Applied Algebra and Geometry Inaugural Meeting
21 February 2017.
The Council Room, Trent Building,
University of Nottingham,
Nottingham, NG7 2RD
10:00-10:30 Welcome and refreshments
10:30-11:30 Peter Giblin: What can singularity theory tell us about viewing surfaces?
11:30-12:30 Evelyne Hubert: Invariants of ternary quartics under the orthogonal group
14:00-15:00 Heather Harrington: Applications of Numerical Algebraic Geometry
15:30-16:30 Henry Wynn: An introduction to Algebraic Statistics
Emilie Dufresne, University of Nottingham.
Peter Giblin: What can singularity theory tell us about viewing surfaces?
I will talk about two pieces of research where geometry and singularity theory have been applied to areas of computer vision. One is the recovery of surface shape, and perhaps camera/viewer motion, from apparent contours (also called profiles or outlines), and the other is a recent classification of the local interactions between features of piecewise smooth surfaces produced by illumination from one light source and apparent contours. I shall not attempt to enter into technical details but will attempt to give a flavour of both topics.
Evelyne Hubert: Invariants of ternary quartics under the orthogonal group
Classical invariant theory has essentially addressed the action of the general linear group on homogeneous polynomials. Yet the orthogonal group arises in applications as the relevant group of transformations, especially in 3 dimensional space. Having a complete set of invariants for its action on quartics is, for instance, relevant in determining biomarkers for white matter from diffusion MRI. We characterize a generating set of rational invariants of the orthogonal group by reducing the problem to the action of a finite subgroup on a slice. The invariants of the orthogonal group can then be obtained in an explicit way. But their numerical evaluation can be achieved more straightforwardly. These results can furthermore be generalised to sextics and higher even degree forms.
(Joint work with Paul Görlach, University of Bonn and Inria Méditerannée)