**Tutorial on Manifolds of Diffeomorphisms, EPDiff**

**Martin Bauer**, University of Vienna, Austria

- Riemannian geometries on the space of curves I
- 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. - 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. - 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)

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

distances). - 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.

**Tutorial on Manifolds of Diffeomorphisms, EPDiff**

**Martin Bauer**, University of Vienna, Austria

- Riemannian geometries on the space of curves I
- 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. - 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. - 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)

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

distances). - 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.