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Geodesic X-ray transform and streaking artifacts on simple surfaces or on spaces of constant curvature

Hiroyuki Chihara

Abstract

The X-ray transform on the plane or on the three-dimensional Euclidean space can be considered as the measurements of CT scanners for normal human tissue. If the human body contains metal regions such as dental implants, stents in blood vessels, metal bones, etc., the beam hardening effect for the energy level of the X-ray causes streaking artifacts in its CT image. More precisely, if there are two strictly convex metal regions contained in the cross-section of normal human tissue, then streaking artifacts occur along the common tangent lines of the two regions. In this paper we study the geodesic X-ray transform and streaking artifacts on nontrapping simple compact Riemannian manifolds with strictly convex boundaries. We show that the streaking artifacts result from the propagation of conormal singularities on the boundary of metal regions along the common tangent geodesics under the strong and seemingly strange assumption that the manifolds are two dimensional or spaces of constant curvature. This condition ensures that every Jacobi field takes the form of the product of a scalar function and parallel transport along the geodesic. Our results clarify the geometric meaning of the theory, which was imperceptible in the known results on the Euclidean space.

Geodesic X-ray transform and streaking artifacts on simple surfaces or on spaces of constant curvature

Abstract

The X-ray transform on the plane or on the three-dimensional Euclidean space can be considered as the measurements of CT scanners for normal human tissue. If the human body contains metal regions such as dental implants, stents in blood vessels, metal bones, etc., the beam hardening effect for the energy level of the X-ray causes streaking artifacts in its CT image. More precisely, if there are two strictly convex metal regions contained in the cross-section of normal human tissue, then streaking artifacts occur along the common tangent lines of the two regions. In this paper we study the geodesic X-ray transform and streaking artifacts on nontrapping simple compact Riemannian manifolds with strictly convex boundaries. We show that the streaking artifacts result from the propagation of conormal singularities on the boundary of metal regions along the common tangent geodesics under the strong and seemingly strange assumption that the manifolds are two dimensional or spaces of constant curvature. This condition ensures that every Jacobi field takes the form of the product of a scalar function and parallel transport along the geodesic. Our results clarify the geometric meaning of the theory, which was imperceptible in the known results on the Euclidean space.
Paper Structure (6 sections, 26 theorems, 253 equations)

This paper contains 6 sections, 26 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi Fields and Parallel Transport
  4. Geometry of Metal Region
  5. Main Theorem
  6. Proof of the Main Theorem

Key Result

Lemma 2.3

In the same setting as in Definition theorem:pushforwardpullback1, we have

Theorems & Definitions (50)

  • Definition 2.1: Geodesic X-ray transform of a compactly supported half density
  • Definition 2.2: Pushforward and pullback induced by a submersion
  • Lemma 2.3: Holman and Uhlmann HolmanUhlmann
  • Lemma 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • Theorem 2.7
  • Lemma 3.1: Gauss Lemma
  • proof
  • ...and 40 more