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Fundamental Mathematical Tools
In this chapter, we introduce the mathematical tools used in this thesis. First order di eren-tial equations are the basic tool we use to construct invertible geometrical transformations. The theory of Lie groups is particularly useful to analyze the properties of transformations groups such as rigid or a ne transformations. And Riemannian geometry is a very powerful framework used more and more to process data belonging to non-linear spaces.
Although some of the notions and mathematical properties presented here are quite ad-vanced, we have tried to make our presentation as intuitive as possible. Rather than technical details, we hope that the usefulness and vision behind each tool will be the most salient items in this short presentation.
First Order Ordinary Di erential Equations
A classical way of obtaining invertible smooth geometrical transformations is to use rst or-der ordinary di erential equations (ODEs) [Tenenbaum 85]. The fundamental idea is that composing iteratively small invertible deformations guarantees that the resulting (possibly large) deformations will remain invertible. This was for example noted in the work of Chris-tensen et al. on uid deformations, who used the expression `topology-preserving’ rather than `invertible’. Indeed, on page 1439 of [Christensen 96], one can read:
We note that the above procedure assures that the concatenated transformation of the template into the study preserves the topology of the template. This is because each of the propagated template transformations preserve topology (due to the fact that the Jacobian of the transformation is positive globally) and the concatenation of topology-preserving transfor-mations produces a topology-preserving transformation.
Definition
To obtain invertible smooth geometrical deformations with rst order ODEs, any point x of Rn can be displaced little by little (in fact, in nitesimally) and in a reversible manner via the continuous integration during one unit of time of an evolution equation. A rst order ODE is an equation of the following form:
x˙(s) = V (x, s). (2.1)
Of course, the integration of (2.1) is a well-de ned mathematical operation only if the velocity vector eld V (x, s) is smooth enough (for instance C1) with respect to x. The principle of obtaining geometrical transformations via the continuous addition of in nitesimally small deformations during one unit of time is illustrated in Fig. 2.1.
A particular case of interest is when V (x, s) does not depend on s. Then, the ODE is called stationary or equivalently autonomous. For instance, the ODE de ning Log-Euclidean polya ne transformations in Chapter 6 is stationary.
In the work of TrouvÈ, Younes, Miller, Joshi and others [TrouvÈ 98, TrouvÈ 00, Camion 01, Beg 03, Miller 03, Guo 04, Guo 04, Joshi 00, Beg 05, TrouvÈ 05b, Glaunes 04, AllassonniËre 05], ODEs are used to generate very general di eomorphisms1, with a very high number of degrees of freedom (theoretically in nite).
On the contrary, we mainly use ODEs in this work to de ne novel classes of di eomor-phisms with a small number of exible degrees of freedom. These transformations, called polyrigid and polya ne, are described in Chapters 5 (original polyrigid transformations), 6 (Log-Euclidean polya ne transformations) and 7 (left-invariant polya ne transformations). We rely in Chapter 8 on autonomous rst order ODEs to generalize the notion of logarithm to di eomorphisms.
Life Span of a Solution to an ODE
In order to de ne a geometrical transformation via the ODE (2.1), it is necessary to prove rst that the position at time 1 exists, whatever the initial position may be. This is what we do in the next chapters for polyrigid and polya ne transformations. This is a necessary precaution since for an arbitrary ODE, the existence is not always insured, however smooth the velocity function V may be. Consider for instance, the 1D evolution y˙(s) = V (y) = y2.
Flow of a First Order ODE
Definition 2.1. The ow associated to a rst order ODE is the family of mappings Φ(., s) : Rn → Rn parameterized by a time parameter s ∈ R, such that for a xed x0, s Φ(x0, s) is the unique solution of x˙ = V (x) with initial condition x0 at time 0.
Intuitively, for a xed s, the mapping x Φ(x, s) gives the way the ambient space is deformed by the integration of the ODE during s units of time. It is always a di eomorphism, i.e. a di erentiable one-to-one mapping between the ambient space and itself, whose inverse is also di erentiable. This invertibility property simply comes from the fact that all deformations induced by the ODE are reversible since one can go back in time by simply multiplying the velocity vector V (x) by −1! The smoothness of the ow comes from the smoothness of V (x).
One-Parameter Subgroups
Definition 2.2. Let (G, .) be a group (i.e., the multiplication `.’ is associative and there exists a neutral element e and each element of G has a unique inverse). Then a family of elements g(s) of G parameterized by s ∈ R is called a one-parameter subgroup of G if and only if:
(a) g(0) = e,
i.e. the neutral element of G
(b)
for all s, t in R :
g(s).g(t) = g(s + t).
Furthermore, if one knows how to di erentiate functions valued in G (e.g., G is a Lie group), and if g(s) is di erentiable at 0, then dgdt (0) is called the in nitesimal generator of the subgroup.
One-parameter subgroups are particularly useful to describe a crucial property of the ow of stationary rst order ODEs. Furthermore, they are closely linked to the notion of group exponential, described in the next Section. These remarkable subgroups will also be at the heart of our de nition of bi-invariant means in Lie groups in Chapter 7.
Interestingly, the in nitesimal generator of a one-parameter subgroup contains all the information about the subgroup, and can generate it entirely, which explains its name. See for example [Godement 82] for more details on this topic in the case of Lie groups.
We have the following result: the ow Φ(., s) of an autonomous rst-order ODE is a one-parameter subgroup of the group of di eomorphisms. In other words: Φ(., s) ◦ Φ(., t) = Φ(., s + t). This implies in particular that the deformations of space given at time 1 by Φ(., 1) are twice that observed at time 0.5 via Φ(., 0.5). The in nitesimal generator of the ow is simply V (x). This is not surprising since it is quite clear how V (x) in nitesimally generates the ow: this is done precisely by integrating the associated ODE. These results will be used in Chapter 6 to prove the particularly nice properties of Log-Euclidean polya ne transformations and in Chapter 8 to generalize the notion of logarithm to di eomorphisms.
Lie Groups
Definition of Lie groups
We start by recalling the basic properties of Lie groups, along with the convenient notions which are classically used to describe these properties. Typical examples of such groups are groups of geometrical transformations (e.g., rigid or a ne transformations), where the multiplication is the composition of mappings. Such groups are particularly important in medical imaging, since the two most basic types of transformations used to register medical images, i.e. rigid and a ne transformations, are both Lie groups.
In simple terms, a Lie group is rst a group in the algebraic sense, i.e. a set of elements in which a multiplication between elements is de ned. This multiplication is assumed to have neat and intuitive properties: it is associative (i.e., (a.b).c = a.(b.c)), has a neutral element e and each element a has a unique inverse a−1.
Second, a Lie group has a structure of (smooth, i.e. C∞) di erential manifold. This means that it is locally similar to a vector space, but can be quite `curved’ globally.
Third, the algebraic and di erential structures are compatible: inversion and (left- and right-) multiplications are smooth mappings. This means that it is possible to inde nitely di erentiate them. For more formal de nitions and more details, please refer to classical di erential geometry books like [Sternberg 64] or [Gallot 93].
Examples
Many usual spaces can be viewed as Lie groups. Namely:
à Vector spaces (with their addition)
à Multiplicative matrix groups: GL(n), O(n), SO(n), etc., with the usual matrix multi-plication.
à Geometric transformation groups such as rigid transformations, similarities, a ne trans-formations… which can anyway also be looked upon as matrix groups via their `faithful’ representation based on homogeneous coordinates.
Table of contents :
Introduction
1.1 Data Processing in Lie Groups and Medical Imaging
1.1.1 Tensor Processing
1.1.2 Parameterization of Geometrical Transformations
1.1.3 Statistics on Invertible Linear Transformations
1.1.4 Statistics on Dieomorphisms
1.2 Manuscript Organization
1.3 Global Picture
2 Fundamental Mathematical Tools
2.1 Notations
2.2 First Order Ordinary Dierential Equations
2.2.1 Denition
2.2.2 Life Span of a Solution to an ODE
2.2.3 Flow of a First Order ODE
2.2.4 One-Parameter Subgroups
2.3 Lie Groups
2.3.1 Denition of Lie groups
2.3.2 Examples
2.3.3 Lie Algebra and Adjoint Representation
2.3.4 Matrix Exponential and Logarithm
2.3.5 Lie Group Exponential and Logarithm
2.3.6 Baker-Campbell-Hausdor Formula
2.4 Riemannian Geometry
2.4.1 Riemannian Metrics
2.4.2 Riemannian Geodesics
2.4.3 Riemannian Exponential and Logarithm
2.4.4 Fréchet Mean
3 Log-Euclidean Metrics on Tensors
3.1 Introduction
3.2 Preliminaries
3.2.1 Dierential Properties of Matrix Exponential
3.2.2 Algebraic Properties of SPD Matrices
3.2.3 Dierential Properties of SPD Matrices
3.2.4 Compatibility Between Algebraic and Dierential Properties
3.3 Log-Euclidean Means
3.3.1 Multiplication of SPD Matrices
3.3.2 Log-Euclidean Metrics on the Lie Group of SPD Matrices
3.3.3 A Vector Space Structure on SPD Matrices
3.3.4 Log-Euclidean Mean
3.4 Comparison with the Ane-Invariant Mean
3.4.1 Elementary Metric Operations and Invariance
3.4.2 Ane-Invariant Means
3.4.3 Geometric Interpolation of Determinants
3.4.4 Criterion for the Equality of the Two Means
3.4.5 Larger Anisotropy in Log-Euclidean Means
3.4.6 Linear and Bilinear Interpolation of SPD Matrices
3.5 Probabilities and Statistics with Log-Euclidean Metrics
3.5.1 General Riemannian Statistical Framework
3.5.2 Random Tensors
3.5.3 Fréchet Means and Covariances with Log-Euclidean Metrics
3.5.4 General Log-Euclidean Statistical Framework
3.6 Conclusions
4 Log-Euclidean Processing of Diusion Tensors
4.1 Introduction
4.1.1 The Defects of Euclidean Calculus
4.1.2 Riemannian Metrics
4.2 Theory
4.2.1 Matrix Exponential, Logarithm and Powers
4.2.2 Denition of Log-Euclidean Metrics
4.2.3 Invariance Properties of Log-Euclidean Metrics
4.2.4 Log-Euclidean Computations on Tensors
4.2.5 Comparison of the Ane-Invariant and Log-Euclidean Frameworks
4.3 Methods
4.3.1 Interpolation
4.3.2 Regularization
4.3.3 Absolute Value of a Symmetric Matrix
4.3.4 Materials
4.4 Results
4.4.1 Interpolation
4.4.2 Regularization
4.5 Discussion and Conclusions
4.5.1 The Defects of Euclidean Calculus
4.5.2 Conclusions and Perspectives
5 Original Polyrigid and Polyane Transformations
5.1 Introduction
5.2 Theory of Polyrigid and Polyane Transformations
5.2.1 Regions of Inuence and Interpolation of Sparse Data
5.2.2 A Framework with ODEs
5.2.3 Theoretical Properties of Polyrigid Transformations
5.2.4 Extension to Polyane Transformations
5.2.5 Summary of the properties of Polyrigid Transformations
5.3 Implementation of Polyrigid Transformations
5.3.1 Discretization Schemes
5.3.2 Implementation with the Insight Toolkit
5.4 Registration of Histological Slices
5.4.1 Object of the Study and Experimental Setup
5.4.2 Limitations of the First-Order Gradient Descent
5.4.3 Registration Results using a Levenberg-Marquardt Algorithm
5.4.4 Alternating Optimization
5.5 Results with more Complex Regions
5.5.1 The Shape of the Regions of Inuence
5.5.2 Results Obtained with Three Anchor Points
5.6 Conclusions and Perspectives
5.7 Appendix: First Derivatives of Original Polyrigid Transformations
6 Log-Euclidean Polyane Transformations
6.1 Introduction
6.2 A Log-Euclidean Polyane Framework
6.2.1 Previous Polyane Framework
6.2.2 Simpler Velocity Vectors for Ane Transformations
6.2.3 Log-Euclidean Polyane Transformations
6.3 Fast Polyane Transform
6.3.1 Matrix Exponential and the ‘Scaling and Squaring’ Method
6.3.2 A ‘Scaling and Squaring’ Method for LEPTs
6.3.3 2D Synthetic Experiments
6.3.4 3D Registration Example
6.4 A Log-Euclidean Framework for Rigid and Ane Transformations
6.4.1 Log-Euclidean Metrics
6.4.2 Invariance Properties
6.4.3 Regularization of Locally Ane Transformations
6.4.4 Log-Euclidean Framework for General Lie Groups
6.4.5 Numerical Implementation of Matrix Logarithm
6.5 Conclusion and Perspectives
7 Bi-Invariant Means in Lie Groups
7.1 Introduction
7.2 Means in Lie Groups
7.2.1 Means and Algebraic Invariance
7.2.2 Bi-Invariant Fréchet Means via Invariant Metrics in Lie Groups
7.2.3 Absence of Bi-Invariant Metrics for Rigid Transformations
7.3 Advanced Properties of the Exponential and Logarithm
7.3.1 Preliminary Result
7.3.2 Group Geodesics
7.4 Bi-Invariant Means in Lie Groups
7.4.1 A Geometric Denition of the Mean
7.4.2 Stability of the Classical Iterative Scheme
7.4.3 Convergence: Special Case
7.4.4 Convergence: General Case
7.4.5 Higher Order Moments
7.5 Bi-Invariant Means in Simple Cases
7.5.1 Bi-Invariant Mean of Two Points
7.5.2 Scalings and Translations in 1D
7.5.3 The Heisenberg Group
7.5.4 On a Subgroup of Triangular Matrices
7.6 Linear Transformations
7.6.1 General Rigid Transformations
7.6.2 2D Rigid Transformations
7.6.3 General Linear Transformations
7.6.4 Tensors
7.7 Left-Invariant Polyane Transformations
7.7.1 Polyane Transformations
7.7.2 A Novel Type of Polyane Transformations
7.8 Conclusions and Perspectives
8 Statistics on Dieomorphisms in a Log-Euclidean Framework
8.1 Introduction
8.2 A Log-Euclidean Framework for Dieomorphisms
8.3 Computation of Exponentials and Logarithms
8.4 Statistics on 3D Dieomorphisms
8.5 Conclusions and Perspectives
9 Conclusions and Perspectives
9.1 Contributions and Publications
9.1.1 General Log-Euclidean Framework for Tensors
9.1.2 Diusion Tensor Processing
9.1.3 Structure Tensor Processing
9.1.4 Statistics on Anatomical Variability via Tensors
9.1.5 Statistics on Deformation Tensors in Non-Linear Registration
9.1.6 Polyrigid and Polyane Transformations
9.1.7 Statistics in Finite-Dimensional Lie Groups
9.1.8 Statistics on Dieomorphisms
9.2 Short Term Perspectives
9.2.1 Tensor Processing
9.2.2 Locally Rigid or Ane Transformations
9.2.3 General Statistics in Lie Groups
9.2.4 Statistics on Dieomorphisms
9.3 More Global Perspectives
9.3.1 Natural Extensions of this Work
9.3.2 Bi-Invariant Framework vs. Riemannian Geometry
9.3.3 Medical Image Registration
9.4 Epilogue .
