Program and abstracts

[Geo-Sci-Info]

Schedule and Abstracts PDF

Tutorial on Manifolds of Diffeomorphisms, EPDiff
Martin Bauer, University of Vienna, Austria

1. Riemannian geometries on the space of curves I
2. Riemannian geometries on the space of curves II
   Abstract (1) and (2): The space of curves is of importance in the field of shape analysis. I will provide
   an overview of various Riemannian metrics that can be defined thereon, and what is known about the
   properties of these metrics. I will put particular emphasis on the induced geodesic distance, the
   geodesic equation and its well-posedness, geodesic and metric completeness and properties of the
   curvature. In addition I will present selected numerical examples illustrating the behaviour of these
   metrics.
3. Right invariant metrics on the diffeomorphism group
   The interest in right invariant metrics on the diffeomorphism group is fuelled by its relations to
   hydrodynamics. Arnold noted in 1966 that Euler's equations, which govern the motion of ideal,
   incompressible fluids, can be interpreted as geodesic equations on the group of volume preserving
   diffeomorphisms with respect to a suitable Riemannian metric. Since then other PDEs arising in
   physics have been interpreted as geodesic equations on the diffeomorphism group or related spaces.
   Examples include Burgers' equation, the KdV and Camassa-Holm equations or the Hunter-Saxton
   equation.
   Another important motivation for the study of the diffeomorphism group can be found in its
   appearance in the field of computational anatomy and image matching: the space of medical images
   is acted upon by the diffeomorphism group and differences between images are encoded by
   diffeomorphisms in the spirit of Grenander's pattern theory. The study of anatomical shapes can be
   thus reduced to the study of the diffeomorphism group.
   Using these observations as a starting point, I will consider the class of Sobolev type metrics on the
   diffeomorphism group of a general manifold M. I will discuss the local and global well-posedness of
   the corresponding geodesic equation, study the induced geodesic distance and present selected
   numerical examples of minimizing geodesics.
4. The space of densities
   I will discuss various Riemannian metrics on the space of densities. Among them is the Fisher--Rao
   metric, which 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. 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.

Tutorial on Manifolds of Diffeomorphisms, EPDiff
Martins Bruveris, Brunel University London, UK

Lecture I - Mapping spaces as manifolds
This lecture will give an introduction to differential geometry in infinite dimensions. The main objects of
shape analysis - the diffeomorphism group, the spaces of curves, surfaces, densities - can all be
modelled as infinite-dimensional manifolds.
Lecture II - Riemannian geometry in infinite dimensions
Parts of Riemannian geometry generalise easily from finite to infinite dimensions. These include the
definition of metric, covariant derivative, geodesic equations and curvature. But there are also
qualitative differences, in particular with the distinction between strong and weak Riemannian metrics.
This lecture will show some of the purely behaviour that can be encountered in infinite dimensions.
Lectures III and IV - Riemannian metrics induced by the diffeomorphism group
The purpose of these lectures is to explore the geometry of Riemannian metrics on the space of
curves and landmarks that are induced by the action of the diffeomorphism group. These metrics
correspond to exact matching of curves and landmarks via LDDMM. We will look at the induced
metrics, geodesic equations and the geodesic distance.

Introduction to the Differential Geometry
Joan Alexis Glaunès (Université Paris Descartes, France) and
Sergey Kushnarev (Singapore University of Technology and Design)

1. Definition of a manifold, Tangent Vectors and Tangents Spaces, Pushforwards, Vector Fields.
2. Tangent bundle and a Cotangent Bundle, Pullbacks, Tensors, Differential Forms.
3. Submersions, Immersions, Embeddings, Submanifolds (Embedded, Immersed)
4. Integral Curves and Flows, Lie Derivatives.
5. Riemannian Metrics.
6. Connections.
7. Riemannian Geodesics and Distance (exp map, normal coordinates, geodesics and minimizing
   distances).
8. Curvature.

Diffeomorphic Models and Matching Problems in the Discrete Case
Joan Alexis Glaunès, Université Paris Descartes, France
This talk will be an introduction and on overview of the framework of diffeomorphic mappings
(LDDMM) for estimating deformations between shapes, and its formulation for discrete problems via
reproducing kernels. I will present the classical construction of the group of diffeomorphisms, and
explain how by considering different types of actions on this group, it can be used to estimate
deformations between different types of geometric data: images, points, surfaces, etc. I will show
some experiments and studies to illustrate.

Geodesic Equations and Shooting Algorithms for Matching and Template Estimation
Joan Alexis Glaunès, Université Paris Descartes, France
In this talk I will explain the link between diffeomorphic mappings and shape spaces, i.e. Riemannian
metrics on sets of shapes. I will explain how the metric on the group of diffeomorphisms induces a
metric on the space of shapes, and detail the geodesic equations in the finite dimensional case
(manifold of landmarks), which is the case in use in practice for many problems once data has been
discretised. I will present different algorithms which are based on these equations (geodesic shooting
algorithms): matching, template estimation, geodesic regression, and explain how all this can be
actually implemented.

Models for Diffeomorphic Mappings between Submanifolds: measures, currents, varifolds
Joan Alexis Glaunès, Université Paris Descartes, France
This talk will focus on some models for defining data attachment terms for matching problems
between submanifolds (curves or surfaces) which are widely used for diffeomorphic mappings. These
are all based on the same idea of defining dual RKHS spaces and using the corresponding norm as a
data attachment term between shapes. This uses mathematical concepts such as currents or
varifolds, which come from geometric measure theory and which I will introduce. I will present both
continuous and discrete forms of these models, and show some outputs of algorithms

Reproducing Kernels in the Vectorial Case
Joan Alexis Glaunès, Université Paris Descartes, France
The theory of reproducing kernels and Reproducing Kernel Hilbert Spaces (RKHS) is extensively
used in the discrete formulation of the LDDMM setting, and in corresponding algorithms. It is also a
fundamental concept in other areas, such as statistical learning. I will present some basic concepts of
this theory in the general case of RKHS of vector fields, and explain how this theory can be used for
interpolation problems, and how it is linked to the LDDMM setting. I will also present shortly a recent
study about translation and rotation invariant kernels, which allows in particular to consider spaces of
divergence free or irrotational vector fields for deformation analysis.

Lie Groups and Lie Group Actions
Richard Hartley, Australian National University, Australia
I will talk about Lie groups and Lie group actions on manifolds, with particular consideration for
applications in Computer Vision. Lie groups play a significant role in Computer Vision, particularly
groups such as SO(3), the group of 3-D rotations, SE(3), the group of Euclidean motions, and
PGL(2,R) and PGL(3, R), the groups of 2 and 3-dimensional projective transformations. In addition,
Lie group actions on such manifolds as the Stiefel manifolds (yielding Grassman manifolds as a space
of orbits) and the action of O(2) on SO(3) x SO(3), yielding the Essential manifold, as well as Shape
manifolds as an orbit space of an action of similarity transforms, are common examples where Lie
group actions give rise to Riemannian manifold structures. Applications are in the areas of Lie group
tracking, averaging (for instance rotation averaging), and kernels on manifolds such as shape
manifolds and Grassman manifolds, all with important applications in computer vision and robotic
vision.

Tutorial on Wavelets
Hui Ji, National University of Singapore
This lecture focuses on the introduction to wavelet frame and its applications in imaging and vision.
The goal to expose audience to important topics in wavelet frames with strong relevance to visual
data processing, in particular image processing/analysis. The audience will also learn how to apply
these methods to solve real problems in imaging and vision. The lecture is an inter-disciplinary one
that emphasizes both rigorous treatment in mathematics and motivations from real-world applications.