THE ROLE OF FOURIER ANALYSIS IN TWO DIMENSIONAL TOMOGRAPHY

Andre Mas; Fatma Terzioglu; Ilse C.F. Ipsen

arxiv: 2604.18541 · v1 · submitted 2026-04-20 · ⚛️ physics.optics · cs.NA· math.NA

THE ROLE OF FOURIER ANALYSIS IN TWO DIMENSIONAL TOMOGRAPHY

Andre Mas , Fatma Terzioglu , Ilse C.F. Ipsen This is my paper

Pith reviewed 2026-05-10 03:44 UTC · model grok-4.3

classification ⚛️ physics.optics cs.NAmath.NA

keywords Fourier transformtomographyinversion formulasdivergent beam transformV-line transformoptical tomographyintegral transforms

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The pith

Fourier analysis supplies explicit inversion formulas for the divergent beam and V-line transforms used in two-dimensional tomography.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the Fourier transform can be used to obtain inversion formulas for integral transforms arising in tomographic imaging. It demonstrates the method on the divergent beam transform and on the V-line transform that appears in models of single-scattering optical tomography. A reader cares because these formulas turn measured data directly into reconstructed images without requiring iterative numerical schemes or hidden approximations. The work therefore presents Fourier analysis as a systematic tool for recovering functions from their line or beam integrals in the plane.

Core claim

The Fourier transform plays a central role in deriving inversion formulas for the integral transforms of tomographic imaging. Explicit formulas are obtained for the divergent beam transform and for the V-line transform, the latter appearing in contemporary models of single-scattering optical tomography.

What carries the argument

Application of Fourier transform properties to the integral transforms to produce explicit inversion formulas.

If this is right

The divergent beam transform admits an explicit inversion formula obtained via Fourier analysis.
The V-line transform arising in single-scattering optical tomography also admits an explicit inversion formula via the same route.
These formulas supply a direct reconstruction method for image data collected under the respective geometries.
The same Fourier approach can serve as a template for other integral transforms that appear in two-dimensional tomography.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit formulas could be discretized for direct numerical reconstruction algorithms in optical imaging systems.
Similar Fourier reductions might apply to related transforms in three-dimensional or limited-angle tomography settings.
If the V-line formula proves stable, it could reduce the need for iterative solvers in single-scattering optical tomography.

Load-bearing premise

Fourier methods can be applied directly to these transforms to produce exact inversion formulas without extra approximations or domain restrictions.

What would settle it

Apply the derived V-line inversion formula to a known test function whose V-line transform can be computed exactly; if the output does not recover the test function, the claim is false.

Figures

Figures reproduced from arXiv: 2604.18541 by Andre Mas, Fatma Terzioglu, Ilse C.F. Ipsen.

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

We highlight the important role of the Fourier transform in deriving inversion formulas for the integral transforms of tomographic imaging. We demonstrate this principle by deriving inversion formulas for the divergent beam transform and the V-line transform, the latter arising in contemporary models of single-scattering optical tomography.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows how Fourier properties yield explicit inversion formulas for the divergent-beam and V-line transforms in 2D, but the advance is mainly an application of known methods rather than a new framework.

read the letter

The core point is that standard Fourier-slice ideas can be pushed through to produce closed-form inverses for the V-line transform that appears in single-scattering optical tomography. The authors walk through the divergent-beam case first, then adapt the same steps to the V-line geometry. That demonstration is the main concrete output. If the derivations are clean and the formulas are stable under the stated conditions, they give people working on optical forward models a direct tool they can code up without having to reinvent the inversion each time. The paper stays tightly focused on two dimensions and two specific operators, which keeps the claims manageable. The writing is straightforward and the logic follows the usual projection-slice route without unnecessary detours. The soft spot is the handling of the V-line geometry itself. Because the two rays meet at a vertex with a fixed opening angle, the operator is not translation-invariant in the same way the Radon or parallel-beam transforms are. Any Fourier representation therefore probably requires either a fixed scattering angle across all measurements or compact support away from the vertices. The abstract does not flag these restrictions explicitly, so the reader has to check the full derivations to see whether the formulas remain exact only inside those limits or whether extra approximation steps creep in. Without numerical checks or stability estimates in the text, it is also hard to judge how sensitive the inversion is to noise or discretization. This is the kind of note that helps specialists who already know the Radon and fan-beam literature and now need the V-line version for an optics application. A reader outside mathematical tomography will not find much new machinery. It is worth sending to referees because the derivations, if they hold up, are reproducible and directly usable; a short revision to spell out the domain restrictions and add a small numerical test would make it a solid, citable reference rather than a borderline one.

Referee Report

1 major / 2 minor

Summary. The manuscript highlights the role of the Fourier transform in deriving inversion formulas for integral transforms in two-dimensional tomography. It demonstrates the approach by deriving explicit inversion formulas for the divergent-beam transform and the V-line transform (arising in single-scattering optical tomography models).

Significance. If the derivations hold without hidden restrictions, the work provides a unified Fourier-based framework for these transforms, which could aid reconstruction algorithms in optical tomography. The explicit derivations for both transforms are a strength, offering potential for parameter-free or stable inversions under appropriate conditions.

major comments (1)

[V-line transform derivation] The V-line transform derivation (central to the second example) integrates along two rays with a fixed opening angle at the vertex. This geometry lacks the translation invariance of the Radon or divergent-beam transforms, so the Fourier-slice theorem application requires either a vertex-tied coordinate system or the assumption of constant angle across all measurements. The manuscript should explicitly state these conditions and verify stability/exactness outside compact support away from the vertex; otherwise the claimed generality for contemporary optical tomography models is overstated.

minor comments (2)

[Abstract] The abstract asserts derivations exist but provides no equations or steps; the full manuscript should include at least one key intermediate equation (e.g., the Fourier representation of the V-line operator) in the main text for immediate verification.
[Introduction] Notation for the V-line transform (e.g., opening angle parameter) should be introduced consistently with the divergent-beam case to facilitate comparison.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The manuscript aims to illustrate the utility of Fourier analysis in deriving inversion formulas, and we address the specific concern on the V-line transform below.

read point-by-point responses

Referee: The V-line transform derivation (central to the second example) integrates along two rays with a fixed opening angle at the vertex. This geometry lacks the translation invariance of the Radon or divergent-beam transforms, so the Fourier-slice theorem application requires either a vertex-tied coordinate system or the assumption of constant angle across all measurements. The manuscript should explicitly state these conditions and verify stability/exactness outside compact support away from the vertex; otherwise the claimed generality for contemporary optical tomography models is overstated.

Authors: We agree that the V-line geometry lacks the translation invariance of the classical Radon transform. In the derivation, the Fourier transform is taken with respect to the vertex position variable while holding the opening angle fixed, which corresponds to a vertex-tied coordinate system for each measurement. This is the standard setting in single-scattering optical tomography models that motivate the V-line transform. We will revise the manuscript to state these assumptions explicitly in the relevant section and to add a short paragraph clarifying the domain of exactness and stability for functions whose support is separated from the vertex. With these clarifications the claimed applicability to contemporary models remains accurate rather than overstated. revision: yes

Circularity Check

0 steps flagged

Fourier derivations for divergent-beam and V-line transforms rely on standard properties without self-referential reduction

full rationale

The provided abstract and context describe deriving inversion formulas via Fourier transform properties applied to the divergent beam and V-line transforms. No equations, self-citations, fitted parameters, or ansatzes are quoted that reduce any claimed prediction or uniqueness result to its own inputs by construction. The central claim uses external Fourier analysis as an independent tool rather than redefining or fitting the transforms internally. This is the expected non-circular outcome for a methods paper highlighting standard transform techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard Fourier analysis for integral transforms with no free parameters, ad-hoc axioms, or invented entities identified.

axioms (1)

standard math Fourier transform properties apply to derive inverses for divergent beam and V-line integral transforms
Invoked implicitly as the core tool for inversion formulas in tomographic imaging.

pith-pipeline@v0.9.0 · 5337 in / 1135 out tokens · 39843 ms · 2026-05-10T03:44:42.706104+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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V ondˇrejc, J

G. Ambartsoumian , Inversion of the V-line Radon transform in a disc and its appl ications in imaging, Comput. Math. Appl., 64 (2012), pp. 260–265, https://doi.org/10.1016/j.camwa. 2012.01.059

work page doi:10.1016/j.camwa 2012

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G. Ambartsoumian , Generalized Radon Transforms and Imaging by Scattered Part icles: Bro- ken Rays, Cones, and Stars in Tomography , no. 6 in Contemporary Mathematics and Its Applications, W orld Scientiﬁc, Hackensack, NJ, 2023, https://doi.org/10.1142/13113

work page doi:10.1142/13113 2023

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Ambartsoumian, M

G. Ambartsoumian, M. J. L. Jebelli, and R. K. Mishra , Generalized V-line transforms in 2D vector tomography , Inverse Problems, 36 (2020), pp. 104002, 25, https://doi.org/10. 1088/1361-6420/abaa2a. 12 A. MAS, F. TERZIOGLU, AND I. C.F. IPSEN

2020

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Ambartsoumian, M

G. Ambartsoumian, M. J. L. Jebelli, and R. K. Mishra , Numerical implementation of generalized V-line transforms on 2D vector ﬁelds and their i nversions, SIAM J. Imaging Sci., 17 (2024), pp. 366–396, https://doi.org/10.1137/23M1580700

work page doi:10.1137/23m1580700 2024

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Ambartsoumian and M

G. Ambartsoumian and M. J. Latifi , Inversion and symmetries of the star transform , J. Geom. Anal., 31 (2021), pp. 11270–11291, https://doi.org/10.1007/s12220-021-00680-5

work page doi:10.1007/s12220-021-00680-5 2021

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Ambartsoumian and M

G. Ambartsoumian and M. J. Latifi Jebelli , The V-line transform with some generalizations and cone diﬀerentiation , Inverse Problems, 35 (2019), pp. 034003, 26, https://doi.org/10. 1088/1361-6420/aafe8a

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Florescu, V

L. Florescu, V. A. Markel, and J. C. Schotland , Single-scattering optical tomogra- phy: simultaneous reconstruction of scattering and absorp tion, Phys. Rev. E, 81 (2010), pp. 016602, 11, https://doi.org/10.1103/PhysRevE.81.016602

work page doi:10.1103/physreve.81.016602 2010

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Florescu, V

L. Florescu, V. A. Markel, and J. C. Schotland , Inversion formulas for the broken-ray Radon transform , Inverse Problems, 27 (2011), pp. 025002, 13, https://doi.org/10.1088/ 0266-5611/27/2/025002

2011

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Florescu, J

L. Florescu, J. C. Schotland, and V. A. Markel , Single-scattering optical tomography , Phys. Rev. E, 79 (2009), pp. 036607, 7, https://doi.org/10.1103/PhysRevE.79.036607

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2013

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Gouia-Zarrad and G

R. Gouia-Zarrad and G. Ambartsoumian , Exact inversion of the conical Radon transform with a ﬁxed opening angle , Inverse Problems, 30 (2014), pp. 045007, 12, https://doi.org/ 10.1088/0266-5611/30/4/045007

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Katsevich and R

A. Katsevich and R. Krylov , Broken ray transform: inversion and a range condition , Inverse Problems, 29 (2013), pp. 075008, 20, https://doi.org/10.1088/0266-5611/29/7/075008

work page doi:10.1088/0266-5611/29/7/075008 2013

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Krylov and A

R. Krylov and A. Katsevich , Inversion of the broken ray transform in the case of energy- dependent attenuation , Phys. Med. Biol., 60 (2015), pp. 4313–4335, https://doi.org/10. 1088/0031-9155/60/11/4313

2015

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Kuchment and F

P. Kuchment and F. Terzioglu , Inversion of weighted divergent beam and cone transforms , Inverse Probl. Imaging, 11 (2017), pp. 1071–1090, https://doi.org/10.3934/ipi.2017049

work page doi:10.3934/ipi.2017049 2017

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