Lecturers

Anuj Srivastava, Dept. of Statistics, Florida State University, Tallahassee, USA
Web page

Schedule and place

This 15-hour course will take place in 5 sessions over five days on June 20, 22, 25, 27, 29, 2018, at UCL, Bâtiment Maxwell, 4 Avenue Georges Lemaître, 1348 Louvain-la-Neuve (room A002, ground floor).

Schedule : 3 hours/day, from 9:30 to 12:45 (including a 15-minute coffee break).

Travel instructions are available here.

Description

Functional and shape data analysis are important research areas, due to their broad applications across many disciplines. An essential component in solving them is the registration of points across functional objects. Without proper registration the results are often inferior and difficult to interpret. The current practice in functional data analysis and shape communities is to treat registration as a pre-processing step, using off-the-shelf alignment procedures, and follow it up with statistical analysis of the resulting data. In contrast, an elastic framework is a more comprehensive approach, where one solves for the registration and statistical inferences in a simultaneous fashion. The key idea here is to use Riemannian metrics with appropriate invariance properties, to form objective functions for alignment and to develop statistical models involving functional data. While these elastic metrics are complicated in general, we have developed a family of square-root transformations that map these metrics into simpler Euclidean metrics, thus enabling more standard statistical procedures. Specifically, we have developed techniques for elastic functional PCA and elastic regression models involving functional variables. This course will demonstrate these ideas using imaging data in neuroscience where anatomical structures can often be represented as functions (curves or surfaces) on intervals or spheres. Examples of curves include DTI fiber tracts and sulcal folds while examples or surfaces include subcortical structures (hippocampus, thalamus, putamen, etc). Statistical goals here include shape analysis and modeling of these structures and to use their shapes in medical diagnosis. As an extension, we will also cover shape analysis of 3D objects by considering shapes of their boundaries (surfaces). A prominent example of this kind of data is full body scans of humans, and we will discuss elastic shape analysis of human body shapes.

List of topics:

  • Functional Data Analysis
    • Estimating functions from discrete data
    • Geometries of some relevant function spaces; L2 metric and functional PCA
    • Function registration problem; use of L2 norms and its limitations
    • Square-root slope function; elastic metric and Fisher-Rao metric; functional alignment under Fisher-Rao metric
  • Shape Analysis of Curves
    • Past techniques in shape analysis - Kendall's approach and active shape models
    • Registration problem
    • Elastic Riemannian metric, shape metric; square-root velocity function
    • Shape Geodesics and deformations
    • Shape statistics
    • Mean and covariances; principal modes of variability in shape sets; simple shape models
  • Shape Analysis of 3D objects
    • Geometric representations of surfaces
    • Elastic Riemannian metric; square-root normal fields
    • Case studies involving human body shapes

Prerequisites: Linear algebra, advanced calculus, introductory geometry and introductory statistics.

Course material

The material will largely be based on the textbook "Functional and Shape Data Analysis" by Srivastava and Klassen, Springer Series in Statistics, 2016.

Evaluation

To be determined.