Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysis of beam hardening streaks in tomography

Alexander Katsevich

TL;DR

The paper analyzes beam hardening streaks in tomography under the widely used two-term Alvarez–Macovski energy model, showing that nonlinearity of CT data induces microlocal artifacts whose singularities can be precisely described. Using microlocal analysis, conormal and paired Lagrangian distributions, and the Radon transform framework, the authors derive the leading singular behavior of the after-logs data $g$ and of the reconstructed image $f^{rec}$, identifying when streaks arise along generalized Legendre transform lines $L_{\mathfrak p}$ and how tangencies and corners of the object boundaries contribute. The main contributions include explicit classifications of $WF(g)$ across smooth, corner, and double-tangent geometries, and closed-form leading-order singularities (e.g., $|h|$, $h|h|$, and $h\log|h|$ types) that explain beam hardening streaks. The numerical experiments with a two-ball phantom validate the theoretical predictions, demonstrating the practical relevance for understanding and potentially mitigating streak artifacts in CT reconstructions.

Abstract

The mathematical foundation of X-ray CT is based on the assumption that by measuring the attenuation of X-rays passing through an object, one can recover the integrals of the attenuation coefficient $μ(x)$ along a sufficiently rich family of lines $L$, $\int_L μ(x) \text{d} x$. This assumption is inaccurate because the energy spectrum of an X-ray beam in a typical CT scanner is wide. At the same time, the X-ray attenuation coefficient of most materials is energy-dependent, and this dependence varies among materials. Thus, reconstruction from X-ray CT data is a nonlinear problem. If the nonlinear nature of CT data is ignored and a conventional linear reconstruction formula is used, which is frequently the case, the resulting image contains beam-hardening artifacts such as streaks. In this work, we describe the nonlinearity of CT data using the conventional model accepted by all CT practitioners. Our main result is the characterization of streak artifacts caused by nonlinearity. We also obtain an explicit expression for the leading singular behavior of the artifacts. Finally, a numerical experiment is conducted to validate the theoretical results.

Analysis of beam hardening streaks in tomography

TL;DR

The paper analyzes beam hardening streaks in tomography under the widely used two-term Alvarez–Macovski energy model, showing that nonlinearity of CT data induces microlocal artifacts whose singularities can be precisely described. Using microlocal analysis, conormal and paired Lagrangian distributions, and the Radon transform framework, the authors derive the leading singular behavior of the after-logs data and of the reconstructed image , identifying when streaks arise along generalized Legendre transform lines and how tangencies and corners of the object boundaries contribute. The main contributions include explicit classifications of across smooth, corner, and double-tangent geometries, and closed-form leading-order singularities (e.g., , , and types) that explain beam hardening streaks. The numerical experiments with a two-ball phantom validate the theoretical predictions, demonstrating the practical relevance for understanding and potentially mitigating streak artifacts in CT reconstructions.

Abstract

The mathematical foundation of X-ray CT is based on the assumption that by measuring the attenuation of X-rays passing through an object, one can recover the integrals of the attenuation coefficient along a sufficiently rich family of lines , . This assumption is inaccurate because the energy spectrum of an X-ray beam in a typical CT scanner is wide. At the same time, the X-ray attenuation coefficient of most materials is energy-dependent, and this dependence varies among materials. Thus, reconstruction from X-ray CT data is a nonlinear problem. If the nonlinear nature of CT data is ignored and a conventional linear reconstruction formula is used, which is frequently the case, the resulting image contains beam-hardening artifacts such as streaks. In this work, we describe the nonlinearity of CT data using the conventional model accepted by all CT practitioners. Our main result is the characterization of streak artifacts caused by nonlinearity. We also obtain an explicit expression for the leading singular behavior of the artifacts. Finally, a numerical experiment is conducted to validate the theoretical results.
Paper Structure (16 sections, 1 theorem, 47 equations, 14 figures, 1 table)

This paper contains 16 sections, 1 theorem, 47 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Assumptions
  4. Spaces of symbols and distributions
  5. Preliminaries
  6. Main results
  7. Singularities of data -- locally smooth $\mathcal{S}$
  8. Tangency at a single point
  9. Tangency at two points
  10. Corner cases.
  11. End of proof of Theorem \ref{['thm:main']}
  12. Strength of added singularities.
  13. The $++$ case
  14. Example
  15. $+-$ case
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $f_{1,2}$ and $\mathcal{S}$ satisfy the assumptions in section sec:assump. Select some $\mathfrak p_0\in \Gamma$ and let $M_0$ be a sufficiently small neighborhood of $\mathfrak p_0$. One has

Figures (14)

  • Figure 1: The case when local segments of $\mathcal{S}$ near $x_1$ and $x_2$ are on one side of the double tangent $L_{\mathfrak p_0}$.
  • Figure 2: The case $\varkappa(x_0)=0$.
  • Figure 3: One corner case.
  • Figure 4: $L_{\mathfrak p_0}$ is tangent to $\mathcal{S}$ in the generalized sense at two corner points.
  • Figure 5: The line segment case.
  • ...and 9 more figures

Theorems & Definitions (5)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 3.1
  • Remark 4.1