Basic Mathematical Methods for Imaging and Visualization

Basic and most commonly applied techniques will be presented in the lectures and demonstrated in example applications from Image Processing and Computer Vision. The same mathematical methods are also applied in other engineering disciplines such as artificial intelligence, machine learning, computer graphics, robotics etc.

The module IN2124 is covering topics such as:
- Linear Algebra
++ linear spaces and bases
++ linear mappings and matrices
++ linear equation systems, solving linear equation systems
++ least squares problems
++ eigen value problems and singular value decomposition
- Analysis
++ metric spaces and topology
++ convergence, compactness
++ continuity and differentiability in multiple dimension, taylor expansion
- Optimization
++ existence and uniqueness of minimizers, identification of minimizers
++ gradient descent, conjugate gradient
++ Newton method, fixed point iteration
- Probability theory
++ probability spaces, random variables
++ expectation and conditional expectation
++ estimators, expectation maximization method

In the exercises the participants have the opportunity to gain deeper understanding and to collect practical experience while implementing or applying the methods in order to solve real problems, .

Upon successful completion of the module, participants understand the basic mathematical techniques and methods. They are then able to formulate real problems in the field of imaging and visualization mathematically, and to select methods for solving the problem, to optimize them and to evaluate them. They will also be able to apply these techniques and methods to other engineering disciplines such as artificial intelligence, machine learning, computer graphics, robotics etc.

IN0015 Discrete Structures, IN0018 Discrete Probability Theory, IN0019 Numerical Programming, MA0901 Linear Algebra for Informatics, MA0902 Analysis for Informatics

The module consists of lectures and tutorial sessions. The content of the lectures is conveyed in presentations of scientific material via slides and blackboard. By solving homework assignments, the students are encouraged to work intensively on the respective topics and their applications. The solutions of the assignments are discussed in the tutorial sessions.

slide presentation, blackboard

MATLAB
- Cleve Moler, first chapter of Numerical Computing with MATLAB, SIAM Linear Algebra
- Yousef Saad, Iterative Methods for Sparse Linear Systems, SIAM
- Lloyd N. Trefethen and David Bau, Numerical Linear Algebra, SIAM
- Gilbert Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press Analysis
- Walter Rudin, Real and Complex Analysis, McGraw-Hill Optimization
- Ake Björck, Numerical Methods for Least Squares Problems, SIAM
- Jonathan Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
- Uri Ascher, A first course in numerical methods, SIAM Probability Theory
- Heinz Bauer, Measure and Integration Theory, deGruyter
- Sheldon Ross, Introduction to probability and statistics for engineers and scientists, Elsevier PDEs
- Lloyd Nick Trefethen , Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations
- Cleve Moler, chapter 11 of Numerical Computing with MATLAB, SIAM

Type of Assessment: written exam.

The exam takes the form of a 75-minute written test, in which the students, based on the questions posed, are intended to demonstrate their knowledge of the basic mathematical methods as well as their ability to apply those methods successfully when solving basic abstract mathematical problems. In addition, by answering questions about concrete applications in image processing and computer vision, the students are expected to show that they can formulate applied problems mathematically, that they can analyze their mathematical properties, and that they can solve them using suitable methods.