MAT-52506 Inverse Problems, fall 2007, 6 credits
MAT-52500 Inversio-ongelmat, syksy 2007, 6 opintopistettä

Introduction

Inverse problems is an active area of contemporary mathematics.
Applications include engineering, biology, geophysics, medicine, finance, and chemistry.

Examples of inverse problems include
- sharpening a blurred photograph (more),
- imaging inner organs of patients using X-ray images taken from different directions (more),
- determining the structure of the Earth using seismic data,
- locating cracks inside materials using electric surface probes,
- finding the shape of a distant asteroid from light intensity measurements only (more).
The common feature of all these problems is that they are very sensitive to measurement noise, and their solution is not straightforward.

This course teaches how to recognize an inverse problem and how to solve it in practice even when the data is noisy.

The first part of the course consists of lectures and exercises, and the latter part is a project work.
Main emphasis is on practical solution of problems arising in applications;
theory is introduced only to the extent needed to understand and implement solution methods.

Central themes of the course are singular value decomposition (SVD) of a matrix, Tikhonov regularization, total variation regularization, and statistical inversion.
These solution methods will be demonstrated in detail in the cases of deblurring and tomography.

The material is in part based on the lecturer's experience in medical imaging industry (Instrumentarium Imaging, General Electric, Palodex Group).

This course is part of the activity of the Finnish Centre of Excellence in Inverse Problems Research.

Lectures

Lecturer: professor Samuli Siltanen. Lectures in periods 1 and 2: Tuesday 12:15-14:00 (hall TC133) and Wednesday 8:30-10:00 (hall TC133).
Lecturer´s office hour Tuesday 16-17 (room TD 321). Lectures are given in Finnish or English depending on the audience.

Schedule for period 2:

  • 16.10. Assignment of project work topics
  • 24.10. Guest lecture CANCELLED due sickness!
  • 30.10. X-ray tomography and the Radon transform (this material belongs to final exam material)
  • Other lecture times: meetings with project work groups; please agree on times by email.

    Lecture notes are collected to this page.

    Project works

    Project works should consist of a written report (6-12 pages long) and commented Matlab codes. The work can be done either in Finnish or English.

    The completed project work should be returned at latest November 16, 2007. Please send the report and codes to me as email attachments. Project works returned later than November 16 will not be considered.

    Accepted project works are graded as 6-12 points. Please note that an accepted project work is necessary for passing the course with the combination interim exam + exercise points + project work points.

    Weekly exercises

    Exercise session is on Fridays at 14:15-16:00 in hall TA203. Course assistant is Juho Linna.

    Students can collect credit points by presenting solutions to the weekly problems. The credit points will be taken into account in the final grading.

    Exercise 1 (7.9.2007)
    Exercise 2 (14.9.2007) Matlab files are available on the course material page. Some correct answers are available in Exercises_2.m
    Exercise 3 (21.9.2007) Matlab files are available on the course material page. Some correct answers are available in Exercises_3.m
    Exercise 4 (28.9.2007) Image data is in the file lobster.mat, and the rest of the Matlab files are available on the course material page. Some correct answers are available in Exercises_4.m
    Corrected Exercise 5 (5.10.2007) Some correct answers are available in Exercises_5.m

    Collected exercise points can be viewed here.

    Internet resources

    Chapters 1-3 in the book Kaipio and Somersalo: Statistical and Computational Inverse Problems (Springer 2005) covers most of the course.
    However, the book is quite condensed. More accessible material is available at
  • Kaipio (in Finnish): http://venda.uku.fi/studies/virtual/KON/KON1/lectures/main.pdf
  • Somersalo (in Finnish): http://www.math.hut.fi/teaching/invvanha/indexvanha.html.fi
  • Tan, Fox and Nicholls (in English): http://www.math.auckland.ac.nz/%7Ephy707/
  • Class page of the inverse problems course given by Jennifer Mueller at Colorado state university.
  • Web page of the 2006 course is available here.

    How to pass the course?

    First alternative has three parts:
  • Pass interim exam (max 24 points). The exam was held on October 10.
    Results are here.
    The exam covered all material lectured about so far except Morozov discrepancy principle and Gibbs sampler.
  • Collect exercise points (80% of the exercises done gives the maximum 12 points).
  • Complete the project work (12 points). The project work is mandatory.
    For students who returned an acceptable project work, grading is based on the following score:
    (exam points + exercise points + project work points)/48.

    Second alternative: Pass one final exam. The exam is on November 27 at 9-12 in a hall that will be announced later. Exercise activity will be taken into account in this final exam if it leads to a higher grade.

    A collection of past exams is available here.

    Prerequisites

    Students entering the course are expected to know
  • basic linear algebra and matrix computations (equivalent to the course MAT-31090 Matriisilaskenta),
  • the concept of least squares solution of a set of linear equations,
  • some Matlab programming skills,
  • basic properties of the Fourier transform,
  • rudiments of probability theory (concepts of probability distribution and random variable).

    Information on the lecturer´s research interests can be found here.

    Student feedback and development of the course

    Student feedback and the corresponding changes in the course are reported here.
    This page was last updated on October 23, 2007.