Computed Tomography in Multivariable Calculus (UMAP)

This module seeks to: 1) strengthen students' intuition in multidimensional geometry; 2) consolidate students' command of multivariable integrals; 3) provide exercises at a level between mechanical and theoretical; and 4) show applications of calculus that are important but rarely appear in calculus texts. This module provides exercises with line integrals and multivariable integrals for use in multivariable calculus courses, at a level intermediate between mechanical problems and abstract proofs. The exercises do not require any background on any applications.

Table of Contents:

INTRODUCTION

PHYSICAL MODEL OF CT SCANNERS

THE ALGEBRAIC RECONSTRUCTION TECHNIQUE (ART)
Reconstructions from Projections Are Already in Linear Algebra
Planar ART
Spatial ART

RADON'S TRANSFORM
The Set of All Straight Lines in the Plane
Integrals of Functions over Straight Lines

APPROXIMATE IDENTITIES
Weight Functions with One Dimension
Approximate Identities with One Dimension
Approximate Identities with Several Dimensions

INVERSION OF THE RADON TRANSFORM
The Adjoint Radon Transform
Filtered Back-Projection

FURTHER TOPICS
Fan-Beam Tomography
Nuclear Magnetic Resonance (NMR)
Historical Notes on Computed Tomography

SAMPLE PROBLEMS FOR EXAMINATIONS

SOLUTIONS TO THE ODD-NUMBERED EXERCISES

SOLUTIONS TO THE SAMPLE PROBLEMS FOR EXAMINATIONS

REFERENCES

ACKNOWLEDGMENTS

ABOUT THE AUTHOR