Syllabus and Reading Material
Please find syllabus and suggested reading for the different sessions of the course below.
Session 1: Riemannian geometry basics
Monday Morning 9:00-11:00am
- Motivation: intro to manifold statistics
- Metrics, geodesics
- Covariant derivative, parallel translation
- Lie groups, homogeneous spaces
Session 2: Manifold Statistics
Monday Afernoon 14:00-16:00pm
- Frechet mean
- Geometric median
- Principal geodesic analysis (PGA)
- Regression models
Session 3: Diffeomorphisms and Image Registration
- Diffeomorphism metrics and geodesic equations
- Image registration
- Atlas building
- PGA and regression models for diffeomorphisms
The main textbook used in this part will be “Functional and Shape Data Analysis”, Anuj Srivastava and Eric Klassen, Springer New York, 2016. See this flyer for 20% discount information.
The exercises for practicals will be based on Computational Exercises provided in this textbook.
Session 1: Functional Data Analysis:
Tuesday Morning 9:00-11:00am
Section 4.2 -- Estimating functions from data
Section 4.3 -- Geometries of Function Spaces
Section 4.4 -- Function registration problem; use of L2 norms and its limitations;
Section 4.6, 4.8, 4.9 -- square-root slope function; elastic metric and Fisher-Rao metric; functional alignment under Fisher-Rao metric.
Session 2: Shape Analysis of Euclidean Curves
Wednesday Morning 9:00-11:00am
Section 2.2 -- Past techniques in shape analysis; registration problem;
Section 5.2 to 5.6 -- Elastic Riemannian metric, Square-root velocity function; shape metric; shape geodesics.
Section 9.3, 9.4 -- mean and covariances; simple shape models.
Session 3: Additional Topics:
Thursday Afternoon 14:00-16:00pm
Section 11.4 -- Shapes of trajectories on manifolds; TSRVF representations;
(From Notes) -- Shape analysis of surfaces; square-root normal fields representations.
Wednesday Afternoon 14:00-16:00pm
- Introduction to non-smooth examples and to phylogenetics
- BHV tree-space and geodesics
- More general CAT(0) geometry and orthant spaces
Suggested reading before lecture: L.J. Billera, S.P. Holmes, & K. Vogtmann (2001). Geometry of the space of phylogenetic trees. Advances in Applied Mathematics 27, 733–767.
Thursday Morning 9:00-11:00am
- The Fréchet mean: algorithms; stickiness; central limit theorems
- Correlations and the shape of distributions: use of the log map; principal geodesics; principal surfaces
Suggested reading before lecture: M. Bacak (2014). Computing medians and means in Hadamard spaces. SIAM Journal of Optimization 24, 1542-1566
Friday Morning 9:00-11:00am
- Stochastic processes in non-smooth spaces
Suggested reading before lecture: T.M.W. Nye & M.C. White (2014). Diffusion on some simple stratified spaces. Journal of Mathematical Imaging and Vision 50, 115-125
Tuesday Afternoon 14:00-16:00pm
- Differential geometry computations using the Theano framework
Please read the preprint Computational Anatomy in Theano Line Kühnel/Stefan Sommer 2017 before the lecture. The paper refers to the code in the repository theanogeometry that we will use in the session. The papers arXiv:1705.10943 and arXiv:1703.09971 can in addition be read for additional examples of the use of Theano. The stochastic landmark dynamics code for the latter paper is available in the repository stochlandyn.
Theano Geometry installation instructions can be found in the repository README. For the session, we will run the notebook 'papers/paper_Computational_Anatomy_in_Theano.ipynb'. The corpus callosum shape data file can be downloaded here. In order to load it in the 4th cell of the notebook, use 'data=io.loadmat('/vagrant/dataM-corpora-callosa.mat')' after e.g. placing the data file in the directory containing the 'Vagrantfile'.