Introduction to the Radon Transform

Projections and Sinograms (First-Year University Level)

Overview

Original image $f(x,y)$ $\theta$ Line integral $\displaystyle\int_L f\,dl$ Sinogram $p(s,\theta)$ $\theta$ (0° → 180°) $s$ Projection data at each angle $\theta$ forms the sinogram Points trace sinusoidal curves → hence "sinogram"

This introduction covers the fundamental concepts of the Radon transform. The goal is to develop an intuitive understanding of what it means to "convert a two-dimensional object into one-dimensional projection data."

Learning Objectives

  • Understand the geometric meaning of line integrals (projections)
  • Understand the concept of a sinogram
  • Survey the principles of X-ray CT scanning
  • Understand why the original image can be recovered from its projections

Table of Contents

  1. Chapter 1 X-ray CT and Projection

    History, the intuition of X-ray CT, and projection as a line integral

  2. Chapter 2 Geometry of Line Integrals

    The $(s, \theta)$ representation of lines, line integrals, the definition of the Radon transform, and examples

  3. Chapter 3 Sinograms

    Visualization of projection data, interpretation of features

  4. Chapter 4 Principles of X-ray CT

    X-ray attenuation, acquisition of projection data

  5. Chapter 5 Overview of Reconstruction

    The basic idea of recovering an image from its projections

Prerequisites

  • Fundamentals of calculus (computation of integrals)
  • Basic trigonometric functions
  • Vectors and the equation of a line