Math 168. Topics in Applied Math: Mathematics of Medical Imaging - Fall 09 - Hans Lindblad

Text: Epstein Introduction to the Mathematics of Medical Imaging, also used in Epstein's course, Roudenko's course.
Free texts: Kak and Slaney Principles of Computerized Tomographic Imaging, Hornak The Basics of NMR
Survey article Angenent, Pichon, Tannenbaum, Mathematical Methods of Medical Imaging
Lectures: MWF 3-4 in B412. Instructor Hans Lindblad, lindblad@math.ucsd.edu. Office hour M 4-5 Cafe Espresso.
Sections: Tu 2-3. TA Chad Wildman, cwildman@math.ucsd.edu. Office hour Tu 9-10 Th 1-2 APM 6442
Description: Medical imaging technologies, such as X-ray tomography, ultrasound, positron emission tomography and magnetic resonance imaging have fundamentally changed medicine. At the core of each technology is a mathematical model to interpret the measurements and a numerical algorithm to reconstruct an image from the measured data. While each technology relies on different physical principles the mathematics and algorithms used are similar. One would like to have a precise picture of a 2 or 3 dimensional object that cannot be obtained directly. The data that is accessible is typically some collection of averages: E.g. we send an X-ray beam into an object and we measure how much of it comes out on the other side, which gives how much of it was absorbed somewhere along the way but we do not know how much was absorbed where along its path. However, if we send X-rays from each direction and parallel we get enough information that we can reconstruct the damping coefficient of the object at each point. The problem of image reconstruction is to build an object out of the averaged data and then estimate how close the reconstruction is to the actual object.
Syllabus: The course provides a firm foundation in the mathematical tools used in Tomography, such as the Fourier and Radon transforms and Sampling theory. We hope to cover chapters 1-11 of the text.
Prerequisites: Multivariable calculus and linear algebra. (preferable some analysis or pde as well.)
Scedule (preliminary): If you click on the day you might find some lecture notes.
wk  date  Monday  Wednesday  Friday
  0  9/21  No Class  No Class  1.1 Mathematical Models, 1.2
  1  9/28  1.2 Image reconstruction, 2.1 2.1-2 Linear equation-infinite dim  2.2, A.1-4, 2.3 complex numbers
  2  10/5  3.1 Tomography  3.2 Point source device  3.4-5 The Radon transform
  3  10/12  4.1-2 The Fourier Transform  4.3-4 The Fourier transform  4.5 Fourier transform higher dim
  4  10/19  5.1-2 Convolutions  5.2-3 The delta function  6.1 The Radon transform
  5  10/26  6.2 Inversion of Radon transform  6.3-4 Approximate inverse  6.4, 6.8 Inverse Hilbert
  6  11/2  7.1-2 Fourier series Midterm PracticeMidtermSolution  7.3-4 Fourier series-L^2
  7  11/9  7.3-4 Fourier series-L^2  Holiday  7.5 Gibbs Phenom 7.7 higher dim
  8  11/16  8.1 Sampling Nyquist's th.  8.2 The Poisson Summation  8.3 Finite Fourier transform
  9  11/23  9.1-2 Filters  10.1-3 Shift invariant filters  Holiday
 10  11/30  10.5 Fast Fourier Transform  11.1-3 X-Ray Tomography  11.4 Algorithms
 11  12/7  No Class  Review Practice Final  Final 12/11, 3-6pm
Homework:
wk sectiondate  Homework (tentative) due Thursdays after section, 6pm in box 6th floor APM.  Solutions
  1  9/29  1.1: 3, 5, 8, 12, 1.2: 1, 2, 7, 11, 15,    hw1.pdf
  2  10/6  2.1: 1, 3, 5, 8, 10, 2.2: 1, 2, 3, 5, 2.3: 2, 6 A.1: 13, A.4: 1, 4,  hw2.pdf
  3  10/13  3.1: 1, 3.2: 3, 3.4: 2, 3, 4, 5, 9, 12, 13,  hw3.pdf
  4  10/20  4.1:2, 5, 4.2: 1, 2, 4, 5, 6, 9, 16, 18, 20, 21, 4.5:1, 2, 10,  hw4.pdf
  5  10/27  5.1:: 10, 12.1-3, 15, 16, 5.2: 2, 3, 6, 5.3: 1, 6.1: 1, 3, 6.2: 3, 8, 4.3: 1, 3, 10,    hw5.pdf
  6  11/3  6.3: 1, 2, 4, 5.1, 7.3, 6.8: 1, A.4: 8, 12, +PracticeMidterm Solutions  hw6.pdf
  7  11/10  7.1: 3, 5, 6, 7.2 3, 5, 7.3 1, 2, 3,    
  8  11/17  7.3: 7, 8, 9, 12, 14, 7.4: 4, 10, 7.5: 3, 4, 5,  
  9  11/24  8.1: 3, 4, 8, 9, 8.2: 1, 5, 13, 8.1: 1,      
10  12/1      
Exams: One midterm in class, a Final 12/11, 3-6pm. Grade: Midterm 30%, final 50%, homework 20%.