PhD Course and Training School:
Scientific Computing for XRay Computed Tomography (CT)

Three modules in three weeks of January 220, 2023.
All lectures and exercises take place on DTU, Lyngby campus,
in building 308 room 109.
It is not possible to participate online.


Course Description
XRay Computed Tomography (CT) is a wellknown technology that is used
routinely in medicine, materials science and many other applications.
We probe an object with Xrays and record the response from the object;
then using a mathematical model for the interaction between the Xrays
and the object, we can reconstruct the object's interior using
sophisticated mathematical methods and numerical algorithms.
This training school is aimed at participants who are interested in the
formulation, implementation, and use of standard reconstruction methods for
CT such as Filtered Back Projection and Algebraic Reconstruction Methods,
as well as more novel methods such as Total Variation.
We give a rigorous mathematical description of the CT reconstruction problem,
the associated mathematical formulations,
and the underlying computational algorithms  supplemented with
handson MATLAB^{®}
computer exercises that illustrate these methods.
The goal is that the participants will get a basic understanding of
the formulation, implementation, and use of basic CT
reconstruction algorithms,
and thus be able to use them to perform data analysis
for their own CT problems.
The participants are expected to be familiar with MATLAB and with
basic aspects of linear algebra and optimization,
and they must bring their own laptop, preferably with MATLAB.
For participants without a license we provide access to
MATLAB on our servers (you must still bring a laptop).
The course is divided into three standalone modules, each of
one week, which can be followed independently. Each module
will finish with presentation of
a microproject on the last day (Friday); there will
be time every day to work on the microproject.
Participation in the Course
Students and PhD students from DTU must follow this course
over all three weeks as the
DTUcourse 02946
(link to DTU's course description).
Other participants register for the course by sending a mail to
professor Per Christian Hansen
pcha@dtu.dk
indicating which week(s) you wish to attend.
There is no course fee.
Practical Details.
All lectures and exercises take place in building 308, room 109 on
DTU
Lyngby Campus.
There is a lot of construction work going on, so make sure that you
check the
map of the campus.
Here is information about
getting to DTU Lyngby Campus.
We start each day at 9 am and continue to about 4:30 pm, with
several breaks each day.
Requirements for passing the course are active participation in the
daily exercises and microprojects and a short oral presentation
each Friday afternoon (by the group) of the microprojects.
Participants
must arrange travel and accommodation themselves.
The course material consists of our slides and exercises,
which are uploaded here.
As supplementary course material
we recommend the book P. C. Hansen, J. S.
Jørgnsen, and W. R. B. Lionheart (Eds.), Computed Tomography:
Algorithms, Insight, and Just Enough Theory, SIAM,
Philadelphia, 2021,
Fundamentals of Algorithms
FA18.
Buying the book is not required.
For many of the exercises you will need MATLAB's Image Processing Toolbox,
as well as our software package
AIR Tools II.
You will also need a few mfiles available here:
github.com/jakobsj/FA18.
Moreover, you will need access to a python compiler and the opensource
Core Imaging Library (CIL)
software package.
For more background material about Xray CT,
the underlying physics, medical applications, etc. we recommend the
following books):
 T.M. Buzug, Computed Tomography, Springer, 2008.
(Available online to DTU students.)
 A.C. Kak and M. Slaney, Principles of Computerized Tomograhpic
Imaging, reprinted by SIAM, 2001.
Free online version.
 J.L. Prince and J.M. Links, Medical Imaging  Signals and
Systems, 2. Ed., Pearson, 2015.
Module 1 (Jan. 26): CT Problems, Filtered Back Projection,
SVD Analysis
Per Christian Hansen and
Jakob Sauer Jørgensen,
both from DTU Compute.
Course plan:
 Monday
 Introduction to Xray CT
 The forward problem: the Radon transform and LambertBeer's law
 Reconstruction by Filtered Back Projection
 Introduction to the microproject and microCT scanners
 1 PM: CT measurements at DTU's Imaging
Center
 Tuesday
 Reconstruction using a real data.
 Wednesday
 Singular values and singular functions.
 The Picard condition.
 Discretization of the CT problem.
 Limiteddata tomography  lecture by
Prof. Todd Quinto,
Tufts University; starts 3:00 pm.
 Thursday
 The singular value decomposition (SVD) of a matrix.
 SVD analysis.
 Friday
 Microprojet work and presentations.
Course material:
 Slides
 Exercises
 The Radon Transform and the Filtered BackProjection Algorithm.
 Singular values and functions: sorry, no exercises.
 Discretization and the system matrix: Exercises 9.1, 9.2, 9.3, 9.4.
 The Role of The Singular Value Decomposition: Exercises
10.1, (10.2 optional), 10.3, 10.4, 10.5, 10.6.
 Files:
ExWeek1Days1and2.pdf,
Ex9.pdf,
Ex10.pdf,
ExLimData.pdf
 Data for limiteddata exercise: DataWeek1Days1and2.zip
(197 MB file).
 Microproject
Module 2 (Jan. 913): Algebraic Iterative Reconstruction Methods
Per Christian Hansen and
Jakob Sauer Jørgensen,
both from DTU Compute.
Course plan:
 Monday
 A bit of linear algebra and matrix notation.
 Kaczmarz's and Cimmino's methods.
 Least squares problems.
 Introduction to microproject.
 Tuesday
 Noisy data and semiconvergence.
 SVD analysis  iteration error and noise error.
 Stopping rules.
 Wednesday
 Algebraic iterative reconstruction in the Core Imaging Library (CIL)
 Realdata reconstruction with CIL
 Time for microproject
 Thursday
 Advanced topics in CIL
 Time for microproject
 Friday
 Microproject work and presentations.
Course material:
 Slides
 Exercises

Algebraic Methods for CT: Exercises 11.1, 11.2, 11.3 11.4, 11.5, 11.6,
11.8, 11.9, 11.10, 11.11  that is a lot, so feel free to prioritize.

Noisy Data and SemiConvergence: 11.12, 11.14, 11.15,
(11.16 optional)
 Stopping rules: no exercises, it is part of microproject B.
 Files:
Ex11.pdf
 Microproject
Module 3 (Jan. 1620): Optimization
Methods and Their Use in CT Reconstruction
Martin S. Andersen and
Yiqiu Dong,
both from DTU Compute.
Course plan:
 Monday
 Gaussian and Poisson measurement models.
 Variational methods.
 Priors and regularization (Tikhonov, generalized Tikhonov).
 Introduction to microproject.
 Tuesday
 Unconstrained optimization.
 Lipschitz continuity.
 Majorization minimization.
 Convexity.
 Step size rules and Stopping criteria.
 Power iteration.
 Wednesday
 Constrained optimization.
 Convex sets.
 Proximal gradient method.
 Optimality conditions.
 Accelerated proximal gradient methods.
 Smoothing techniques.
 Thursday
 Friday
 Microproject work and presentations.
Course material:
 Slides
 Exercises
 Regularization: Exercises 12.112.8 (all of them).
 Optimization: Exercises 13.113.8 (all of them).
 Files:
Ex12.pdf,
Ex13.pdf
 Microproject