Geometric Tomography - Uniqueness, Statistical Reconstruction and Algorithms

Lecturer: Julia Hörrmann

Thursday 8-10 am, Online via Zoom (Link: https://ethz.zoom.us/j/94532158621)



Course content:
The problem of reconstruction of an object from geometric information like X-ray data is a classical inverse problem on the overlap between applied mathematics, statistics, computer science and electrical engineering.
We focus on various aspects of the problem in the case of prior shape information on the reconstruction object. We will answer questions on uniqueness of the reconstruction and also cover statistical and algorithmic aspects.

Exercises:
We will discuss smaller exercises and open problems during the lectures. Furthermore there will be some exercises using Matlab and its Optimization Toolbox. Please install it from https://itshop.ethz.ch.

Prerequisites:
A sound mathematical background in geometry, analysis and probability is required though a repetition of relevant material will be included. The ability to understand and write mathematical proofs is mandatory.

Learning objective:
Introduction to geometric tomography and understanding of various theoretical aspects of reconstruction problems.

Exam:
There will be an oral exam.

Online office hour:
Friday, 9-10 am (zoom-link upon request via e-mail to julia.hoerrmann@inf.ethz.ch)

Date   Topic   Lecture Notes   Solutions   Recording  
17.09.20   Introduction   2020-09-17_Script   2020-09-17_Recording
24.09.20   Background from convex geometry   2020-09-24_Script 2020-09-24_Solutions 2020-09-24_Recording
01.10.20   Background from empirical process theory   2020-10-01_Script   2020-10-01_Recording
08.10.20   Reconstruction from support function measurements   2020-10-08_Script 2020-10-08_Solutions 2020-10-08_Recording
15.10.20   Reconstruction from support function measurements   2020-10-15_Script 2020-10-15_Solutions 2020-10-15_Recording
22.10.20   Reconstruction from surface area measures   2020-10-22_Script   2020-10-22_Recording
29.10.20   Reconstruction from surface area measures   2020-10-29_Script   2020-10-29_Recording
05.11.20   Reconstruction from brightness function measurements   2020-11-05_Script   2020-11-05_Recording
12.11.20   Reconstruction from brightness function measurements   2020-11-12_Script   2020-11-12_Recording
19.11.20   Reconstruction from the covariogram   2020-11-18_Script   2020-11-18_Recording
26.11.20   Reconstruction from the covariogram   2020-11-26_Script   2020-11-26_Recording
03.12.20   Reconstruction from parallel X-rays        
10.12.20   Reconstruction from parallel X-rays        
17.12.20   Exam repetition        


References:
R. Gardner: Geometric Tomography
F. Natterer: The Mathematics of Computerized Tomography
A. Rieder: Keine Probleme mit inversen Problemen