URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2MMA70&collegejaar=2025&taal=enOmschrijving
Week 1: General introduction to PDE-based image processing I: Scale Space Theory, Linear PDEs, Method of Separation, Fourier Theory, Gaussian Derivatives.Week 2: General introduction to PDE-based image processing II: Image interpolation and representation, Image regularization (Tikhonov- Regularization, Total variation Theory).
Week 3: Locally adaptive frames in image processing, their shortcomings, and the motivation for Orientation Scores:
Vector fields on manifolds: the algebraic and the geometric viewpoint.
Locally optimal differential frames based on structure tensor and Hessian of an image: Their use and their short-comings (motivating Lie group methods). Introduction to Lie Groups.
Week 4: Image Processing via Invertible Orientation Scores.
Functional Analysis and (generalized) continuous Wavelet theory.
Week 5: Design of Invertible Orientation Scores of 2D and 3D images. Design and definition of proper wavelets, and tutorials of the Lie Analysis Mathematica 11 Package www.lieanalysis.nl
Week 6: Tracking via `shortest curves’ in orientation scores:
Optimal geodesics in the homogeneous space M of positions and orientations. Computational schemes, applications, cusps in spatial projections, and variants of the Reeds-Shepp car model.
Week 7: Tracking via `shortest curves’ in orientation score: The Pontryagin Maximum Principle and Hamiltonian flows on SE(d) and M. Exact computations of sub-Riemannian geodesics and Cartan Connections.
Week 8: Catch-up week (no new material will be taught in this week)
unless due to unforseen and special circumstances one of the previous lectures was canceled. In the latter case the catch-up week will be used to replace a canceled lecture.
Doelstellingen
This MSc course aims to provide applied mathematicians and mathematical engineers with advanced differential geometrical knowledge and with effective multi-orientation algorithms. It consists of advanced differential geometry (60%), geometric scientific computing (20%) and (industrially oriented) medical imaging applications (20%).The course is theoretically oriented, does not require pre-knowledge on image processing, nor does it require a high expertise on the programming language Mathematica (Wolfram-Research). There will be self-contained notebooks that illustrate the theory for hands-on experience with modest practical assignments.
The main topics are:
1. PDE-based image processing and locally adaptive differential frames,
2. Multi-Orientation Representations of Images (known as “orientation scores”),
3. Continuous Wavelet Transforms,
4. Lie groups, with focus on the roto-translation group SE(d) (“Special Euclidean Motion Group”),
5. Partial Differential Equations on SE(d),
6. Optimal Geometric Control via Finsler geometry and Ordinary Differential Equations on SE(d).
Although these topics are useful in many applications (optimal control, robotics, physics, mechanics, cortical modeling, crowd-dynamics, image analysis) we illustrate the strength of our theory only on geometric medical imaging applications.