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V-line tensor tomography: numerical results

Gaik Ambartsoumian, Rohit Kumar Mishra, Indrani Zamindar

TL;DR

This work studies numerical inversion algorithms for V-line transforms of symmetric $2$-tensor fields in the plane, including longitudinal, transverse, and mixed V-line transforms $\mathcal{L},\mathcal{T},\mathcal{M}$ and the star transform $\mathcal{S}$, with their first moments. Building on prior theory, it provides explicit reconstruction formulas for special tensor forms (e.g., $\mathbf{f}=\mathrm{d}^2\varphi$, $(\mathrm{d}^\perp)^2\varphi$, $\mathrm{d}\mathrm{d}^\perp\varphi$) and develops PDE-based and direct inversion methods to recover general tensor fields from combinations of transforms, validated through extensive MATLAB simulations on smooth and non-smooth phantoms under varied noise. The numerical results reveal robust recoveries for several tensor configurations, while highlighting ill-conditioning and artifacts associated with first-moment inversions and numerical differentiation, which can be mitigated by domain padding and post-processing. Overall, the paper demonstrates accurate, noise-robust tensor tomography in the V-line setting and clarifies the impact of opening angles and data types on reconstruction quality, with practical implications for imaging modalities using piecewise-linear data paths.

Abstract

This article presents the numerical verification and validation of several inversion algorithms for V-line transforms (VLTs) acting on symmetric 2-tensor fields in the plane. The analysis of these transforms and the theoretical foundation of their inversion methods were studied in a recent work [G. Ambartsoumian, R. K. Mishra, and I. Zamindar, Inverse Problems, 40 (2024), 035003]. We demonstrate the efficient recovery of an unknown symmetric 2-tensor field from various combinations of the longitudinal, transverse, and mixed VLTs, their corresponding first moments, and the star VLT. The paper examines the performance of the proposed algorithms in different settings and illustrates the results with numerical simulations on smooth and non-smooth phantoms.

V-line tensor tomography: numerical results

TL;DR

This work studies numerical inversion algorithms for V-line transforms of symmetric -tensor fields in the plane, including longitudinal, transverse, and mixed V-line transforms and the star transform , with their first moments. Building on prior theory, it provides explicit reconstruction formulas for special tensor forms (e.g., , , ) and develops PDE-based and direct inversion methods to recover general tensor fields from combinations of transforms, validated through extensive MATLAB simulations on smooth and non-smooth phantoms under varied noise. The numerical results reveal robust recoveries for several tensor configurations, while highlighting ill-conditioning and artifacts associated with first-moment inversions and numerical differentiation, which can be mitigated by domain padding and post-processing. Overall, the paper demonstrates accurate, noise-robust tensor tomography in the V-line setting and clarifies the impact of opening angles and data types on reconstruction quality, with practical implications for imaging modalities using piecewise-linear data paths.

Abstract

This article presents the numerical verification and validation of several inversion algorithms for V-line transforms (VLTs) acting on symmetric 2-tensor fields in the plane. The analysis of these transforms and the theoretical foundation of their inversion methods were studied in a recent work [G. Ambartsoumian, R. K. Mishra, and I. Zamindar, Inverse Problems, 40 (2024), 035003]. We demonstrate the efficient recovery of an unknown symmetric 2-tensor field from various combinations of the longitudinal, transverse, and mixed VLTs, their corresponding first moments, and the star VLT. The paper examines the performance of the proposed algorithms in different settings and illustrates the results with numerical simulations on smooth and non-smooth phantoms.
Paper Structure (14 sections, 6 theorems, 34 equations, 30 figures, 16 tables)

This paper contains 14 sections, 6 theorems, 34 equations, 30 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Phantoms, data formation, and numerical solution of PDEs
  4. Description of phantoms
  5. Data formation
  6. Solving PDEs numerically
  7. Numerical Implementation
  8. Numerical implementation for special kinds of tensor fields
  9. Tensor fields of the form f = d2 varphi, f = ddperp varphi, or f = (dperp2 varphi
  10. Tensor fields of the form f = dg or f = dperp g
  11. Full Recovery of f from Lf, Tf and Mf
  12. Recovery of tensor fields from first moment VLTs
  13. Numerical implementation of inverting the tensor star transform
  14. Acknowledgements

Key Result

Theorem 1

Let $\varphi$ be a twice differentiable, compactly supported function, that is, $\varphi \in C^2_c(D_1)$.

Figures (30)

  • Figure 1: The components of the phantoms considered in numerical reconstructions.
  • Figure 2: Recovery of $\varphi$ when $\textbf{f}$=$\mathrm{d}^2\varphi$, using formula \ref{['eq:f=d^2varphi']}.
  • Figure 3: Graphical representation of $\varphi_{\text{original}} -\varphi_{\text{rec}}$. The colormap is scaled according to the minimum and maximum values of all four pictures.
  • Figure 4: Recovery of $\varphi$ when $\textbf{f}$=$\mathrm{d}\mathrm{d}^\perp\varphi$, $u_1 >u_2$ (elliptic), with $10\%$ noise, using \ref{['eq:f=dd^perp_varphi_LT']}, \ref{['eq:f=dd^perp_varphi_M']}.
  • Figure 5: Recovery of $\varphi$ when $\textbf{f}$=$\mathrm{d}\mathrm{d}^\perp\varphi$, $u_1=u_2$ (parabolic), with $10\%$ noise, using \ref{['eq:f=dd^perp_varphi_LT']}, \ref{['eq:f=dd^perp_varphi_M']}.
  • ...and 25 more figures

Theorems & Definitions (21)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Definition 4
  • Remark 3.1
  • Remark 3
  • Remark 4
  • Remark 5
  • ...and 11 more