Third AGM Meeting on Symmetry and Shape Analysis 2015-2016
29 April 2016, Imperial College London
Room 402, Centre for Doctoral Training Hub
[IMPORTANT! -- All non-speaker attendees are requested to register by writing to Tomasz Tyranowski by Thursday, 21 April 2016]
This one-day meeting will cover various disciplines related to symmetry and shape analysis. Specific topics will include:
-- data structure and fusion,
-- multiple resolutions and dynamic images,
-- changes in image topology.
Timetable:
10.00 -- 10.45 Dorje Brody (Brunel University)
10.45 -- 11.15 coffee break
11.15 -- 12.00 Stefan Sommer (University of Copenhagen)
12.00 -- 12.45 Alexis Arnaudon (Imperial College)
12.45 -- 14.15 lunch
14.15 -- 15.00 Martin Bauer (University of Vienna)
15.00 -- 15.45 Jakob Møller-Andersen (Technical University of Denmark)
15.45 -- 16.15 coffee break
Titles and Abstracts
Dorje Brody
What are Typical Shapes?
This talk is concerned with the geometrical and statistical analysis of shapes of finitely many points on the plane. In scenarios where to each shape is assigned a cost or energy function, Kendall's original analysis can be extended in two different and perhaps surprising ways: (i) When the number of points defining a given shape is increased, the Lévy-Gromov measure concentration phenomenon is applicable to show that a randomly sampled shape (under a uniform measure on the shape manifold) will nonetheless have a typical shape; (ii) When there is a large number of shapes such that the total cost or energy function is held fixed, the large deviation theory is applicable to show that (under some measure on the shape manifold) it is exceedingly unlikely to find atypical shapes. A diffusion process on the shape manifold will also be constructed such that an arbitrary initial shape distribution will equilibrate to a typical distribution.
Alexis Arnaudon
Stochastic image matching with landmarks
I will present and discuss a novel stochastic perturbation of the LDDM equations that will be illustrated on landmark matching. Joint work with Darryl Holm and Stefan Sommer.
Stefan Sommer
Anisotropic Stochastic Flows and Nonholonomic Mechanics
We describe the use of frame bundles for mapping stochastic flows with non-isotropic generator to differentiable manifolds. This construction can be used for defining generative statistical models on manifolds that resemble the Euclidean space normal distribution. We will in particular investigate the interaction between a group action on the manifold and the nonholonomic distribution that constrains the horizontal frame bundle flows and describe the resulting links between geometric mechanics, sub-Riemannian geometry and non-linear statistics.
Martin Bauer
The Fisher-Rao metric on the space of smooth densities
The Fisher--Rao metric on the space of probability densities is of importance in the field of information geometry. Restricted to finite-dimensional submanifolds, so-called statistical manifolds, it is called Fisher's information metric. The Fisher--Rao metric has the property that it is invariant under the action of the diffeomorphism group. In this talk I will show, that on a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.
Jakob Møller-Andersen
Shape Optimization on Shape Manifolds
Shape optimization has the general goal of finding shapes that are optimal in some context. A classical example is PDE constrained minimization of e.g. the drag of a wing profile, or compliance of a material. Usually the space of admissible shapes is not a vector space, and different ad-hoc methods are used to reformulate the problem such that standard optimization methods can be applied. To numerically discretize the problem, a common approach is to represent shapes via. parametrizations e.g. by splines. As most of these problems originate from physics, the particular parametrization of a physical shape is not of importance, and thus the problems are invariant under reparametrizations. The space of parametrizations modulo reparametrizations, called a shape space, is then a "natural" setting for these problems. Shape spaces have been studied: they carry a manifold structure and can be equipped with different Riemannian metrics. This opens up the option of using optimization methods on Riemannian manifolds to solve shape optimization problems, and some strides have been made in this direction. This talk will give a comparison of traditional shape optimization and Riemannian shape optimization on shape space. We present a simple shape optimization problem on curves, and a discretization based on isogeometric analysis. We will show how the choice of metric and algorithms influences the optimization, and discuss the advantages/disadvantages of this Riemannian approach.
Directions:
Lecture Room 402 in the EPSRC Centres for Doctoral Training suite is located on the 4th floor of the ICSM building with access via the SHERFIELD Level 2 Lift lobby:
Map to EPSRC Centres for Doctoral Training Suite
SHERFIELD is building 20 on the campus map:
Map of South Kensington Campus
Contacts:
Tomasz Tyranowski (main organizer)