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Analysis and X-ray tomography

Joonas Ilmavirta

TL;DR

The notes address the fundamental problem of recovering a function from its line integrals (the X-ray transform) within the broad field of inverse problems, focusing on injectivity and uniqueness rather than numerical methods.A diverse methodological toolkit is developed, including Fourier analysis on circles and tori, angular Fourier decompositions, Abel and circular-average methods, and geometric transport-equation approaches, all aimed at proving injectivity and, in some cases, providing inversion formulas.Key contributions include multiple independent injectivity proofs (torus/Fourier methods, angular-Fourier/Abel transforms, Radon/circular-average approaches) and the Pestov identity-based transport framework for vector-field tomography, with extensions to manifolds and higher-order tensors.The material emphasizes the deep connections among perspectives—from analytic (Fourier, Abel, Radon) to geometric (sphere bundle, geodesic flow, Pestov identity)—highlighting how symmetry and duality drive injectivity results and, in some settings, explicit inversions.The work lays a conceptual and technical foundation for Euclidean X-ray tomography, the role of normal operators, and vector-field tomography, with implications for both theory and potential applications in imaging.

Abstract

These are lecture notes for the course "Analysis and X-ray tomography". The course is a broad overview of various tools in analysis that can be used to study X-ray tomography. The focus is on tools and ideas, not so much on technical details and minimal assumptions. Only very basic functional analysis is assumed as background. Exercise problems are included.

Analysis and X-ray tomography

TL;DR

The notes address the fundamental problem of recovering a function from its line integrals (the X-ray transform) within the broad field of inverse problems, focusing on injectivity and uniqueness rather than numerical methods.A diverse methodological toolkit is developed, including Fourier analysis on circles and tori, angular Fourier decompositions, Abel and circular-average methods, and geometric transport-equation approaches, all aimed at proving injectivity and, in some cases, providing inversion formulas.Key contributions include multiple independent injectivity proofs (torus/Fourier methods, angular-Fourier/Abel transforms, Radon/circular-average approaches) and the Pestov identity-based transport framework for vector-field tomography, with extensions to manifolds and higher-order tensors.The material emphasizes the deep connections among perspectives—from analytic (Fourier, Abel, Radon) to geometric (sphere bundle, geodesic flow, Pestov identity)—highlighting how symmetry and duality drive injectivity results and, in some settings, explicit inversions.The work lays a conceptual and technical foundation for Euclidean X-ray tomography, the role of normal operators, and vector-field tomography, with implications for both theory and potential applications in imaging.

Abstract

These are lecture notes for the course "Analysis and X-ray tomography". The course is a broad overview of various tools in analysis that can be used to study X-ray tomography. The focus is on tools and ideas, not so much on technical details and minimal assumptions. Only very basic functional analysis is assumed as background. Exercise problems are included.

Paper Structure

This paper contains 84 sections, 37 theorems, 182 equations, 27 figures.

Table of Contents

  1. Foreword
  2. Introduction
  3. Direct and inverse problems
  4. Goals
  5. The X-ray transform
  6. The Radon transform and parametrizations
  7. The Fourier series
  8. Introduction
  9. Fourier transform and inverse Fourier transform on a circle
  10. Multidimensional Fourier series
  11. X-ray tomography on a torus
  12. Geodesics on a torus
  13. Injectivity from a torus to a Euclidean space
  14. Interplay between the X-ray and Fourier transforms on a torus
  15. Injectivity via angular Fourier series
  16. ...and 69 more sections

Key Result

Theorem 2.2

The Fourier transform on $\mathbb{R}/2\pi\mathbb{Z}$ is a unitary isometry $\mathcal{F}\colon L^2(\mathbb{R}/2\pi\mathbb{Z})\to\ell^2(\mathbb{Z})$, given by which is well defined as a Lebesgue integral. The inverse Fourier transform $\mathcal{F}^{-1}\colon\ell^2(\mathbb{Z})\to L^2(\mathbb{R}/2\pi\mathbb{Z})$ is also unitary and isometric, and is given by where the series of functions converges i

Figures (27)

  • Figure 1: In X-ray tomography the aim is to reconstruct a function in a domain from its integrals over all lines through the domain.
  • Figure 2: In electrical impedance tomography (EIT) the task is to find a conductivity $\gamma$ that depends on position from measurements of voltage and current at electrodes placed at the boundary.
  • Figure 3: Two ways to parametrize a line in the plane: giving a point and a direction or giving the closest point to the origin.
  • Figure 4: A periodic function on the real line. The behavior of the function is fully captured by its values in the fundamental domain, an interval whose length equals the period of the function. There are many options for the choice of the fundamental domain, but for $2\pi$-periodic function we may conveniently take it to be $[0,2\pi)$. The two endpoints of the fundamental domain are naturally identified with each other.
  • Figure 5: Different angles correspond to the same direction on the unit circle $\mathbb{S}^1$. Thus the numbers $\alpha,\beta,\gamma\in\mathbb{R}$ belong to the same equivalence class in $\mathbb{R}/2\pi\mathbb{Z}$.
  • ...and 22 more figures

Theorems & Definitions (66)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof : Sketchy proof of lemma \ref{['lma:torus-to-Rn']}
  • proof : Full proof of lemma \ref{['lma:torus-to-Rn']}
  • Lemma 3.3
  • ...and 56 more