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arxiv: 2606.31632 · v1 · pith:U4HSVZKR · submitted 2026-06-30 · math.NA · cs.NA· math.AP

V-Line Tensor Tomography in a Disk: Theoretical and Numerical Reconstruction

Rahul Bhardwaj , Madhu Gupta This is my paper

Reviewed by Pith2026-07-01 04:18 UTCgrok-4.3pith:U4HSVZKRopen to challenge →

classification math.NA cs.NAmath.AP
keywords V-line transformtensor tomographykernel characterizationinversion formulasymmetric tensor fieldsnumerical reconstruction
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The pith

The kernel of V-line transforms on symmetric m-tensor fields inside a disk is explicitly characterized, yielding a new inversion formula via decomposition.

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

The paper characterizes the kernel of the V-line transforms applied to symmetric m-tensor fields supported inside a disk of radius R. It then applies a decomposition result to produce an explicit inversion formula that recovers the tensor field from the transform data. The work also supplies numerical reconstructions for the cases of vector fields and symmetric 2-tensor fields, testing the formulas on phantoms both with and without added noise. A reader cares because V-line data arise in imaging geometries that follow broken lines, and an exact kernel description identifies which tensor features remain invisible to the measurements.

Core claim

We provide an explicit characterization of the kernel of the V-line transforms acting on a symmetric m-tensor field and derive a new inversion formula using a decomposition result. Numerical verification confirms the algorithms for vector fields and symmetric 2-tensor fields on various phantoms, including under noise.

What carries the argument

Explicit kernel characterization of the V-line transform on symmetric m-tensor fields, together with the decomposition result that produces the inversion formula.

If this is right

  • The kernel consists precisely of those symmetric m-tensor fields that produce zero V-line data.
  • The decomposition-based inversion recovers the field uniquely from complete V-line measurements.
  • The same numerical schemes remain stable for both m=1 and m=2 when moderate noise is present.
  • Reconstruction quality holds across multiple phantom geometries tested in the disk.

Where Pith is reading between the lines

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

  • The kernel description may supply stability bounds for the inversion when data are incomplete.
  • The decomposition technique could be tested on V-line transforms defined on other bounded domains.
  • Numerical implementations might extend directly to higher-order symmetric tensors once the kernel formula is available.

Load-bearing premise

The symmetric m-tensor field has compact support strictly inside the disk of radius R centered at the origin.

What would settle it

A symmetric m-tensor field inside the disk whose V-line transform vanishes but lies outside the stated kernel, or a numerical test phantom whose reconstruction by the derived formula deviates from the known ground truth beyond discretization error.

Figures

Figures reproduced from arXiv: 2606.31632 by Madhu Gupta, Rahul Bhardwaj.

Figure 1
Figure 1. Figure 1: (a) Construction of the V-line BR(β, d) & (b) Configuration of broken rays used for data collection. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: displays the results for this smooth test case. The first column shows the ground-truth scalar components f1 and f2. The second column illustrates the simulated forward data Lf (β, d) and T f (β, d) plotted against the angular variable β (in degrees) and the distance parameter d. The final column displays the reconstructed components obtained via the proposed inversion method. We see that the reconstructed… view at source ↗
Figure 3
Figure 3. Figure 3: Reconstructed components of f with different levels of noise. We now consider another phantom, which we refer to as Ph2. In this class, each component of the phantom is represented as a weighted combination of three characteristic functions of disks having different radii r and centers (a, b). The values of these parameters are given in [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Original components of the vector field f (first column), the associated V-line transform data Lf and T f (second column), and the reconstructed vector field components (third column). Extending the validation of the inversion scheme, [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstructed components of f with different levels of noise. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Original components of the vector field f (first column), the associated V-line transform data Lf and T f (second column), and the reconstructed vector field components (third column). Further, [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstructed components of f with different levels of noise. Phantoms f No noise 5% noise 10% noise 20% noise PH1 f1 1.29% 5.33% 10.40% 20.82% PH1 f2 1.57% 8.38% 16.46% 33.04% PH2 f1 8.29% 31.72% 62.25% 123.26% PH2 f2 9.52% 60.84% 120.30% 239.77% PH3 f1 15.62% 39.04% 72.89% 143.92% PH3 f2 16.37% 31.26% 55.52% 106.95% [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reconstructed components of f (Phantom 2) for different angles (θ = 15◦ to 75◦ ) [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reconstructed components of f (Phantom 3) for different angles (θ = 15◦ to 75◦ ). 5.2 Reconstruction of Tensor Fields To generate the data, we will follow the same procedure as discussed in subsection 5.1. The only difference is that we will now consider the three components of f = (f11, f12, f22), and we compute Lf , Mf , and T f . Now the goal is to recover f11, f12 and f22 from the given Lf , Mf , and T… view at source ↗
Figure 10
Figure 10. Figure 10: and 12 shows the reconstruction of the f11, f12 and f22 for Ph2 and Ph3 in the absence of noise. This shows that the algorithm can reconstruct sharp discontinuities for both convex and non-convex geometries with almost no artifacts [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Reconstructed 2-tensor field f (Phantom 2) from Lf , T f , and Mf with noisy data [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reconstructed 2-tensor field f (Phantom 3) from Lf , T f , and Mf . 23 [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Reconstructed 2-tensor field f (Phantom 3) from Lf , T f , and Mf with noisy data. 6 Acknowledgements Rahul Bhardwaj gratefully acknowledges the partial financial support provided by the FIST programme of the Department of Science and Technology (DST), Government of India, under Grant No. SR/FST/MS-I/2018/22(C). Data availability statement. No datasets were generated or analyzed during the current study; … view at source ↗
read the original abstract

In this article, we investigate V-line transforms for symmetric $m$-tensor fields whose support lies inside a disk of radius $R$ and centered at the origin. We provide an explicit characterization of the kernel of the V-line transforms acting on a symmetric $m$-tensor field and derive a new inversion formula using a decomposition result. In addition, we present a comprehensive numerical verification and validation of the inversion algorithms for these V-line transforms for vector fields and symmetric $2$-tensor fields, which were recently developed in \cite{bhardwaj_2024,bhardwaj2025tensor}. The reconstruction results obtained for various phantoms demonstrate the effectiveness and robustness of the proposed numerical methods, including in the presence of noise.

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.

Referee Report

1 major / 2 minor

Summary. The paper investigates V-line transforms acting on symmetric m-tensor fields with compact support inside a disk of radius R centered at the origin. It claims an explicit characterization of the kernel of these transforms, derives a new inversion formula via a decomposition result, and provides numerical verification and validation of inversion algorithms (for vector fields and symmetric 2-tensor fields) taken from the authors' prior work, demonstrating effectiveness and noise robustness on various phantoms.

Significance. If the kernel characterization and inversion formula are correct under the stated hypotheses, the work advances V-line tensor tomography by supplying explicit theoretical tools for higher-order tensors together with concrete numerical evidence of practical reconstruction quality. The numerical experiments on multiple phantoms, including noisy data, constitute a clear strength.

major comments (1)
  1. [Abstract, §2] Abstract and §2 (setting and assumptions): The explicit kernel characterization and the new inversion formula are derived under the hypothesis that the symmetric m-tensor field has compact support strictly inside the disk. The manuscript supplies no analysis, extension, or counter-example for the case in which support reaches or touches the boundary, where V-line integration paths can interact with the boundary in ways excluded by the strict-interior hypothesis; this precondition is load-bearing for the central theoretical claims.
minor comments (2)
  1. [§5] §5 (numerical section): The description of phantom construction and data-selection criteria should be expanded so that the reported reconstruction results can be reproduced from the given information.
  2. Notation: The distinction between the new inversion formula derived in this paper and the algorithms validated numerically (taken from the cited prior works) should be made explicit in the text and in the figure captions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive evaluation of the paper's contributions. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Abstract, §2] Abstract and §2 (setting and assumptions): The explicit kernel characterization and the new inversion formula are derived under the hypothesis that the symmetric m-tensor field has compact support strictly inside the disk. The manuscript supplies no analysis, extension, or counter-example for the case in which support reaches or touches the boundary, where V-line integration paths can interact with the boundary in ways excluded by the strict-interior hypothesis; this precondition is load-bearing for the central theoretical claims.

    Authors: The strict-interior compact support hypothesis is explicitly required for the kernel characterization and the decomposition-based inversion formula, because it guarantees that every V-line segment lies entirely in the open disk where the tensor field is supported and avoids any boundary interaction that would alter the integral geometry. This setting is stated in the abstract and Section 2 and is the natural domain for the theoretical tools developed. Extending the results to the case of support touching the boundary would demand a separate analysis of modified path behaviors and possible singularities, which lies outside the scope of the present work. We therefore do not revise the manuscript to include such an extension or counter-examples. revision: no

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states new theoretical results: an explicit kernel characterization for the V-line transform on symmetric m-tensor fields and a new inversion formula derived via a decomposition result, under the explicit assumption of compact support strictly inside the disk. The self-citations to bhardwaj_2024 and bhardwaj2025tensor apply only to the numerical verification and validation of previously developed algorithms for the vector and 2-tensor cases; they do not supply the kernel characterization or the claimed new inversion formula. No equations, definitions, or steps in the provided abstract reduce the central claims to fitted inputs, self-definitions, or a self-citation chain by construction. The support assumption is stated as a precondition but does not create a circular reduction in the derivation itself. This is a standard extension of prior work with independent theoretical content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are visible from the abstract; the work relies on standard properties of symmetric tensor fields and integral transforms in the plane.

axioms (1)
  • standard math Standard properties of symmetric tensor fields and line integrals in Euclidean space hold without additional restrictions.
    Invoked implicitly as the background for defining V-line transforms on tensors.

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Reference graph

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