Riemannian Geometric Statistics in Medical Image Analysis

By Tom Fletcher

Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data.

Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods.

Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. The foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology

Content includes:

• The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs
• Applications of statistics on manifolds and shape spaces in medical image computing
• Diffeomorphic deformations and their applications

As the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science.

• A complete reference covering both the foundations and state-of-the-art methods
• Edited and authored by leading researchers in the field
• Contains theory, examples, applications, and algorithms
• Gives an overview of current research challenges and future applications

• Language: English
• Publisher: Academic Press
• Release date: Sep 2, 2019
• ISBN: 9780128147269

Book preview

Riemannian Geometric Statistics in Medical Image Analysis

First edition

Xavier Pennec

Stefan Sommer

Tom Fletcher

Table of Contents

Cover image

Title page

Copyright

Contributors

Introduction

Introduction

Part 1: Foundations of geometric statistics

1: Introduction to differential and Riemannian geometry

Abstract

1.1. Introduction

1.2. Manifolds

1.3. Riemannian manifolds

1.4. Elements of analysis in Riemannian manifolds

1.5. Lie groups and homogeneous manifolds

1.6. Elements of computing on Riemannian manifolds

1.7. Examples

1.8. Additional references

References

2: Statistics on manifolds

Abstract

2.1. Introduction

2.2. The Fréchet mean

2.3. Covariance and principal geodesic analysis

2.4. Regression models

2.5. Probabilistic models

References

3: Manifold-valued image processing with SPD matrices

Abstract

Acknowledgements

3.1. Introduction

3.2. Exponential, logarithm, and square root of SPD matrices

3.3. Affine-invariant metrics

3.4. Basic statistical operations on SPD matrices

3.5. Manifold-valued image processing

3.6. Other metrics on SPD matrices

3.7. Applications in diffusion tensor imaging (DTI)

3.8. Learning brain variability from Sulcal lines

References

4: Riemannian geometry on shapes and diffeomorphisms

Abstract

4.1. Introduction

4.2. Shapes and actions

4.3. The diffeomorphism group in shape analysis

4.4. Riemannian metrics on shape spaces

4.5. Shape spaces

4.6. Statistics in LDDMM

4.7. Outer and inner shape metrics

4.8. Further reading

References

5: Beyond Riemannian geometry

Abstract

5.1. Introduction

5.2. Affine connection spaces

5.3. Canonical connections on Lie groups

5.4. Left, right, and biinvariant Riemannian metrics on a Lie group

5.5. Statistics on Lie groups as symmetric spaces

5.6. The stationary velocity fields (SVF) framework for diffeomorphisms

5.7. Parallel transport of SVF deformations

5.8. Historical notes and additional references

References

Part 2: Statistics on manifolds and shape spaces

6: Object shape representation via skeletal models (s-reps) and statistical analysis

Abstract

Acknowledgements

6.1. Introduction to skeletal models

6.2. Computing an s-rep from an image or object boundary

6.3. Skeletal interpolation

6.4. Skeletal fitting

6.5. Correspondence

6.6. Skeletal statistics

6.7. How to compare representations and statistical methods

6.8. Results of classification, hypothesis testing, and probability distribution estimation

6.9. The code and its performance

6.10. Weaknesses of the skeletal approach

References

7: Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications

Abstract

Acknowledgements

7.1. Introduction

7.2. Riemannian geometry of the hypersphere

7.3. Weak consistency of iFME on the sphere

7.4. Experimental results

7.5. Application to the classification of movement disorders

7.6. Riemannian geometry of the special orthogonal group

7.7. Weak consistency of iFME on so(n)

7.8. Experimental results

7.9. Conclusions

References

8: Statistics on stratified spaces

Abstract

Acknowledgements

8.1. Introduction to stratified geometry

8.2. Least squares models

8.3. BHV tree space

8.4. The space of unlabeled trees

8.5. Beyond trees

References

9: Bias on estimation in quotient space and correction methods

Abstract

Acknowledgement

9.1. Introduction

9.2. Shapes and quotient spaces

9.3. Template estimation

9.4. Asymptotic bias of template estimation

9.5. Applications to statistics on organ shapes

9.6. Bias correction methods

9.7. Conclusion

References

10: Probabilistic approaches to geometric statistics

Abstract

10.1. Introduction

10.2. Parametric probability distributions on manifolds

10.3. The Brownian motion

10.4. Fiber bundle geometry

10.5. Anisotropic normal distributions

10.6. Statistics with bundles

10.7. Parameter estimation

10.8. Advanced concepts

10.9. Conclusion

10.10. Further reading

References

11: On shape analysis of functional data

Abstract

11.1. Introduction

11.2. Registration problem and elastic approach

11.3. Shape space and geodesic paths

11.4. Statistical summaries and principal modes of shape variability

11.5. Summary and conclusion

Appendix. Mathematical background

References

Part 3: Deformations, diffeomorphisms and their applications

12: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles

Abstract

Acknowledgements

12.1. Introduction

12.2. General setting and notations

12.3. Currents

12.4. Varifolds

12.5. Normal cycles

12.6. Computational aspects

12.7. Conclusion

References

13: A discretize–optimize approach for LDDMM registration

Abstract

13.1. Introduction

13.2. Background and related work

13.3. Continuous mathematical models

13.4. Discretization of the energies

13.5. Discretization and solution of PDEs

13.6. Discretization in multiple dimensions

13.7. Multilevel registration and numerical optimization

13.8. Experiments and results

13.9. Discussion and conclusion

References

14: Spatially adaptive metrics for diffeomorphic image matching in LDDMM

Abstract

14.1. Introduction to LDDMM

14.2. Sum of kernels and semidirect product of groups

14.3. Sliding motion constraints

14.4. Left-invariant metrics

14.5. Open directions

References

15: Low-dimensional shape analysis in the space of diffeomorphisms

Abstract

Acknowledgements

15.1. Introduction

15.2. Background

15.3. PPGA of diffeomorphisms

15.4. Inference

15.5. Evaluation

15.6. Results

15.7. Discussion and conclusion

References

16: Diffeomorphic density registration

Abstract

Acknowledgements

16.1. Introduction

16.2. Diffeomorphisms and densities

16.3. Diffeomorphic density registration

16.4. Density registration in the LDDMM-framework

16.5. Optimal information transport

16.6. A gradient flow approach

References

Index

Copyright

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Contributors

Martin Bauer     Florida State University, Department of Mathematics, Tallahassee, FL, United States

Rudrasis Chakraborty     University of Florida, CISE Department, Gainesville, FL, United States

Benjamin Charlier

IMAG, Univ. Montpellier, CNRS, Montpellier, France

Institut du Cerveau et de la Moëlle Épinière, ARAMIS, Paris, France

Nicolas Charon     Johns Hopkins University, Center of Imaging Sciences, Baltimore, MD, United States

Hyo-young Choi     UNC, Chapel Hill, NC, United States

James Damon     UNC, Chapel Hill, NC, United States

Loic Devilliers     Université Côte d'Azur and Inria, Epione team, Sophia Antipolis, France

Aasa Feragen     University of Copenhagen, Department of Computer Science, Copenhagen, Denmark

Tom Fletcher     University of Virginia, Departments of Electrical & Computer Engineering and Computer Science, Charlottesville, VA, United States

Joan Glaunès     MAP5, Université Paris Descartes, Paris, France

Polina Golland     Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Lab, Cambridge, MA, United States

Pietro Gori     Télécom ParisTech, LTCI, équipe IMAGES, Paris, France

Junpyo Hong     UNC, Chapel Hill, NC, United States

Sarang Joshi     University of Utah, Department of Bioengineering, Scientific Computing and Imaging Institute, Salt Lake City, UT, United States

Sungkyu Jung     Seoul National University, Seoul, Republic of Korea

Zhiyuan Liu     UNC, Chapel Hill, NC, United States

Marco Lorenzi     Université Côte d'Azur and Inria, Epione team, Sophia Antipolis, France

J.S. Marron     UNC, Chapel Hill, NC, United States

Stephen Marsland     Victoria University of Wellington, School of Mathematics and Statistics, Wellington, New Zealand

Nina Miolane

Université Côte d'Azur and Inria, Epione team, Sophia Antipolis, France

Stanford University, Department of Statistics, Stanford, CA, United States

Jan Modersitzki

Institute of Mathematics and Image Computing, University of Lübeck, Lübeck, Germany

Fraunhofer MEVIS, Lübeck, Germany

Klas Modin     Chalmers University of Technology and the University of Gothenburg, Department of Mathematical Sciences, Göteborg, Sweden

Marc Niethammer

Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC, United States

Biomedical Research Imaging Center (BRIC), Chapel Hill, NC, United States

Tom Nye     Newcastle University, School of Mathematics, Statistics and Physics, Newcastle upon Tyne, United Kingdom

Beatriz Paniagua     UNC, Chapel Hill, NC, United States

Xavier Pennec     Université Côte d'Azur and Inria, Epione team, Sophia Antipolis, France

Stephen M. Pizer     UNC, Chapel Hill, NC, United States

Thomas Polzin     Institute of Mathematics and Image Computing, University of Lübeck, Lübeck, Germany

Laurent Risser     Institut de Mathématiques de Toulouse, CNRS, Université de Toulouse, UMR CNRS 5219, Toulouse, France

Pierre Roussillon     ENS Cachan, CNRS, Université Paris-Saclay, CMLA, Cachan, France

Jörn Schulz     Arctic University of Norway, Tromsø, Norway

Ankur Sharma     UNC, Chapel Hill, NC, United States

Stefan Sommer     University of Copenhagen, Department of Computer Science, Copenhagen, Denmark

Anuj Srivastava     Florida State University, Tallahassee, FL, United States

Liyun Tu     UNC, Chapel Hill, NC, United States

Baba C. Vemuri     University of Florida, CISE Department, Gainesville, FL, United States

François-Xavier Vialard     Laboratoire d'informatique Gaspard Monge, Université Paris-Est Marne-la-Vallée, UMR CNRS 8049, Champs sur Marne, France

Jared Vicory     UNC, Chapel Hill, NC, United States

Jiyao Wang     UNC, Chapel Hill, NC, United States

William M. Wells III     Harvard Medical School, Department of Radiology, Boston, MA, United States

Miaomiao Zhang     Washington University in St. Louis, Computer Science and Engineering, St. Louis, MO, United States

Ruiyi Zhang     Florida State University, Tallahassee, FL, United States

Introduction

Xavier Pennec; Stefan Sommer; Tom Fletcher     University Côte d'Azur and Inria, Sophia Antipolis, France

DIKU, University of Copenhagen, Copenhagen, Denmark

University of Virginia, Charlottesville, VA, United States

Introduction

Over the last two decades, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Typical examples of data in this domain include organ shapes and deformations resulting from segmentation and registration in computational anatomy, and symmetric positive definite matrices in diffusion imaging. In this context, Riemannian geometry has gradually been established as one the most powerful mathematical and computational paradigms.

This book aims at being an introduction to and a reference on Riemannian geometric statistics and its use in medical image analysis for researchers and graduate students. The book provides both descriptions of the core methodology and presentations of state-of-the-art methods used in the field. We wish to present this combination of foundational material and current research together with examples, applications, and algorithms in a volume that is edited and authored by the leading researchers in the field. In addition, we wish to provide an overview of current research challenges and future applications.

Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. We hope that both the foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology.

Contents

Part 1 of this edited volume describes the foundations of Riemannian geometric computing methods for statistics on manifolds. The book here emphasizes concepts rather than proofs with the goal of providing graduate students in computer science the mathematical background needed to start in this domain. Chapter 1 presents an introduction to differential, Riemannian and Lie group geometry, and chapter 2 covers statistics on manifolds. Chapters 3–5 present introductions to geometry of SPD matrices, shape analysis through the action of the diffeomorphism group, and geometry and statistical analysis beyond the Riemannian setting when an affine connection, not a metric, is available.

Part 2 includes contributions from leading researchers in the field on applications of statistics on manifolds and shape spaces in medical image computing. In chapter 6, Stephen Pizer, Steve Marron, and coauthors describe shape representation via skeletal models and how this allows application of nonlinear statistical methods on shape spaces. Chapter 7 by Rudrasis Chakraborty and Baba Vemuri concerns estimation of the iterative Riemannian barycenter, a candidate for the generalization of the Euclidean mean value on selected manifolds. In chapter 8, Aasa Feragen and Tom Nye discuss the statistics on stratified spaces that generalize manifold by allowing variation of the topological structure. Estimation of templates in quotient spaces is the topic of chapter 9 by Nina Miolane, Loic Devilliers, and Xavier Pennec. Stefan Sommer discusses parametric statistics on manifolds using stochastic processes in chapter 10. In chapter 11, Ruiyi Zhang and Anuj Srivastava consider shape analysis of functional data using elastic metrics.

Part 3 of the book focuses on diffeomorphic deformations and their applications in shape analysis. Nicolas Charon, Benjamin Charlier, Joan Glaunès, Pierre Roussillon, and Pietro Gori present currents, varifolds, and normal cycles for shape comparison in chapter 12. Numerical aspects of large deformation registration is discussed in chapter 13 by Thomas Polzin, Marc Niethammer, François-Xavier Vialard, and Jan Modersitzki. Francois-Xavier and Laurent Risser present spatially varying metrics for large deformation matching in chapter 14. Chapter 15 by Miaomiao Zhang, Polina Golland, William M. Wells, and Tom Fletcher presents a framework for low-dimensional representations of large deformations and its use in shape analysis. Finally, in chapter 16, Martin Bauer, Sarang Joshi, and Klas Modin study densities matching in the diffeomorphic setting.

We are extremely grateful for this broad set of excellent contributions to the book by leading researchers in the field, and we hope that the book in its entirety will inspire new developments and research directions in this exciting intersection between applied mathematics and computer science.

The editors

February, 2019

Part 1

Foundations of geometric statistics

Outline

1. Introduction to differential and Riemannian geometry

2. Statistics on manifolds

3. Manifold-valued image processing with SPD matrices

4. Riemannian geometry on shapes and diffeomorphisms

5. Beyond Riemannian geometry

1

Introduction to differential and Riemannian geometry

Stefan Sommera; Tom Fletcherb; Xavier Pennecc    aUniversity of Copenhagen, Department of Computer Science, Copenhagen, Denmark

bUniversity of Virginia, Departments of Electrical & Computer Engineering and Computer Science, Charlottesville, VA, United States

cUniversité Côte d'Azur and Inria, Epione team, Sophia Antipolis, France

Abstract

This chapter introduces the basic concepts of differential geometry: Manifolds, charts, curves, their derivatives, and tangent spaces. The addition of a Riemannian metric enables length and angle measurements on tangent spaces giving rise to the notions of curve length, geodesics, and thereby the basic constructs for statistical analysis of manifold-valued data. Lie groups appear when the manifold in addition has smooth group structure, and homogeneous spaces arise as quotients of Lie groups. We discuss invariant metrics on Lie groups and their geodesics.

The goal is to establish the mathematical bases that will further allow to build a simple but consistent statistical computing framework on manifolds. In the later part of the chapter, we describe computational tools, the Exp and Log maps, derived from the Riemannian metric. The implementation of these atomic tools will then constitute the basis to build more complex generic algorithms in the following chapters.

Keywords

Riemannian Geometry; Riemannian Metric; Riemannian Manifold; Tangent Space; Lie Group; Geodesic; Exp and Log maps

1.1 Introduction

When data exhibit nonlinearity, the mathematical description of the data space must often depart from the convenient linear structure of Euclidean vector spaces. Nonlinearity prevents global vector space structure, but we can nevertheless ask which mathematical properties from the Euclidean case can be kept while still preserving the accurate modeling of the data. It turns out that in many cases, local resemblance to a Euclidean vector space is one such property. In other words, up to some approximation, the data space can be linearized in limited regions while forcing a linear model on the entire space would introduce too much distortion.

The concept of local similarity to Euclidean spaces brings us exactly to the setting of manifolds. Topological, differential, and Riemannian manifolds are characterized by the existence of local maps, charts, between the manifold and a Euclidean space. These charts are structure preserving: They are homeomorphisms in the case of topological manifolds, diffeomorphisms in the case of differential manifolds, and, in the case of Riemannian manifolds, they carry local inner products that encode the non-Euclidean geometry.

The following sections describe these foundational concepts and how they lead to notions commonly associated with geometry: curves, length, distances, geodesics, curvature, parallel transport, and volume form. In addition to the differential and Riemannian structure, we describe one extra layer of structure, Lie groups that are manifolds equipped with smooth group structure. Lie groups and their quotients are examples of homogeneous spaces. The group structure provides relations between distant points on the group and thereby additional ways of constructing Riemannian metrics and deriving geodesic equations.

Topological, differential, and Riemannian manifolds are often covered by separate graduate courses in mathematics. In this much briefer overview, we describe the general concepts, often sacrificing mathematical rigor to instead provide intuitive reasons for the mathematical definitions. For a more in-depth introduction to geometry, the interested reader may, for example, refer to the sequence of books by John M. Lee on topological, differentiable, and Riemannian manifolds [17,18,16] or to the book on Riemannian geometry by do Carmo [4]. More advanced references include [15], [11], and [24].

1.2 Manifolds

, and collections of charts denoted atlases. We will discuss this construction shortly, however, we first focus on the case where the manifold is a subset of a larger Euclidean space. This viewpoint is often less abstract and closer to our natural intuition of a surface embedded in our surrounding 3D Euclidean space.

. On the other hand, when using maps and piecing the global surface together using the compatibility of the overlapping parts, we take the abstract view using charts and atlases.

1.2.1 Embedded submanifolds

, that is, the set

(1.1)

. We can generalize this way of constructing a manifold to the following definition.

Definition 1.1

Embedded manifold

is an embedded manifold of dimension d.

The map F (see Fig. 1.1).

Figure 1.1 is of dimension 3 − 1 = 2.

has a manifold structure as constructed with charts and atlases. In addition, the topological and differentiable structure of M letting us denote M as embedded . For now, we will be somewhat relaxed about the details and use the construction as a working definition of what we think of as a manifold.

The map F can be seen as a set of m linearizes the constraints around x, additional examples of commonly occurring manifolds that we will see in this book arise directly from embedded manifolds or as quotients of embedded manifolds.

Example 1.1

d-dimensional spheres . Here we express the unit length equation generalizing (1.1) by

(1.2)

.

Example 1.2

Orthogonal matrices by the equation

(1.3)

. We will see in Section .

1.2.2 Charts and local euclideaness

We now describe how charts, local parameterizations of the manifold, can be constructed from the implicit representation above. We will use this to give a more abstract definition of a differentiable manifold.

When navigating the surface of the earth, we seldom use curved representations of the surface but instead rely on charts that give a flat, 2D representation of regions limited in extent. It turns out that this analogy can be extended to embed manifolds with a rigorous mathematical formulation.

Definition 1.2

A chart on a d.

The definition exactly captures the informal idea of representing a local part of the surface, the open set U(see Fig. 1.2).

Figure 1.2 , respectively. The compatibility condition ensures that ϕ and ψ agree on the overlap U  ∩  V between U and V in the sense that the composition ψ ∘ ϕ −1 is a differentiable map.

When using charts, we often say that we work in coordinates, we implicitly imply that there is a chart ϕ .

having Jacobian with full rank m. Recall the setting of the implicit function theorem (see e.g. such that x denotes the first d coordinates and y the last m denote the last m columns of the Jacobian matrix dF, that is, the derivatives of F taken with respect to variations in yhas full rank m of x .

of full rank. With this in mind, the map g .

1.2.3 Abstract manifolds and atlases

We now use the concept of charts to define atlases as collections of charts and from this the abstract notion of a manifold.

Definition 1.3

Atlas

such that

,

is a differentiable map.

to the manifold. In order for this construction to work, we must ensure that there is no ambiguity in the structure we get if the domain of multiple charts cover a given point. The compatibility condition ensures exactly that.

Definition 1.4

Manifold

is a manifold of dimension d.

Remark 1.1

for some integer ris smooth is a topological manifold with no differentiable structure.

Because of the implicit function theorem, embedded submanifolds in the sense of Definition 1.1 have charts and atlases. Embedded submanifolds are therefore particular examples of abstract manifolds. In fact, this goes both ways: The Whitney embedding theorem states that any dembedding and not a global smooth embedding.

Example 1.3

The projective space . Depending on the properties of the equivalence relation, the quotient space of a manifold may not be a manifold in general (more details will be given in Chapter 9). In the case of the projective space, we can verify the above abstract manifold definition. Therefore the projective space cannot be seen as an embedded manifold directly, but it can be seen as the quotient space of an embedded manifold.

1.2.4 Tangent vectors and tangent space

As the name implies, derivatives lies at the core of differential geometry. The differentiable structure allows taking derivatives of curves in much the same way as the usual derivatives in Euclidean space. However, spaces of tangent vectors to curves behave somewhat differently on manifolds due to the lack of the global reference frame that the Euclidean space coordinate system gives. We here discuss derivatives of curves, tangent vectors, and tangent spaces.

. For each t, the curve derivative is

(1.4)

, denoted the tangent vector to γ at t, we can regard γ . As illustrated on Fig. 1.3, the tangent vectors of γ span a dto the first order at xdenotes the kernel (null-space) of the Jacobian matrix of Fat the point x.

Figure 1.3 The curve γ maps the interval [0, T ] to the manifold. Using a chart ϕ , we can work in coordinates with the curve ϕ ∘ γ is embedded, then γ . It can be written in coordinates using ϕ is the affine d .

On abstract manifolds, the definition of tangent vectors becomes somewhat more intricate. Let γ . By the continuity of γ and openness of Ufor s sufficiently close to tis defined for such sby definition. However, we would like to be able to define tangent vectors independently of the underlying curve. In addition, we need to ensure that the construction does not depend on the chart ϕ.

whose derivative is

(1.5)

This operation is clearly linear in f in the sense that

when g of f and g. Operators on differentiable functions satisfying these properties are called derivations. It can now be checked that the curve derivative using a chart above defines derivations. By the chain rule we can see that these derivations are independent of the chosen chart.

as derivations is rather abstract. In practice, it is often most convenient to just remember that there is an abstract definition and otherwise think of tangent vectors as derivatives of curves. In fact, tangent vectors and tangent spaces can also be defined without derivations using only the derivatives of curves. However, in this case, we must define a tangent vector as an equivalence class of curves because multiple curves can result in the same derivative. This construction, although in some sense more intuitive, therefore has its own complexities.

.

for the i.

Remark 1.2

Einstein summation convention

is implicit because the index i .

Just as a Euclidean vector space V . For each xof a covector ξ on a vector v . Note that the latter notation with brackets is similar to the notation for inner products used later on.

1.2.5 Differentials and pushforward

The interpretation of tangent vectors as derivations allows taking derivatives of functions. If X and hence a derivation that acts on functions. If instead f being the jth component of f. The differential df is often denoted the pushforward of f because it uses f to map, that is, pushis often used. When f .

As a particular case, consider a map f itself, and we can consider df . Though the differential df is also a pushforward, the notation df is most often used because of its interpretation as a covector field.

1.3 Riemannian manifolds

when considering embedded manifolds, or via charts and atlases with the abstract definition of manifolds. We now start including geometric and metric structures.

(Fig. 1.4). In the embedding case, tangent spaces are affine spaces of the embedding vector space, and the simplest way to specify this mapping is through an affine transformation, hence the name affine connection introduced by Cartan [3]. A connection operator also describes how a vector is transported from a tangent space to a neighboring one along a given curve. Integrating this transport along the curve specifies the parallel transport along this curve. However, there is usually no global parallelism as in Euclidean space. As a matter of fact, transporting the same vector along two different curves arriving at the same point in general leads to different vectors at the endpoint. This is easily seen on the sphere when traveling from north pole to the equator, then along the equator for 90 degrees and back to north pole turns any tangent vector by 90 degrees. This defect of global parallelism is the sign of curvature.

Figure 1.4 Tangent vectors along the red (light gray in print version) and blue (dark gray in print version) curves drawn on the manifold belong to different tangent spaces. To define the acceleration as the difference of neighboring tangent vectors, we need to specify a mapping to connect a tangent space at one point to the tangent spaces at infinitesimally close points. In the embedding case, tangent spaces are affine spaces of the embedding vector space, and the simplest way to specify this mapping is through an affine transformation.

, we define the equivalent of straight lines in the manifold, geodesics. We should notice that there exists many different choices of connections on a given manifold, which lead to different geodesics. However, geodesics by themselves do not quantify how far away from each other two points are. For that purpose, we need an additional structure, a distance. By restricting to distances that are compatible with the differential structure, we enter into the realm of Riemannian geometry.

1.3.1 Riemannian metric

A Riemannian metric at points x of the manifold. For each x; see . The ij. This matrix is called the local representation of the Riemannian metric in the chart x, and the dot product of two vectors v and w is called a cometric.

Figure 1.5 along the curve γ , the metric g . Contrary to the Euclidean case, g can only be compared by g evaluated at y .

1.3.2 Curve length and Riemannian distance

on the manifold, then we can compute at each t . To compute the length of the curve, the norm is integrated along the curve:

.

The distance between two points of a connected Riemannian manifold is the minimum length among the curves γ joining these points:

(1.6)

The topology induced by this Riemannian distance is the original topology of the manifold: open balls constitute a basis of open sets.

. In section to the tangent space at each point of the manifold. Embedded manifolds thus inherit also their geometric structure in the form of the Riemannian metric from the embedding space.

1.3.3 Geodesics

In Riemannian manifolds, locally length-minimizing curves are called metric geodesics. The next subsection will show that these curves are also autoparallel for a specific connection, so that they are simply called geodesics in general. A curve is locally length minimizing if for all t and sufficiently small s. It turns out that critical points for the energy also optimize the length functional. Moreover, they are parameterized proportionally to their arc length removing the ambiguity of the parameterization.

We now define the Christoffel symbols from the metric g by

(1.7)

Using the calculus of variations, it can be shown that the geodesics satisfy the second-order differential system

(1.8)

We will see the Christoffel symbols again in coordinate expressions for the connection below.

1.3.4 Levi-Civita connection

The fundamental theorem of Riemannian geometry states that on any Riemannian manifold, there is a unique connection which is compatible with the metric and which has the property of being torsion-free. This connection is called the Levi-Civita connection. For that choice of connection, shortest curves have zero acceleration and are thus geodesics in the sense of being straight lines. In the following we only consider the Levi-Civita connection unless explicitly stated.

The connection allows us to take derivatives of a vector field Y in the direction of another vector field X . This is also denoted the covariant derivative of Y along X. The connection is linear in X and obeys the product rule in Y being the derivative of f in the direction of X . With vector fields X and Y , we can use this to compute the coordinate expression for derivatives of Y along X:

Using this, the connection allows us to write the geodesic equation (1.8) as the zero acceleration constraint:

is parallel transported if it is extended to a tfor each tlinking tangent spaces. The parallel transport inherits linearity from the connection. It follows from the definition that γ .

along two curves γ and ϕ is flat, that is, has zero curvature.

1.3.5 Completeness

The Riemannian manifold is said to be geodesically complete is geodesically complete. This is a consequence of the Hopf–Rinow–de Rham theorem, which also states that geodesically complete manifolds are complete metric spaces with the induced distance and that there always exists at least one minimizing geodesic between any two points of the manifold, that is, a curve whose length is the distance between the two points.

From now on, we will assume that the manifold is geodesically complete. This assumption is one of the fundamental properties ensuring the well-posedness of algorithms for computing on manifolds.

1.3.6 Exponential and logarithm maps

Let x be a point of the manifold that we consider as a local reference point, and let v are defined for each t starting at x with tangent vector v. This mapping

is called the exponential map at point x. Straight lines passing 0 in the tangent space are transformed into geodesics passing the point x on the manifold, and the distances along these lines are conserved (Fig. 1.6).

Figure 1.6 (Left) Geodesics starting at x are images of the exponential map γ ( t )=Exp x ( tv for sufficiently small t and Exp x by Exp x ( v ). The cut locus of x is its antipodal point, and the injectivity radius is π .

, but it is generally one-to-one only locally around 0 in the tangent space corresponding to a local neighborhood of x . In this chart the geodesics going through x . Moreover, the distance with respect to the base point x is preserved:

Thus the exponential chart at x gives a local representation of the manifold in the tangent space at a given point. This is also called a normal coordinate system or normal chart is therefore only locally defined, that is, for points y near x.

The exponential and logarithm maps are commonly referred to as the Exp and Log maps.

1.3.7 Cut locus

to infinity, then it is either always minimizing for all tis called a cut pointis called a tangential cut point. The set of all cut points of all geodesics starting from x is the cut locus , and the set of corresponding vectors is the tangential cut locus containing 0 and delimited by the tangential cut locus.

is transformed into the unique minimizing geodesic from x to y:

From a computational point of view, it is often interesting to extend this representation to include the tangential cut locus. However, we have to take care of the multiple representations: Points in the cut locus where several minimizing geodesics meet are represented by several points on the tangential cut locus as the geodesics are starting with different tangent vectors (e.g. antipodal points on the sphere and rotation of π around a given axis for 3D rotations). This multiplicity problem cannot be avoided as the set of such points is dense in the cut locus.

is quantified by the injectivity radius ).

Example 1.4

), the geodesics are the great circles, and the cut locus of a point x . The exponential chart is obtained by rolling the sphere onto its tangent space so that the great circles going through p ; see Fig. 1.6.

where antipodal points are identified.

1.4 Elements of analysis in Riemannian manifolds

We here outline further constructions on manifolds relating to taking derivatives of functions, the intrinsic Riemannian measure, and defining curvature. These notions will be used in the following chapters of this book, for instance, for optimization algorithms.

1.4.1 Gradient and musical isomorphisms

Let f . This mapping corresponds to the transpose operator that is implicitly used in Euclidean spaces to transform derivatives of functions (row vectors) to column vectors. On manifolds, the Riemannian metric must be specified explicitly since the coordinate system used may not be orthonormal everywhere.

The mapping works for any covector and is often denoted the sharp . It has an inverse in the flat are denoted musical isomorphisms because they raise or lower the indices of the coordinates.

We can use the sharp map to define the Riemannian gradient as a vector:

of the gradient.

1.4.2 Hessian and Taylor expansion

The covariant derivative of the gradient, the Hessian, arises from the connection ∇:

Here the two expressions on the right are given using the action of the connection on the differential form df . Its expression in a local coordinate system is

be the expression of f in a normal coordinate system at x. Its Taylor expansion around the origin in coordinates is

corresponds to the Hessian Hess f. Thus the Taylor expansion can be written in any coordinate system:

(1.9)

1.4.3 Riemannian measure or volume form

induces an infinitesimal volume element on each tangent space, and thus a measure on the manifold that in coordinates has the expression

or in any exponential chart. If f is its image in the exponential chart at x, then we have

1.4.4 Curvature

in a way in which it inherits a flat geometry. In both cases the periodicity of the torus remains, which prevents it from being a vector space.

. It is defined from the covariant derivative by evaluation on vector fields X, Y, Z:

(1.10)

denotes the anticommutativity of the fields X and Y. If f , then the new vector field produced by the bracket is given by its application to f. The curvature tensor R ; see . The curvature tensor when evaluated at X, Ygiven by this difference.

Figure 1.7 (Left) The curvature tensor describes the difference in parallel transport of a vector Z around an infinitesimal parallelogram spanned by the vector fields X and Y (dashed vectors). (Right) The sectional curvature measures the product of principal curvatures in a 2D submanifold given as the geodesic spray of a subspace V . The principal curvatures arise from comparing these geodesics to circles as for the Euclidean notion of curvature of a curve.

The reader should note that two different sign conventions exist for the curvature tensor: definition (1.10) is used in a number of reference books in physics and mathematics [20,16,14,24,11]. Other authors use a minus sign to simplify some of the tensor notations [26,21,4,5,1] and different order conventions