Introduction to the Radon Transform

Projections and Sinograms (High School to First-Year University Level)

Overview

Original image $f(x,y)$ line $L$ 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 sufficiently many 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 sufficiently many projections

Prerequisites

  • Fundamentals of calculus (computation of integrals)
  • The idea of a function of two variables (how to read $f(x,y)$)
  • Basic trigonometric functions
  • Vectors and the equation of a line

Frequently Asked Questions

What is the Radon transform?

The Radon transform is a mathematical operation that converts a two-dimensional function $f(x,y)$ into the collection of its line integrals (projections). Each line is specified by a position $s$ and an angle $\theta$, and the resulting values $\mathcal{R}f(s,\theta)$ form the transform. It was introduced in 1917 by the Austrian mathematician Johann Radon and is widely applied as the mathematical foundation of CT scanning.

What is a sinogram?

A sinogram is a two-dimensional image that displays the result of the Radon transform as a function of the two variables angle $\theta$ and position $s$. It is so named because a point-like feature in the original image traces out a sinusoidal curve in the sinogram.

How is the Radon transform related to CT scanning?

What an X-ray CT scanner measures directly is the intensity $I$ of the X-rays after they pass through the body. The logarithm of the ratio to the incident intensity, $-\ln(I/I_0)$, corresponds to the line integral of the attenuation coefficient, and this is what approximates the Radon transform. Applying the inverse Radon transform (e.g. filtered back projection) to that data reconstructs cross-sectional images of the body.