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Reconstruction of scalar functions and vector fields from weighted V-line transforms with swinging branches

Gaik Ambartsoumian, Rohit Kumar Mishra, Indrani Zamindar

TL;DR

The article develops a unified framework for reconstructing scalar functions and vector fields from weighted V-line transforms with swinging branches in the plane, allowing vertex locations inside the support and without restricting branch directions or weights. It provides kernel descriptions and inversion formulas for diverse regimes: divergent-beam limits ($\alpha=0$), general scalar V-lines ($\alpha\neq0$), and vector V-lines at $\alpha=1$, as well as mixed cases with first moments. Key contributions include transport-equation based inversion for scalars, explicit kernel characterizations for vector data, and Poisson-system based reconstructions for vector fields, with explicit procedures for constant-branch settings and extensions to integral moments. The results generalize several prior findings in the literature and lay groundwork for numerical algorithms and higher-dimensional tensor extensions, with open questions on non-constant branches and practical implementation. The work has potential impact on single-scattering tomography and related imaging modalities by enabling robust recovery under more realistic, flexible data-collection geometries.

Abstract

Weighted V-line transforms map a symmetric tensor field of order $m\ge0$ to a linear combination of certain integrals of those fields along two rays emanating from the same vertex. A significant focus of current research in integral geometry centers on the inversion of V-line transforms in formally determined setups. Of particular interest are the restrictions of these operators in which the vertices of integration trajectories can be anywhere inside the support of the field, while the directions of the pair of rays, often called branches of the V-line, are determined by the vertex location. Such transforms have been thoroughly investigated when the branch directions are either constant or radial. In addition to that, in most of the prior research on this subject, it was assumed that the weights of integration along each branch are the same. In this paper we analyze the transforms defined on scalar functions and vector fields, satisfying a much weaker assumption on the branch directions. The weights restriction is also lifted in all but one setup. Consequently, we extend multiple previously known results on the kernel description, injectivity, and inversion of the transforms with simplifying assumptions and prove pertinent statements for more general setups not studied before.

Reconstruction of scalar functions and vector fields from weighted V-line transforms with swinging branches

TL;DR

The article develops a unified framework for reconstructing scalar functions and vector fields from weighted V-line transforms with swinging branches in the plane, allowing vertex locations inside the support and without restricting branch directions or weights. It provides kernel descriptions and inversion formulas for diverse regimes: divergent-beam limits ($\alpha=0$), general scalar V-lines ($\alpha\neq0$), and vector V-lines at $\alpha=1$, as well as mixed cases with first moments. Key contributions include transport-equation based inversion for scalars, explicit kernel characterizations for vector data, and Poisson-system based reconstructions for vector fields, with explicit procedures for constant-branch settings and extensions to integral moments. The results generalize several prior findings in the literature and lay groundwork for numerical algorithms and higher-dimensional tensor extensions, with open questions on non-constant branches and practical implementation. The work has potential impact on single-scattering tomography and related imaging modalities by enabling robust recovery under more realistic, flexible data-collection geometries.

Abstract

Weighted V-line transforms map a symmetric tensor field of order to a linear combination of certain integrals of those fields along two rays emanating from the same vertex. A significant focus of current research in integral geometry centers on the inversion of V-line transforms in formally determined setups. Of particular interest are the restrictions of these operators in which the vertices of integration trajectories can be anywhere inside the support of the field, while the directions of the pair of rays, often called branches of the V-line, are determined by the vertex location. Such transforms have been thoroughly investigated when the branch directions are either constant or radial. In addition to that, in most of the prior research on this subject, it was assumed that the weights of integration along each branch are the same. In this paper we analyze the transforms defined on scalar functions and vector fields, satisfying a much weaker assumption on the branch directions. The weights restriction is also lifted in all but one setup. Consequently, we extend multiple previously known results on the kernel description, injectivity, and inversion of the transforms with simplifying assumptions and prove pertinent statements for more general setups not studied before.

Paper Structure

This paper contains 15 sections, 17 theorems, 63 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Listing of the main results
  4. Divergent beam transforms ()
  5. V-line transform of scalar functions ()
  6. Longitudinal and transverse V-line transforms with their first moments (a1)
  7. Reconstruction of a vector field f from Lf and Tf
  8. Proof of Theorem \ref{['th:kernel of L and T V-line']}
  9. Proof of Theorem \ref{['th:Inversion of L and T V-line']}
  10. Recovery of a vector field f from Lf/Lf and Lf/Tf
  11. Longitudinal/transverse V-line transforms, when and u,v are constant vector fields
  12. Full recovery of a vector field from and
  13. Full recovery of using integral moments
  14. Open questions and future work
  15. Acknowledgements

Key Result

Theorem 1

For any $\textbf{f} \in C^2_c(S^1;\mathbb{D})$, there exist unique smooth functions $\varphi$ and $\psi$ such that

Figures (1)

  • Figure 1: The sketch on the left describes a simple setup of single scattering tomography. The source $S$ emits radiation along certain rays. The single scattered photons are then captured by either a convex ($\Gamma_1$) or a concave ($\Gamma_2$) array of collimated detectors. Concave-type detectors are often used in CT, while convex detectors are used in pin-hole cameras in nuclear imaging. Under certain assumptions, the knowledge of intensity of incoming and scattered radiation for each source-detector pair provides the VLT of the attenuation coefficient $\mu_t(\textbf{x})$ of the medium (e.g. see amb-book). It is easy to see that the integral curves of the corresponding vector fields $\textbf{u}(\textbf{x})$ and $\textbf{v}(\textbf{x})$ here are straight line segments. In particular, if the detector arrays are placed along circular arcs, then the resulting vector fields are focal. That is, for each detector $\Gamma_i$ there is a fixed point $\textbf{x}_0^i$ (the focus of $\Gamma_i$), such that the rays detected by the detector $\Gamma_i$ pass through $\textbf{x}_0^i$. In other words, for all $\textbf{x}$ inside the image domain, one can define the vector fields as $\textbf{u}(\textbf{x}) = \frac{\textbf{x}_0^1 - \textbf{x}}{|\textbf{x}_0^1 - \textbf{x}|}$ and $\textbf{v}(\textbf{x}) = \frac{\textbf{x}_0^2 - \textbf{x}}{|\textbf{x}_0^2 - \textbf{x}|}$. In the case when $\Gamma_1$ are $\Gamma_2$ are flat (i.e. when the foci of the detector arrays are at infinity), the vector fields $\textbf{u}$ and $\textbf{v}$ are constant.

Theorems & Definitions (44)

  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 2
  • Theorem : derevtsov3
  • Theorem 1
  • proof
  • Theorem 2: Kernel Description
  • ...and 34 more