This course offers an introduction to the mathematical principles underpinning medical imaging techniques. Designed for students with a foundational understanding of calculus and linear algebra, the course explores the mathematical frameworks that enable imaging technologies like X-ray and CT to visualize internal structures in the body. The course concludes with an exploration of inverse problems. Through this course students will develop an understanding of how mathematical concepts are applied in reconstructing images from raw data, ultimately providing insights into the field of medical imaging and its real-world applications.
Overview
Syllabus
Weeks Weekly Lecture Topics (Module Titles)
1 Day 1 FUNCTION- LIMIT AND CONTINUITY Day 2 DERIVATIVE OF A FUNCTION Day 3 INTEGRATION 1 Day 4 INTEGRATION 2 Day 5 ASSIGNMENT SUBMISSION
2 Day 1 IMPROPER INTEGRALS Day 2 DIFFERENTIAL EQUATIONS- INTRODUCTION Day 3 DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE Day 4 LINEAR DIFFERENTIAL EQUATIONS Day 5 ASSIGNMENT SUBMISSION
3 Day 1 EXACT LINEAR EQUATIONS Day 2 COMPLEX NUMBERS Day 3 FOURIER SERIES REPRESENTATION OF A PERIODIC FUNCTION Day 4 FOURIER TRANSFORMS 1-INTRODUCTION Day 5 ASSIGNMENT SUBMISSION
4 Day 1 FOURIER TRANSFORMS 2- HALF RANGE FOURIER SERIES Day 2 FOURIER TRANSFORMS 3- SPECIAL FUNCTIONS Day 3 FOURIER TRANSFORMS 4-INVERSE Day 4 FOURIER TRANSFORM 5 – MULTIVARIABLE FUNCTIONS Day 5 ASSIGNMENT SUBMISSION
5 Day 1 CONVOLUTION AND FOURIER TRANSFORM Day 2 MATRICES Day 3 SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS Day 4 SYSTEM OF LINEAR HOMOGENEOUS EQUATIONS Day 5 ASSIGNMENT SUBMISSION
6 Day 1 INDUSTRIAL MATHEMATICS-INTRODUCTION- X-RAY Day 2 X-RAY BEHAVIOR AND BEER’S LAW Day 3 LINE IN THE PLANE Day 4 X-RAY PROBLEMS Day 5 ASSIGNMENT SUBMISSION
7 Day 1 RADON TRANSFORM 1 Day 2 RADON TRANSFORM 2 Day 3 RADON TRANSFORM 3 Day 4 RADONTRANSFORM PROBLEMS 1 Day 5 ASSIGNMENT SUBMISSION
8 Day 1 PHANTOMS 1 Day 2 PHANTOMS 2 Day 3 VISUALIZATION Day 4 BACK PROJECTIONS Day 5 ASSIGNMENT SUBMISSION
9 Day 1 THE CENTRAL SLICE THEOREM Day 2 FILTERS AND CONVOLUTION Day 3 ALGEBRAIC RECONSTRUCTION TECHNIQUES Day 4 CONVOLUTION AND FILTER Day 5 ASSIGNMENT SUBMISSION
10 Day 1 DISCRETE IMAGE RECONSTRUCTION 1 Day 2 DISCRETE IMAGE RECONSTRUCTION 2 Day 3 DISCRETE IMAGE RECONSTRUCTION 3 Day 4 EXAMPLES OF LEAST SQUARE APPROXIMATION Day 5 ASSIGNMENT SUBMISSION
11 Day 1 KACZMARZ’S METHOD Day 2 MRI- AN OVERVIEW Day 3 IDENTIFICATION PROBLEM Day 4 INVERSE PROBLEMS Day 5 ASSIGNMENT SUBMISSION
12 Day 1 INVERSE PROBLEMS IN CALCULUS - 1 Day 2 INVERSE PROBLEMS IN CALCULUS - 2 Day 3 INVERSE PROBLEMS IN MATRICES Day 4 INVERSE PROBLEMS IN SPRING OSCILLATIONS Day 5 ASSIGNMENT SUBMISSION
13 Day 1 INVERSE PROBLEMS IN DIFFERENTIAL EQUATION Day 2 MIXING PROBLEM & PROJECTILE MOTION Day 3 ASSIGNMENT SUBMISSION Day 4 Day 5
1 Day 1 FUNCTION- LIMIT AND CONTINUITY Day 2 DERIVATIVE OF A FUNCTION Day 3 INTEGRATION 1 Day 4 INTEGRATION 2 Day 5 ASSIGNMENT SUBMISSION
2 Day 1 IMPROPER INTEGRALS Day 2 DIFFERENTIAL EQUATIONS- INTRODUCTION Day 3 DIFFERENTIAL EQUATIONS OF FIRST ORDER AND FIRST DEGREE Day 4 LINEAR DIFFERENTIAL EQUATIONS Day 5 ASSIGNMENT SUBMISSION
3 Day 1 EXACT LINEAR EQUATIONS Day 2 COMPLEX NUMBERS Day 3 FOURIER SERIES REPRESENTATION OF A PERIODIC FUNCTION Day 4 FOURIER TRANSFORMS 1-INTRODUCTION Day 5 ASSIGNMENT SUBMISSION
4 Day 1 FOURIER TRANSFORMS 2- HALF RANGE FOURIER SERIES Day 2 FOURIER TRANSFORMS 3- SPECIAL FUNCTIONS Day 3 FOURIER TRANSFORMS 4-INVERSE Day 4 FOURIER TRANSFORM 5 – MULTIVARIABLE FUNCTIONS Day 5 ASSIGNMENT SUBMISSION
5 Day 1 CONVOLUTION AND FOURIER TRANSFORM Day 2 MATRICES Day 3 SOLUTIONS TO SYSTEM OF LINEAR EQUATIONS Day 4 SYSTEM OF LINEAR HOMOGENEOUS EQUATIONS Day 5 ASSIGNMENT SUBMISSION
6 Day 1 INDUSTRIAL MATHEMATICS-INTRODUCTION- X-RAY Day 2 X-RAY BEHAVIOR AND BEER’S LAW Day 3 LINE IN THE PLANE Day 4 X-RAY PROBLEMS Day 5 ASSIGNMENT SUBMISSION
7 Day 1 RADON TRANSFORM 1 Day 2 RADON TRANSFORM 2 Day 3 RADON TRANSFORM 3 Day 4 RADONTRANSFORM PROBLEMS 1 Day 5 ASSIGNMENT SUBMISSION
8 Day 1 PHANTOMS 1 Day 2 PHANTOMS 2 Day 3 VISUALIZATION Day 4 BACK PROJECTIONS Day 5 ASSIGNMENT SUBMISSION
9 Day 1 THE CENTRAL SLICE THEOREM Day 2 FILTERS AND CONVOLUTION Day 3 ALGEBRAIC RECONSTRUCTION TECHNIQUES Day 4 CONVOLUTION AND FILTER Day 5 ASSIGNMENT SUBMISSION
10 Day 1 DISCRETE IMAGE RECONSTRUCTION 1 Day 2 DISCRETE IMAGE RECONSTRUCTION 2 Day 3 DISCRETE IMAGE RECONSTRUCTION 3 Day 4 EXAMPLES OF LEAST SQUARE APPROXIMATION Day 5 ASSIGNMENT SUBMISSION
11 Day 1 KACZMARZ’S METHOD Day 2 MRI- AN OVERVIEW Day 3 IDENTIFICATION PROBLEM Day 4 INVERSE PROBLEMS Day 5 ASSIGNMENT SUBMISSION
12 Day 1 INVERSE PROBLEMS IN CALCULUS - 1 Day 2 INVERSE PROBLEMS IN CALCULUS - 2 Day 3 INVERSE PROBLEMS IN MATRICES Day 4 INVERSE PROBLEMS IN SPRING OSCILLATIONS Day 5 ASSIGNMENT SUBMISSION
13 Day 1 INVERSE PROBLEMS IN DIFFERENTIAL EQUATION Day 2 MIXING PROBLEM & PROJECTILE MOTION Day 3 ASSIGNMENT SUBMISSION Day 4 Day 5
Taught by
Dr. Mansoor P
