., horizontal-ray integrals, which are not recoverable from T_k on ψ∈(0,π/2). For a generic bump, these terms are nonzero. So formula (4) appears to be missing an additive boundary contribution. This is not the same as the principal-value or domain-extension issues the reader flagged; those are matters of convention. This is an omitted term in an identity.

Maybe the boundary terms cancel after the β and γ integrations, but no argument is given. As written, the proof of the main theorem is incomplete.

That said, the paper is not a wreck. The problem is important, the strategy is sensible, and with a careful treatment of the boundary terms—either showing they cancel or subtracting them—the result might be salvageable. The citation pattern looks fine, and the claimed novelty is real.

For peer review: I would accept it because the question is worthwhile and the gap is specific enough to be fixable. But the referee needs to check Lemma 2.5 carefully. I would not cite it in its current form.

Referee Report

3 major / 4 minor

Summary. The paper considers the weighted conical Radon transform T_k(f)(z,beta,psi) over all one-sided circular cones whose vertices lie on the z-axis and whose symmetry axes are horizontal, with weight ||x-b(z)||^{k-1}. It claims an explicit inversion formula: for compactly supported smooth f in R^3, f(x) is recovered by a triple integral involving T_k and its derivatives. The strategy is to relate T_k to the vertical slice transform of the weighted X-ray transform (Lemma 2.4), to invert the vertical slice transform using Gindikin's theorem (Lemma 2.5), and then to invert the resulting V-line transform via the two-dimensional Radon transform (Theorem 2.6).

Significance. If the main theorem were correct, it would provide an exact inversion formula for a three-dimensional family of cones, corresponding to Compton camera data with detectors on a line, and would complement existing results for other cone manifolds. The paper is concise and the overall reduction to established inversions is appealing. The derivation has no fitted parameters, and the reliance on known inversion theorems (Gindikin's vertical slice inversion and the Radon inversion) is transparent. However, the central derivation contains a serious omitted-boundary-term issue in Lemma 2.5, and the statement of Theorem 2.6 also leaves the domain of psi and the interpretation of singular integrals unspecified. These issues are load-bearing for the claimed formula.

major comments (3)

[§2, Lemma 2.5, proof of (4)] The passage from the q-integral to the psi-integral silently discards boundary terms. The function Gamma(chi_k f)_e is extended by zero for |t|>1, so its distributional derivative in q contains delta contributions at q = 1 - sin(eta) cos(beta-phi) and q = -1 - sin(eta) cos(beta-phi), corresponding to t = 1 and t = -1. The change of variables cos(psi) = sin(eta) cos(beta-phi) + q with psi in [0,pi] captures only the classical interior derivative and omits these endpoint deltas. The omitted contribution is of the form involving chi_k f(z, e_beta) and chi_k f(z, -e_beta), i.e. weighted X-ray data along horizontal rays. Such data are not present as an additive term in the displayed integrand of (4), and for a generic compactly supported f (for example, a positive bump on the x-axis) the omitted term is nonzero. Unless a cancellation is proved, or the boundary term is explicitly expressed through T_k (for instance via the limit of T_k/sin(psi) as psi -> 0+), formula (4) and hence Theorem 2.6 are incomplete.
[§2, Theorem 2.6] The inversion formula integrates psi over [0,pi], whereas T_k is defined only for psi in (0,pi/2). The manuscript implicitly relies on an extension of T_k to obtuse cones, presumably through the symmetry T_k(z, beta, pi-psi) = T_k(z, beta+pi, psi), which follows from the relation e_{beta+pi} = -e_beta and the definition of the cone. This extension is never stated or proved. Since the formula uses values of T_k at psi > pi/2, the statement of Theorem 2.6 is currently not well-defined and needs an explicit definition or a symmetry argument.
[§2, Theorem 2.6 and Lemma 2.5] The singular integrals in the formula are not given a precise interpretation. The inner psi-integral has the kernel 1/(cos(psi) - sin(gamma) cos(beta-phi)), which vanishes for generic gamma, beta, phi, psi, and the outer p-integral has the kernel 1/(r cos(eta) + x_3 sin(eta) - p), which is a Hilbert-type singular integral after the derivatives act on T_k. The paper neither states that these are principal-value integrals nor proves that the displayed iterated integral converges in the ordinary or principal-value sense for f in C_0^infinity(R^3). This is essential for the formula to be meaningful as an inversion formula.

minor comments (4)

[§2, Theorem 2.3] The statement of Gindikin's inversion formula mixes notation: the integrand contains 1/t and derivatives with respect to t, while the proof uses 1/q and derivatives with respect to q. The variables should be made consistent.
[§1, Definition of T_k] The domain of T_k is given as R x [0,2pi) x (0,pi/2), but Lemma 2.4 states an identity for psi in (0,pi). Please reconcile the domain of definition with the range of psi used throughout the paper.
[§2, Lemma 2.5] The notation phi in [0,2pi] should presumably be phi in [0,2pi), and the interval notation in the statement should be made uniform.
[§2, proof of Theorem 2.6] The sentence 'The inversion such transform can be reduced to that of the X-ray transform' contains a grammatical error and should read 'The inversion of such a transform...'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inversion is a composition of independent external inversion theorems and transform identities, with no fitted inputs or self-citation chain.

full rationale

The derivation chain is not circular. T_k(f) is defined directly as a weighted cone-surface integral (Section 1). Lemma 2.4 derives an exact identity relating T_k(f) to the vertical slice transform of the weighted X-ray transform chi_k f, using only spherical-coordinate substitution and integration over the fiber. Lemma 2.5 then applies Gindikin's vertical slice inversion theorem (Theorem 2.3) to recover (chi_0 f)_e from T_k(f), and uses an independent relation from Moon-Haltmeier [17] connecting chi_0 f to chi_k f. Theorem 2.6 finishes by passing from (chi_0 f)_e to the 2D Radon transform of a reflected restriction and applying the standard Radon inversion formula [13]. Each load-bearing step invokes an established external inversion theorem or a transform identity; no target quantity f or T_k(f) is fed back into the derivation as an assumption, no fitted parameter is renamed as a prediction, and no load-bearing step is justified solely by a self-citation. The concerns noted by the reader and skeptic about the psi-range extension, principal-value interpretation, and possible omitted boundary terms in Lemma 2.5 are potential rigor or correctness issues, not circularity: even if the formula were incomplete, that would not make the argument equivalent to its inputs by construction. Thus the circularity score is 0.

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

The reconstruction rests on three external inversion results and two unstated technical conventions; none are fitted and none are self-citations.

assumptions (5)

standard math Gindikin's vertical slice inversion formula (Theorem 2.3) for even continuous functions on S^2.
Used in Lemma 2.5 to recover (chi_k f)_e from its vertical slice transform; accepted from [9].
standard math Standard 2D Radon inversion formula f = (1/2 pi^2) integral integral partial_p R f / (x dot theta - p) dp dtheta.
Used in Theorem 2.6 to invert the V-line and Radon data on each half-plane; cited to [13].
domain assumption Relation (chi_0 f)_e(z, omega) = 1/(k-1)! integral (omega_3 - s)^(k-1) partial_z^k (chi_k f)_e(z, omega_1, omega_2, s) ds.
Quoted from [17] and used to reduce weighted X-ray data to unweighted V-line data; not proved in the paper.
domain assumption The conical transform data is known for all half-angles psi in (0, pi), or can be obtained from acute-angle data by the unstated symmetry T_k(z, beta, pi-psi) = T_k(z, beta+pi, psi).
Definition restricts psi to (0, pi/2) but the inversion formula integrates psi over [0, pi].
domain assumption Singular integrals in the inversion formula are interpreted as principal values.
The kernels 1/(cos psi - sin gamma cos(beta - phi)) and 1/(r cos eta + x3 sin eta - p) have zeros; the paper does not state the principal-value interpretation.

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Cite this review

Pith. "Pith review of An Inversion Formula for Horizontal Conical Radon Transform." pith.science (2026). https://pith.science/paper/AWHKEZRT

@misc{pith2026190808255,
  author       = {Pith},
  title        = {Pith review of: An Inversion Formula for Horizontal Conical Radon Transform},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AWHKEZRT}},
  note         = {Machine review of arXiv:1908.08255}
}

read the original abstract

In this paper, we consider the conical Radon transform on all cones with horizontal central axis whose vertices are on a straight line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane.

Figures

Figures reproduced from arXiv: 1908.08255 by the authors.

**Figure 1.** Our setups: the conical Radon transform over all cones with horizontal central line and the vertex in z-axis. In order to invert the conical introduced in the previous section, we introduce the weighted X-ray and vertical slice transforms. Definition 2.1 (The weighted X-Ray transform). Let k ∈ N and f ∈ C∞ 0 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Detecting small low emission radiating sources

Moritz Allmaras, David P Darrow, Yulia Hristova, Guido Kanschat, and Peter Kuchment. Detecting small low emission radiating sources. arXiv preprint arXiv:1012.3373 , 2010

arXiv 2010

[2] [2]

Inversion of the v-line radon transform in a disc and its applications in imaging

Gaik Ambartsoumian. Inversion of the v-line radon transform in a disc and its applications in imaging. Computers & Mathematics with Applications , 64(3):260–265, 2012

work page 2012

[3] [3]

Analytical reconstruction formula for one- dimensional compton camera

Roman Basko, Gengsheng L Zeng, and Grant T Gullberg. Analytical reconstruction formula for one- dimensional compton camera. IEEE Transactions on Nuclear Science , 44(3):1342–1346, 1997

work page 1997

[4] [4]

Application of spherical harmonics to image reconstruction for the compton camera

Roman Basko, Gengsheng L Zeng, and Grant T Gullberg. Application of spherical harmonics to image reconstruction for the compton camera. Physics in Medicine & Biology , 43(4):887, 1998

work page 1998

[5] [5]

Towards direct reconstruction from a gamma camera based on compton scattering

Michael J Cree and Philip J Bones. Towards direct reconstruction from a gamma camera based on compton scattering. IEEE transactions on medical imaging , 13(2):398–407, 1994

work page 1994

[6] [6]

Gamma-radiation imaging system based on the compton eﬀect

DB Everett, JS Fleming, RW Todd, and JM Nightingale. Gamma-radiation imaging system based on the compton eﬀect. In Proceedings of the Institution of Electrical Engineers , volume 124, pages 995–1000. IET, 1977. 5

work page 1977

[7] [7]

Inversion formulas for the broken-ray radon transform

Lucia Florescu, Vadim A Markel, and John C Schotland. Inversion formulas for the broken-ray radon transform. Inverse Problems, 27(2):025002, 2011

work page 2011

[8] [8]

Single-scattering optical tomography

Lucia Florescu, John C Schotland, and Vadim A Markel. Single-scattering optical tomography. Physical Review E, 79(3):036607, 2009

work page 2009