Mathematics of Shapes and Applications - IMS

Mathematics of Shapes and Applications
(4 - 31 Jul 2016)

~ Abstracts ~

Constrained and multiple shape analysis
Sylvain Arguillère, Johns Hopkins University, USA

The general purpose of shape analysis is to compare different shapes in a way that takes into account their geometric properties, such as smoothness, number of self-intersection points, convexity... One way to do this is to find a flow of diffeomorphisms that brings one (template) shape as close as possible to the other (target) shape while minimizing a certain energy. This is the so-called LDDMM method (Large Deformation Diffeomorphic Metric Matching). In this talk, I will recall this framework, and extend it to the case where several shapes are considered, ad where constraints are added to the shapes, in order to better describe the objects they represent, and give some examples and applications in computational anatomy.

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 behavior of these metrics.

3) Right invariant metrics on the diffeomorphism group
The interest in right invariant metrics on the diffeomorphism group is fueled 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.

Computational eye/brain anatomy
Mirza Faisal Beg, Simon Fraser University, Canada

I will be presenting some recent work from our group in applying the toolbox of CA in the setting of retinal layer morphometry, and applications in glaucoma and Alzheimer's disease. If time permits, I will also talk about network-based structural biomarkers as well as a unified voxel-based morphometry/tensor-based morphometry method for generating structural measures for discriminating between different dementias.

Shape recognition and the eigenvalues of linear operators
Mohamed Ben Haj Rhouma, Qatar University, Tunisia

The purpose of this talk is to provide a theoretical overview of the characteristics of the eigenvalues of four well-known linear operators and assess their usefulness as reliable tools for shape recognition. In particular, we revisit the eigenvalues of the Dirichlet and Neumann problems and examine the recent literature on shape recognition using the Laplace operator. In addition, we will also examine the ratios of eigenvalues of the bi-laplacian which is a fourth order operator. In particular this will be done for the clamped plate and buckling of clamped plate problems. Our work shows that features based on these physical eigenvalue problems are reliable tools for shape recognition. Time permitting we will also discuss the relationship between the ratios of eigenvalues and some invariant moments of 2D shapes.

Explaining decisions of deep neural networks
Alexander Binder, Singapore University of Technology and Design

Deep neural networks excel in a number of tasks, however explanations are not common about what makes them arrive at a decision for one given input sample, e.g. one image. Notably, for the existing methods of explanations, there are open questions about what they achieve, e.g. what information backpropagation really does provide, in particular among computer scientists. In this talk I am going to present layer-wise relevance propagation, an approach to decompose the prediction of a deep neural network for one image in terms of single pixels and regions.Theoretical motivations, explanations of differences versus deconvolution and backpropagation will be given. A framework will be introduced for the evaluation of the computed explanations based on the implied ordering of regions and pixels. Results of numerical evaluations of the computed visualizations will be shown on Imagenet, MIT Places and SUN397 test data. Finally its ability to identify biases in your training data will be demonstrated.

Regularity of the geodesic boundary value problem on the diffeomorphism group
Martins Bruveris, Brunel University London, UK

I will talk about a result that is inspired from and a generalization of results on the smoothness of geodesics for right-invariant Riemannian metrics on the diffeomorphism group.

Under some natural assumptions, a right-invariant metric gives rise to a smooth, right-invariant exponential map on the group $D^q(R^d)$ of Sobolev diffeomorphisms with $q$ large enough. The right invariance leads to the following property: if the initial conditions of a geodesic are of class $H^{q+k}$, then so is the whole geodesic. This implies that smooth initial conditions lead to smooth geodesics.

In this talk I will show how to generalize this regularity principle to show the corresponding statement about the boundary value problem: if two diffeomorphisms and are nonconjugate along a geodesic, then the geodesic is as smooth as the boundary points.

This result also holds on diffeomorphism groups of compact manifolds and spaces of curves and surfaces.

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.

On the discrete approximation of continuous functional shapes equipped with regular signals
Benjamin Charlier, Université de Montpellier, France

A full theoretical and numerical setting for fshapes using $L^2$ signals has been presented in [CCT]. Nevertheless, some difficulties arise in the applications from this choice of $L^2$ norm. In this work, we consider signals of higher regularity such as $H^1$ or $BV$ signals. These functional norms prevent the signal from oscillating in the numerical simulations.
In this talk, we will focus on the "functional matching" problem. It means that we optimize the signal supported on an initial surface with respect to a target signal supported on a different surface. We then provide conditions on the discretization procedure implying a $\Gamma$-convergence result for the discrete matching energy towards the continuous one.
This is a joint work with G. Nardi and A. Trouvé [CNT]
bibliography :
[CCT] B. Charlier, N. Charon, A. Trouvé - The fshape framework for the variability analysis of functional shapes (2015)
[CNT] B. Charlier, G. Nardi, A. Trouvé - The matching problem between functional shapes via a BV -penalty term: a $\Gamma$-convergence result (preprint)

Metamorphoses of textured submanifolds with Sobolev regularity
Nicolas Charon, Johns Hopkins University, USA

This talk will describe in detail a joint model of geometric-functional variability between signals defined on deformable manifolds. These objects called fshapes were first introduced in [Charon et al. 2015]. Building on this work, we extend the original L2 model to treat signals of higher regularity on their geometrical support with stronger Hilbert norms (typically Sobolev). We describe the Hilbert bundle structure of such spaces and construct metrics based on metamorphoses, that groups into one common framework both shape deformation metrics and usual (flat) image metamorphoses. We then propose a formulation of matching between any two fshapes from the optimal control perspective, study existence of optimal controls and derive Hamiltonian equations describing the dynamics of solutions. Secondly, we tackle the discrete counterpart of these problems with adequate finite element methods and show a few results and applications of this fshape metamorphosis approach.

Bayesian statistics on manifolds: beyond MAP
Tom Fletcher, University of Utah, USA

Much of the work on manifold statistics has focused on point estimates, e.g., means, variances, regression functions, etc. Such estimates can be formulated probabilistically as maximum-likelihood estimates, Fréchet expectations, or in the Bayesian setting, maximum a posteriori (MAP) estimates. However, reducing the posterior distribution to a single point loses much of the advantage of the Bayesian point of view. In this talk, I will present some ideas on how to more fully use the posterior distribution. The basic tools for doing this are Markov Chain Monte Carlo sampling methods for manifold-valued random variables. I will show two examples of how sampling from the posterior distribution can be useful. First, in diffeomorphic image registration, posterior sampling provides a method for estimating parameters to the metric that control the level of regularization of the transformations. Second, a Bayesian analysis of a normal distribution law on manifolds provides a method for understanding the uncertainty of the Fréchet mean.

Tutorial on statistics on Riemannian manifolds
Tom Fletcher, University of Utah, USA

Manifold representations are useful for many different types of data, including directional data, geometric transformations, tensors, and shape. Statistical analysis of these data is an important problem in a wide range of image analysis and computer vision applications. These lectures will cover the fundamentals for statistical analysis of data that are represented as points on a Riemannian manifold. Throughout, the focus will be on statistical methods that have probabilistic formulations. Motivating examples will include spheres, shape spaces, and diffeomorphic image registration. An outline is as follows:

1. Basic statistics for manifold data:

This will include intrinsic and extrinsic definitions of the mean, the median, and variance of a set of points on a Riemannian manifold. The concept of Fréchet expectation plays a critical role for intrinsic statistics, and it can be thought of as a generalization of least-squares estimation to metric spaces. I will explain how this can be linked to a notion of a normal distribution on manifolds. This provides a probabilistic framework for statistics on manifolds, making possible maximum-likelihood estimation and Monte Carlo sampling.

2. Regression models on manifolds:

This lecture will cover regression models where the independent variable is a real number and the dependent variable is manifold-valued. The motivating example is regression analysis of shape changing over time.
First, I will cover parametric regression functions, the simplest of which is a geodesic, which gives leads to a generalization of linear regression to manifolds. More flexible parametric curves include Riemannian polynomials, which are defined as curves whose covariant derivatives vanish beyond some order. Finally, I will discuss a kernel regression model for manifolds that is nonparametric.

3. Dimensionality reduction on manifolds:

In this lecture I will discuss various approaches for dimensionality reduction on Riemannian manifolds. First, we will see principal geodesic analysis (PGA), which is a generalization of principal components analysis (PCA) to manifolds. This will include tangent space approximations to the fitting problem as well as more recent exact computations. Next, we will see that PGA can be given a probabilistic interpretation, using the Riemannian normal distribution. Finally, I will show a Bayesian formulation of PGA that provides automatic selection of the number of inherent dimensions in the data.

4. Diffeomorphisms and image matching:

This lecture will adapt many of the statistical models presented for finite-dimensional manifolds to the setting of diffeomorphic transformations of images. First, we will cover the basics of image matching, including LDDMM geodesic relaxation and shooting. The average from before now becomes the problem of computing an image atlas, or template. The various PGA models of dimensionality reduction can also be generalized. Finally, I will demonstrate how the metric on diffeomorphisms can be interpreted as a Bayesian prior, which leads to a method for inferring the parameters controlling the regularization of the transformations.

Longitudinal neuroimaging: analysis of shape and appearance to study group- and subject-specific growth and disease processes
Guido Gerig, New York University, USA

Clinical assessment routinely uses terms such as development, growth trajectory, aging, degeneration, disease progress, recovery or prediction. This terminology inherently carries the aspect of dynamic processes, suggesting that measurement of dynamic spatiotemporal changes may provide information not available from single snapshots in time.

Image processing of 4-D data embedding time-varying anatomical objects and functional measures requires a new class of analysis methods and tools that makes use of the inherent correlation and causality of repeated acquisitions. This talk will discuss progress in the development of advanced 4-D image and shape analysis methodologies that carry the notion of linear and nonlinear regression, now applied to complex, high-dimensional data such as images, image-derived shapes and structures, or a combination thereof. Methods include joint segmentation of serial 3D data enforcing temporal consistency, building of 4-D models of tissue diffusivity via longitudinal diffusion imaging, and 4-D shape models. We will demonstrate that statistical concepts of longitudinal data analysis such as linear and nonlinear mixed-effect modeling, commonly applied to univariate or low-dimensional data, can be extended to structures and shapes modeled from longitudinal image data.

We will show results from ongoing clinical studies such as analysis of early brain growth in subjects at risk for mental illness, early prediagnostic brain changes in autism, analysis of neurodegeneration in normal aging and Huntington's disease, and quantitative assessment of recovery in severe TBI.

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.

Introduction to the differential geometry
Joan Alexis Glaunès, Université Paris Descartes, France

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.

Joint work with Sergey Kushnarev, Singapore University of Technology and Design.

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 present both continuous and discrete forms of these models, and show some outputs of algorithms

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.

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.

An analysis of shape ensembles through modular diffeomorphisms
Barbara Gris, CMLA - ENS Cachan, France

A population of geometrical shapes can be studied through diffeomorphisms transforming one shape into another. We present a mathematical framework in which we build these diffeomorphisms thanks to a small number of user-defined generators of deformations called deformation module. This parametric setting enables to incorporate constraints in the deformation model and aims at building a vocabulary to describe the variability amongst a population. We present a method to build easily complex deformation modules by combining base-modules in a hierarchical framework.

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 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.

Stochastic landmark dynamics 1: soliton interactions
Darryl Holm, Imperial College London, UK

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (landmarks) of the stochastically perturbed EPDiff equation derived using this method. These numerical simulations show that landmark solutions of the stochastically perturbed EPDiff equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to stochastic partial differential equations (SPDE). One lesson is that a certain amount of care must be taken when incorporating stochasticity into the EPDiff equation. In particular, some choices of stochastic perturbations of the landmark dynamics by Wiener noise allow landmarks to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincaré structure of the EPDiff equation, and it also does not occur for the original landmark solutions of the unperturbed deterministic EPDiff equation. The discussion raises issues about the science of stochastic deformations of evolutionary PDE and the sensitivity of the resulting solutions of the SPDE to the choices made in stochastic modelling.

Magnetic resonance imaging (MRI) - based electrical property mapping
Shaoying Huang, Singapore University of Technology and Design

To obtain an in vivo electrical property map of human body in a non-invasive way has been motivated by a numbers of reasons. First, electrical property of human tissue has high contrast, therefore an electrical property map provides anatomical information of human tissues. Secondly, electrical property can be used for early cancer detections because the electrical property of cancerous tissues is highly distinguishable from that of healthy ones. Moreover, an electrical property map is crucial for accurate calculation of specific absorption rate (SAR) which is a main parameter for assessing risk of radiofrequency and microwave radiation. We process data from a magnetic resonance imaging (MRI) scanner for retrieving electrical property maps for human body.The MRI-based electrical property retrievals will be reviewed. The work done in my group at Singapore University of Technology and Design will be presented in detail. Two types of data are used, one is RF magnetic fields (B1-maps) and the other is phase maps. For B1-maps, we have proposed a few methods including taking approaches of analyzing Maxwell equations analytically and numerically. For phase maps as inputs, our contributions will be detailed. Corresponding results for different methods will be shown and discussed. Discussions and future challenges in this area will be presented as well.

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.

Brainnetome atlas: a new brain atlas based on connectional profiles
Tianzi Jiang, Chinese Academy of Sciences, China

Brain atlas is considered to be the cornerstone of basic neuroscience and clinical researches. Brainnetome atlas is constructed with brain connectivity profiles. It is in vivo, with finer-grained brain subregions, and with anatomical and functional connection profiles. Using the brainnetome atlas, researchers could simulate and model brain networks using informatics and simulation technologies to elucidate the basic organizing principles of the brain. Others could use this same atlas to design novel neuromorphic systems that are inspired by the architecture of the brain. Therefore, this cutting-edge brainnetome atlas paves the way for constructing an even more fine-grained atlas of the human brain and offers the potential for applications in brain-inspired computing. In this lecture, we will summarize the advance of the human brainnetome atlas and its applications. We first give a brief introduction on the history of the brain atlas development. Then we present the basic ideas of the human brainnetome atlas and the procedure to construct this atlas. After that, some parcellation results of representative brain areas will be presented. We also give a brief presentation on how to use the human brainnetome atlas to address issues in neuroscience and clinical research. Finally, we will give a brief perspective on monkey brainnetome atlas and the related neurotechniqes.

Weighted diffeomorphic density matching with applications to thoracic image registration
Sarang Joshi, University of Utah, USA

In this talk we will study the problem of thoracic image registration, in particular the estimation of complex anatomical deformations associated with the breathing cycle. Using the intimate link between the Riemannian geometry of the space of diffeomorphisms and the space of densities, we develop an image registration framework that incorporates both the fundamental law of conservation of mass as well as spatially varying tissue compressibility properties. By exploiting the geometrical structure, the resulting algorithm is computationally efficient, yet widely general.

Geometric representation and analysis of curve shapes on manifolds and its applications to biology
Shantanu Joshi, University of California Los Angeles, USA

Shapes are compact signal representations of objects from images, and are remarkably useful for understanding and modeling biological morphology. Although there have been several advances in developing analytical techniques for shape analysis, the ensuing statistical inference still uses linear signal processing approaches that rely on standard inner products on Euclidean spaces. With the increasing high resolution biomedical imaging acquisitions, it is becoming clear that this discrepancy in the approaches between statistical analysis and shape representation is a major limitation. Additionally with the acquisition of imaging modalities such as structural magnetic resonance imaging, or diffusion tensor imaging, we need novel approaches for shape signal representation and analysis.

In my talk, I will outline a geometric approach for representing and analyzing parameterized shapes of boundaries. Shapes are represented as elements of an infinite-dimensional, non-linear, quotient space, and statistics of shapes are defined and computed using differential geometry of this shape space. This approach enables an intrinsic statistical analysis of both open and closed, parameterized curves. Due to a special square-root velocity parameterization, the shape space turns out to be a infinite-dimensional sphere, and geodesics can be analytically specified. Additionally, the geodesics are also be computed in a parameterization-invariant manner. This allows invariant elastic matching of shapes with biologically homologous results. The underlying geometric approach has also been shown to be extensible to 3D surface shapes as well as functional data. In this talk, I will present applications to biomedical imaging, computer vision, paleontology, and most recently to brain morphometry.

Time-varying dynamics: framework and cost functions
Irene Kaltenmark, CMLA - ENS Cachan, France

The Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework has proved to be highly efficient for addressing the problem of modeling and analysis of the variability of populations of shapes, allowing for the direct comparison and quantization of diffeomorphic morphometric changes. However, the analysis of medical imaging data also requires the processing of more complex changes, which especially appear during growth or aging phenomena. The observed organisms are subject over time to transformations that are no longer diffeomorphic, at least in a biological sense. One reason might be a gradual creation of new material uncorrelated to the preexisting one. For this purpose, we offer to extend the LDDMM framework to address the problem of non diffeomorphic structural variations in longitudinal scenarios during a growth or degenerative process. We keep the geometric central concept of a group of deformations acting on a shape space. This action induces a pointwise expression of the dynamic of the shape. The introduction of partial mappings leads yet to a time-varying dynamic that modifies the action of the group of deformations. In growth scenarios, the shape evolves via inner partial mappings induced by a growth dynamic. The underlying minimization problem requires an adapted framework to consider a new set of cost functions (penalization term on the deformation).

Introduction to the differential geometry
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.

Computing distances and geodesics between manifold-valued curves in the SRV framework
Alice Le Brigant, Mathematical Institute of Bordeaux, France

Many applications involve the study of curves in the euclidean space, but some require to consider curves in non flat manifolds. Simple examples in positive curvature include trajectories on the sphere where points represent positions on the earth, and a space of interest in negative curvature is the hyperbolic upper-half plane, which coincides with the statistical manifold of Gaussian densities. We are motivated by the study of curves in the latter for signal processing purposes, however our framework is more general. We study a first-order Sobolev metric on the space of manifold-valued curves using a generalization of the Square Root Velocity (SRV) function introduced by Srivastava et al. This framework allows us to characterize geodesics for our metric and, after discretization of our model, to build them using a geodesic shooting algorithm. Mean curves can also be computed with a gradient-descent type method.

Interpolation on symmetric spaces and variational discretizations of gauge field theories
Melvin Leok, University of California, San Diego, USA

Many gauge field theories can be described using a multisymplectic Lagrangian formulation, where the configuration manifold is the space of Lorentzian metrics. Group-equivariant interpolation spaces are critical to the construction of geometric structure-preserving discretizations of such problems, since they can be used to construct a variational discretization that exhibits a discrete Noether's theorem. We approach this problem more generally, by considering interpolation spaces for functions taking values in a symmetric space --- a smooth manifold with an inversion symmetry about every point.

Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition -- a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.

Langevin equations for landmark image registration with uncertainty
Stephen Marsland, Massey University, New Zealand

Registration of images parameterised by landmarks provides a useful method of describing shape variations by computing the minimum-energy time-dependent deformation field that flows one landmark set to the other. This is sometimes known as the geodesic interpolating spline and can be solved via a Hamiltonian boundary-value problem to give a diffeomorphic registration between images. However, small changes in the positions of the landmarks can produce large changes in the resulting diffeomorphism. We formulate a Langevin equation for looking at small random perturbations of this registration. The Langevin equation and three computationally convenient approximations are introduced and used as prior distributions. A Bayesian framework is then used to compute a posterior distribution for the registration, and also to formulate an average of multiple sets of landmarks.

Shape discrimination via quantitative homological features
Facundo Mémoli, The Ohio State University, USA

In this talk shapes are modeled as finite metric spaces and the notion of distance between shapes that we consider is the Gromov-Hausdorff distance.
I'll describe some ideas for the estimation of this distance via some metric space invariants induced by constructions derived from simplicial homology with field coefficients. In particular, I'll describe a poly-time computable lower bound for the GH distance between two shapes that arises from computing a certain distance between the homological invariants corresponding to each shape.

Shape optimization on shape manifolds
Jakob Møller-Andersen, Danmarks Tekniske Universitet, Denmark

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.

Image registration: higher oder models, uncertainties, and predictions
Marc Niethammer, The University of North Carolina at Chapel Hill, USA

This talk will primarily focus on some extensions to the large displacement diffeomorphic metric mapping (LDDMM) registration model. In particular, this talk will discuss higher order registration models, how to compute aspects of registration uncertainty, and how to learn models of image appearance and deformation.

A cortical columnar normal coordinate system
Tilak Ratnanather, Johns Hopkins University, USA

We demonstrate a novel surface-based diffeomorphic algorithm to generate a 3D coordinate grid in the cortical ribbon. In the grid, normal coordinate lines from the gray/white (inner) surface to the gray/csf (outer) surface are constrained to be normal at the surfaces. Specifically, the cortical ribbon is described by two triangulated surfaces with open boundaries of a cortical region. Conceptually, one surface sits on top of the white matter structure and the other on top of the gray matter. It is assumed that the cortical ribbon consists of cortical columns which are orthogonal to the white matter surface. This might be viewed as a consequence of the development of the columns from the cortical plate. It is also assumed that the columns are orthogonal to the outer surface. So if we construct a vector field such that the inner surface evolves diffeomorphically towards the outer one, the distance of the resultant trajectories will be a measure of thickness. This approach offers great potential for quantitative functional and histological analysis of cortical activity and anatomy.

(Joint work with Laurent Younes, Sylvain Arguillère and Michael Miller).

Kernel metrics on normal cycles for the matching of geometrical structures
Pierre Roussillon, Ecole Normale Supérieure de Cachan, France

We introduce a new dissimilarity measure for shape registration using the notion of normal cycles, a concept from geometric measure theory which allows to generalize curvature for non smooth subsets of the euclidean space. Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. This approach is closely similar to previous works based on currents and varifolds. We derive the computational setting for discrete curves and surfaces in R3, using the Large Deformation Diffeomorphic Metric Mapping framework as model for deformations. We present synthetic and real data experiments and compare with the currents and varifolds approaches.

An overview of the developing human connectome project (DHCP)
Daniel Rueckert, Imperial College London, UK

Few advances in neuroscience could have as much impact as a precise global description of human brain connectivity (connectome) and its variability. Understanding this connectome in detail will provide insights into fundamental neural processes and intractable neuropsychiatric diseases. Currently, the connectome of the mature adult brain is in progress. The Developing Human Connectome Project (dHCP) aims to make major scientific progress by creating the first 4-dimensional connectome of early life. Our goal is to create a dynamic map of human brain connectivity from 20 to 44 weeks post-conceptional age, which will link together imaging, clinical, behavioural, and genetic information. This unique setting, with imaging and collateral data in an expandable open-source informatics structure, will permit wide use by the scientific community, and to undertake pioneer studies into normal and abnormal development by studying well-phenotyped and genotyped group of infants with specific genetic and environmental risks that could lead to Autistic Spectrum Disorder or Cerebral Palsy.

Mathematical models for controlling seizures and neurosurgical planning
Nishant Sinha, Nanyang Technological University

Neural stimulation and surgical resection are common treatments for intractable epilepsy. However, the efficacy of these methods remains limited; 30% of patients continue to have seizures post-surgery, and neural stimulation reduces seizures in only about 50% of patients. Therefore, more effective and widely applicable stimulation protocols for neural stimulation are needed. Moreover, methods to predict and improve surgical outcomes are needed.

In this talk, I will describe our efforts to address these two research questions.
I will present how we have applied optimal control theory to patient-specific models of epileptic seizures, for the purpose of designing neural stimuli. I will show results for this approach in large-scale models of epilepsy. I will also describe a practical method that we have developed to predict the outcome of neurosurgery for patients with localized epilepsy. We have designed a dynamical network model of transitions to seizure-like dynamics, whereby the connectivity of the model is inferred from patient inter-ictal ECoG data. From this model, we compute the likelihood of surgical success. By applying this method, surgical outcomes were successfully predicted with 81.2% accuracy on a dataset of 16 patients. This result is promising given the simplicity of the computational model. Overall, these results provide us confidence that computational models may prove to be helpful for planning neurosurgical interventions.

This is joint work with Prof. Justin Dauwels (NTU), Dr. Peter Taylor (Newcastle University), and Dr. Justin Ruths (SUTD).

Stochastic landmark dynamics 2: statistics and correlations
Stefan Sommer, University of Copenhagen, Denmark

The dynamics of landmarks following the stochastically pertubed EPDiff equation depend on the spatial correlation in the noise process. We study different approaches for performing estimation of parameters for the noise and initial distribution of positions and momenta given observations of landmarks at discrete time points. Based on approximations of the Fokker-Planck equation and Monte Carlo sampling, we perform inference in the statistical models that arise from parametric families of noise, and we show how the particular structure of the stochastic EPDiff equations is important for the resulting numerical inference algorithms.

Frame-based spline kernels with multiscale properties in diffeomorphic metric image registration
Ming Zhen Tan, National University of Singapore

Deformable diffeomorphic registration of medical images is a widely studied topic with important statistical applications. State-of-the-art approaches build a time-dependent velocity field in a reproducing kernel Hilbert space (RKHS) and enforce smoothness constrains on the deformation via radial-based reproducing kernels, such as the Gaussian kernel. In this work, we propose and discuss the construction of frame-based kernels using b-splines and spline-framelets that have a natural multiscale structure, and thus, incorporate multiscale characteristics in the velocity field. As a proof of concept, we incorporate these kernels in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework for image registration and compared the results to a conventional Gaussian kernel.

Stochastic landmark dynamics 3: Stochastic discrete Hamiltonian variational integrators
Tomasz Tyranowski, Imperial College London, UK

Hamiltonian systems with multiplicative noise are a mathematical model for many physical systems with uncertainty. They may also be applied as a model for stochastic landmark dynamics in image registration. Stochastic Hamiltonian systems, just like their deterministic counterparts, possess several important geometric
features, for instance, their phase flows preserve the symplectic structure. When simulating these systems numerically, it is therefore advisable that the numerical scheme also preserves such geometric structures. In this talk we propose a variational principle for stochastic Hamiltonian systems and use it to construct stochastic Galerkin variational integrators. We show that such integrators are indeed symplectic, preserve integrals of motion related to Lie group symmetries, and demonstrate superior long-time energy behavior compared to nonsymplectic methods. We also analyze their convergence properties and present the results of several numerical experiments.

Cubic splines on the group of diffeomorphisms and the Fisher-Rao metric
Francois-Xavier Vialard, Université Paris-Dauphine, France

In this talk, we will present the analytical study of a higher-order variational problem on the group of diffeomorphisms of the interval [0,1] endowed with a right-invariant Sobolev metric H^2. Namely, we aim at minimizing the Riemannian acceleration of a curve under boundary conditions We study the relaxation of the problem and show how the Fisher-Rao metric on the space of densities appears in the functional. We then discuss optimality conditions associated with a standard Riccati equation and show numerical simulations that tends to prove the existence of minimizers that is not "classical".

Special registration problems: beyond the LDDMM paradigm
Laurent Younes, Johns Hopkins University, USA

The talk will discuss registration problems in which additional features are needed beyond those provided by the standard LDDMM method. These features can be expressed through the introduction of constraints on the registration process, or by modifying the metric, relaxing the right-invariance by diffeomorphism in a problem-dependent way. We will review some of the instances and applications of such approaches, including multi-shapes, atrophy, normal evolution and others.

Change point estimation of brain shape data in relation with Alzheimer's disease
Laurent Younes, Johns Hopkins University, USA

The manifestation of an event, such as the onset of a disease, is not always immediate and often requires some time for its repercussions to become observable. Slowly progressing diseases, and in particular neuro-degenerative disorders such as Alzheimer's disease (AD), fall into this category. The manifestation of such diseases is related to the onset of cognitive or functional impairment and, at the time when this occurs, the disease may have already had been affecting the brain anatomically and functionally for a considerable time. We consider a statistical two-phase regression model in which the change point of a disease biomarker is measured relative to another point in time, such as the manifestation of the disease, which is subject to right-censoring (i.e., possibly unobserved over the entire course of the study). We develop point estimation methods for this model, based on maximum likelihood, and bootstrap validation methods. The effectiveness of our approach is illustrated by numerical simulations, and by the estimation of a change point for atrophy in the context of Alzheimer's disease, wherein it is related to the cognitive manifestation of the disease.
This work is a collaboration with Marilyn Albert, Xiaoying Tang and Michael Miller, and was partially supported by the NIH.

Mapping of large-scale brain networks and statistical connectomics
Andrew Zalesky, The University of Melbourne, Australia

In recent years, attempts to comprehensively map the human connectome have occupied a central position in neuroscience, leading to several large-scale collaborative endeavours that are unprecedented in neuroscience. In this talk, I will outline some of the key computational and statistical challenges in mapping whole-brain networks with neuroimaging techniques. I will specifically focus on diffusion and resting-state functional magnetic resonance imaging. I will present novel network-based methods for performing statistical inference on the connectome and fusing together functional and structural connectivity maps. I will also present empirical and asymptotic results that highlight the importance of connectome specificity. Finally, I will present clinical examples demonstrating the utility of connectomic techniques in understanding brain network organization is psychiatric disorders such as schizophrenia.

Mapping multimodal brain network changes in ageing and neuropsychiatric disorders
Juan, Helen Zhou, Duke-NUS Medical School

Neuropsychiatric disorders target large-scale neural networks. Emerging network-sensitive neuroimaging techniques have allowed researchers to demonstrate that the spatial patterning of each neurodegenerative disease relates closely to a distinct functional intrinsic connectivity network (ICN), mapped in the healthy brain with task-free or "resting-state" functional magnetic resonance imaging (fMRI). This talk will describe our recent work on the salience network connectivity changes in patients with Alzheimer's disease, behavioral variant frontotemporal dementia and persons at-risk for psychosis using multimodal neuroimaging techniques (magnetic resonance imaging (MRI)). Moreover, longitudinal inter-network connectivity changes underlying cognitive decline in healthy aging will be highlighted. Lastly, we will discuss our recent work on linking dynamic functional connectivity states with vigilance fluctuations. Further developed, multimodal connectivity signatures may help us reveal disease mechanism and predict or track disease progression.

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