# Seminars (GFSW03)

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Event When Speaker Title Presentation Material
GFSW03 13th November 2017
09:45 to 10:30
Peter Michor General Sobolev metrics on the manifold of all Riemannian metrics
Based on collaborations with M.Bauer, M.Bruveris, P.Harms. For a compact manifold $M^m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space $\operatorname{Met}_{H^s(\hat g)}(M)$ of all Riemannian metrics of Sobolev class $H^s$ for real $s>\frac m2$ with respect to $\hat g$. The $L^2$-metric on $\operatorname{Met}_{C^\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ were considered in [M.Bauer, P.Harms, and P.W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom., 94(2):187-208, 2013.] In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive real (not integer) order $s$. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping.
GFSW03 13th November 2017
11:00 to 11:30
Jean Feydy An efficient kernel product for automatic differentiation libraries, with applications to measure transport
Authors : Benjamin Charlier, Jean Feydy, Joan Alexis Glaunès and Alain Trouvé This paper presents a memory-efficient implementation of the kernel matrix-vector product, which is suitable for use with automatic differentiation libraries -- in our case, PyTorch. This piece of software alleviates the major bottleneck of autodiff libraries as far as diffeomorphic image registration is concerned: symbolic python code can now scale up to large point clouds and shapes (100,000+ vertices). To showcase the value of automatic differentiation to the LDDMM community, we introduce the "normalized Hamiltonian" setting and show that it corresponds to a spatially regularized optimal transport of mass distributions: made tractable by autodiff libraries, the kernel normalization trick turns an extrinsic image deformation routine into an intrinsic measure transportation program.
GFSW03 13th November 2017
11:30 to 12:15
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GFSW03 13th November 2017
14:00 to 14:45
Klas Modin Riemannian Gradient Flows in Shape Analysis
In this talk I show how the framework of Riemannian gradient flows on Lie group action orbits is connected to several branches of mathematics: optimal transport, information geometry, matrix decompositions, multivariate Gaussians, entropy flows, etc. The framework guides analysis, numerics, and software implementation.
GFSW03 13th November 2017
14:45 to 15:30
Alain Goriely Morphoelasticity and the Geometry of Growth
GFSW03 13th November 2017
16:00 to 16:45
François Gay-Balmaz Towards a geometric variational discretization of compressible fluid dynamics
We present a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Our framework applies to irregular mesh discretizations in 2D and 3D. It systematically extends work previously made for incompressible fluids to the compressible case. We consider in detail the numerical scheme on 2D irregular simplicial meshes and evaluate the behavior of the scheme for the rotating shallow water equations. While our focus is fluid mechanics, our approach is potentially useful for discretizing problems involving evolution equations on diffeomorphism groups. This is a joint work with W. Bauer.
GFSW03 14th November 2017
09:00 to 09:45
Sarang Joshi Bridge Simulation and Metric Estimation on Lie Groups and Orbit Spaces
Joint work with Stefan Sommer Alexis Arnaudon and Line Kuhnel. Performing statistical inference of non-linear Manifold valued data has wide ranging applications in wide ranging fields including bioinformatics, shape analysis, medical imaging, computational anatomy, computer vision, and information geometry. Most common existing statistical inference techniques assume that the Manifold is a Riemannian Manifold with a pre defined canonical metric. In this talk I will present some of our recent work in estimating the Metric structure of the manifold.
GFSW03 14th November 2017
09:45 to 10:30
Ganesh Sundaramoorthi Accelerated optimization on manifolds
GFSW03 14th November 2017
11:00 to 11:30
Sophie Hecht Incompressible limit of a mechanical models for tissue growth
We consider mathematical models for tissue growth. These models describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that some cell population models of this kind converge at the incompressible limit towards a Hele-Shaw type free boundary problem. The first model introduce a non-overlapping constraint of a population choosing a singular pressure law. The second model represents two interacting populations of cells which avoid mixing. Following earlier works, we show that the models approximate a free boundary Hele Shaw type model that we characterise using both analytical and numerical argument.
GFSW03 14th November 2017
11:30 to 12:15
Krastan Blagoev On loss of form in cancer growth
GFSW03 14th November 2017
14:00 to 14:45
Stephen Marsland Differential invariants for the actions of planar Lie groups
Co-authors: Richard Brown (Massey University), Robert McLachlan (Massey University)

A classic problem in image processing is to recognise the similarity or planar objects (point sets, curves, or images) up to transformations from a local planar group such as the Euclidean, similarity, and projective groups. Building on Cartan’s solution to the equivalence problem, an influential new paradigm for this problem was introduced by Calabi et al., the differential invariant signature. The general theory has been developed extensively and many examples computed for planar curves, including the Euclidean, equi-affine, and projective groups. In this talk we demonstrate how to develop and apply differential invariant signatures for planar images.
GFSW03 14th November 2017
14:45 to 15:30
Barbara Gris Shape analysis through a deformation prior
A general approach for matching two shapes is based on the estimation of a deformation (a diffeomorphism) transforming the first one into the second one. We developed a new framework in order to build diffeomorphisms so that a prior on deformation patterns can be easily incorporated. This prior can for instance correspond to an additional knowledge one has on the data under study. Our framework is based on the notion of deformation modules which are structures capable of generating vector fields of a particular chosen type and parametrized in small dimension. Several deformation modules can combine and interact in order to general a multi-modular diffeomorphisms. I will present how this framework allows to incorporate a prior in a deformation model thanks to an adapted deformation module. I will also present how an adapted deformation module can be automatically built given a sequence of data.
GFSW03 14th November 2017
16:00 to 16:45
Irene Kaltenmark Geometrical Growth Models for Computational Anatomy
In the field of computational anatomy, 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 to transformations over time 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. The evolution of the shape can then be described by the joint action of a deformation process and a creation process.
For this purpose, we offer to extend the LDDMM framework to address the problem of non diffeomorphic structural variations in longitudinal data. We keep the geometric central concept of a group of deformations acting on embedded shapes. The necessity for partial mappings leads to a time-varying dynamic that modifies the action of the group of deformations. Ultimately, growth priors are integrated into a new optimal control problem for assimilation of time-varying surface data, leading to an interesting problem in the field of the calculus of variations where the choice of the attachment term on the data, current or varifold, plays an unexpected role.

The underlying minimization problem requires an adapted framework to consider a new set of cost functions (penalization term on the deformation). This new model is inspired by the deployment of animal horns and will be applied to it.

Keywords: computational anatomy, growth model, shape spaces, Riemannian metrics, group of diffeomorphisms, large deformations, variational methods, optimal control.
GFSW03 15th November 2017
09:00 to 09:45
Chris Klingenberg How organisms shape themselves: using geometric morphometrics for understanding evolution and development
Over the last three decades, geometric morphometrics has seen tremendous progress in terms of new techniques for analyzing shape variation. Statistical shape analysis provides a solid mathematical foundation, and a broad range of sophisticated tools is available for characterizing shapes and for extracting specific information that can answer a variety of biological questions. Biological datasets usually have an inherent structure that can potentially reveal important insights about the processes and mechanisms responsible for the observed variation. For instance, many organisms or their organs are symmetric, and the usually slight deviations from perfect symmetry can be characterized with morphometric methods and provide useful biological insight. Such analyses of fluctuating asymmetry can provide information on the developmental basis of integration among traits. Likewise, organisms with modular body plans consisting of repeated parts, such as most plants, provide opportuni ties to examine additional levels of variation within individuals. Many morphometric studies use samples of specimens from multiple taxa, and considering both the variation within taxa and the evolved differences among taxa permits to make inferences about evolutionary mechanisms. Adopting a multi-level approach that considers all the morphological information that can be obtained in a given study design promises rich biological insights, often for little extra effort by the investigator. My lecture will illustrate this approach with examples from animals and plants.
GFSW03 15th November 2017
09:45 to 10:30
Ian Dryden Bayesian analysis of object data using Top Space and Quotient Space models
The analysis of object data is becoming common, where example objects under study include functions, curves, shapes, images or trees. Although the applications can be very broad, the common ingredient in all the studies is the need to deal with geometrical invariances. For the simple example of landmark shapes, one can specify a model for the landmark co-ordinates (in the Top Space) and then consider the marginal distribution of shape after integrating out the invariance transformations of translation, rotation and scale. An alternative approach is to optimize over translation, rotation and scale, and carry out modelling and analysis in the resulting Quotient Space. We shall discuss several examples, including functional alignment of growth curves via diffeomorphisms, molecule matching, and 3D face regression where translation and rotation are removed. Bayesian inference is developed and the Top space versus Quotient space approaches are compared.
GFSW03 15th November 2017
11:00 to 11:30
Arezki Boudaoud Reconstructing leaf morphogenesis using two-dimensional shape analysis
Ref. Biot et al. Multiscale quantification of morphodynamics: MorphoLeaf software for 2D shape analysis. Development 2016
GFSW03 15th November 2017
11:30 to 12:15
Ian Jermyn The elastic metric for surfaces and its use
Shape analysis requires methods for measuring distances between shapes, to define summary statistics, for example, or Gaussian-like distributions. One way to construct such distances is to specify a Riemannian metric on an appropriate space of maps, and then define shape distance as geodesic distance in a quotient space. For shapes in two dimensions, the 'elastic metric' combines tractability with intuitive appeal, with special cases that dramatically simplify computations while still producing state of the art results. For shapes in three dimensions, the situation is less clear. It is unknown whether the full elastic metric admits simplifying representations, and while a reduced version of the metric does, the resulting transform is difficult to invert, and its usefulness has therefore been questionable. In this talk, I will motivate the elastic metric for shapes in three dimensions, elucidate its interesting structure and its relation to the two-dimensional case, and describe what is known about the representation used to construct it. I will then focus on the reduced metric. This admits a representation that greatly simplifies computations, but which is probably not invertible. I will describe recent work that constructs an approximate right inverse for this representation, and show how, despite the theoretical uncertainty, this leads in practice to excellent results in shape analysis problems. This is joint work with Anuj Srivastava, Sebastian Kurtek, Hamid Laga, and Qian Xie.
GFSW03 16th November 2017
09:00 to 09:45
Laurent Younes What can we learn from large deformation diffeomorphic metric mapping on spaces of rigid bodies?
The Large Deformation Diffeomorphic Metric Mapping (LDDMM) algorithms rely on a sub-Riemannian metric on the diffeomorphism group with strong smoothness requirements. This talk will describe a series of simple simulations illustrating the effect of such metrics when considering, in particular, the motion of rigid objects subject to the associated least action principle. It will also show why modifications of this metric can be useful in some cases, with some examples provided.
GFSW03 16th November 2017
09:45 to 10:30
Nina Miolane Template shape estimation: correcting an asymptotic bias
Computational Anatomy studies the normal and pathological variations of organs' shapes, often with respect to a mean organ shape called the template shape. Estimating the template shape is then the first step of the analysis. We use tools from geometric statistics to demonstrate the asymptotic biasedness of the “Frechet mean algorithm”, also called "Max-max algorithm", used for template shape estimation. The geometric intuition provided by this study leads us to suggest two debiasing procedures that we compare. Our results are illustrated on synthetic and real data sets. This is joint work with Dr. Xavier Pennec and Pr. Susan Holmes.
GFSW03 16th November 2017
11:00 to 11:30
Alexandre Bône Learning distributions of shape trajectories: a hierarchical model on a manifold of diffeomorphisms
Co-authors: Olivier Colliot (CNRS), Stanley Durrleman (INRIA)

We propose a mixed effects statistical model to learn a distribution of shape trajectories from longitudinal data, i.e. the collection of individual objects repeatedly observed at multiple time-points. Shape trajectories and their variations are defined via the action of a group of deformations. The model is built on a generic statistical model for manifold-valued longitudinal data, for which we propose to use a finite-dimensional set of diffeomorphisms with a manifold structure, an efficient numerical scheme to compute parallel transport on this manifold and a specific sampling strategy for estimating shapes within a Markov Chain Monte Carlo (MCMC) method. The method allows the estimation of an average spatiotemporal trajectory of shape changes at the group level, and the individual variations of this trajectory in terms of shape and pace of shape changes. This estimation is obtained by a Stochastic Approximation of the Expectation-Maximization (MCMC-SAEM). We show that the algorithm recovers the optimal model parameters with simulated 2D shapes. We apply the method to estimate a scenario of alteration of the shape of the hippocampus 3D brain structure during the course of Alzheimer's disease.
GFSW03 16th November 2017
11:30 to 12:15
Alexis Arnaudon How to deform and shake images?
In recent years, several methods and frameworks have been developed to deform images in the aim of solving problems such as shape analysis or image registration. The application and usefulness of these methods is now well established as well as their mathematical foundation. Nevertheless, because various uncertainties remain at all stages of the image analysis procedure (from data capture to intrinsic variability within a dataset), practical extensions of these deterministic methods should be available. In this talk, I will walk you through one of them, by starting from the geometrical formulation of the theory of large deformation matching, then implementing a particular type of stochastic deformation to preserve the original geometrical structure to end with the description of practical methods for the estimation of the unknown noise parameters of the model. I will illustrate this theory with numerical solutions for a discrete representation of images, where the method of moments can easily be implemented to solve this inverse problem of estimating the noise parameters. This is joint work with Stefan Sommer and Darryl Holm.
GFSW03 16th November 2017
14:00 to 14:45
Stefan Sommer Statistical Inference in Nonlinear Spaces via Maximum Likelihood and Diffusion Bridge Simulation
Co-authors: Darryl D. Holm (Imperial College London), Alexis Arnaudon (Imperial College London) , Sarang Joshi (University of Utah)

An alternative to performing statistical inference in manifolds by optimizating least squares criterions such as those defining the Frechet mean is to optimize the likelihood of data. This approach emphasizes maximum likelihood means over Frechet means, and it in general allows generalization of Euclidean statistical procedures defined via the data likelihood. While parametric families of probability distributions are generally hard to construct in nonlinear spaces, transition densities of stochastic processes provide a geometrically natural way of defining data likelihoods. Examples of this includes the stochastic EPDiff framework, Riemannian Brownian motions and anisotropic generalizations of the Euclidean normal distribution. In the talk, we discuss likeliood based inference on manifolds and procedures for approximating data likelihood by simulation of manifold and Lie group valued diffusion bridges.

GFSW03 16th November 2017
14:45 to 15:30
Alain Trouve Distortion minimizing geodesic subspaces on shape ensembles
We have seen great progress during the last ten years in the understanding of the riemaniann framework on shape spaces, also supported by the steady exponential growth of the computational resources. The outcome is that the modeling possibilities for shapes ensembles are now fairly large. However, it is not that clear how to address concretely the selection of different metric structures for the analysis of a shape ensembles. In this talk, we will discuss some simple geometrical point of view going in that direction and with interesting links with matrix completion. Joint work with David Jacobs, Benjamin Charlier and J. Feydy.
GFSW03 16th November 2017
16:00 to 16:45
Marc Niethammer Machine Learning Approaches for Deformable Image Registration
Image registration is a key tool for medical image analysis. This talk will cover some recent machine learning approaches for deformable image registration.
GFSW03 17th November 2017
09:00 to 09:45
Mads Nielsen Measuring shape change by registration
Longitudinal or cross-sectional differences in shape and volume has traditionally been measured as difference in shape, but recent methodologies use a dense deformation field as a large deformation diffeomorphic metric mapping or a stationary velocity field. For shapes properties of they changes along the finite flow can be obtained using surface integrals with numerical advantages. Vi show examples from longitudinal changes of brain MRIs.
GFSW03 17th November 2017
09:45 to 10:30
Kirsty Wan The Morphology of Cellular Motility
Authors: K.Y. Wan & R.E. Goldstein. Many species of microorganisms such as bacteria, algae, and ciliates self-propel using slender, deformable structures known as cilia and flagella. Great variability exists in the number of flagella, their beating modes, and the basal architecture whence the flagella emanate. For instance, the model alga Chlamydomonas reinhardtii uses two near-identical flagella to pull itself through the fluid, executing a breaststroke. Meanwhile the little-known octoflagellate Pyramimonas octopus exhibits spontaneous switching between a small number of highly reproducible gaits. Here, we show how high resolution spatiotemporal visualisation and analysis of live cell locomotion may be used for behavioural stereotyping at the microscale, and furthermore to reveal the stochastic nature of flagellar beating. Quantitative distance and shape measures are deployed to delineate even subtle changes in behaviour, providing a means by which perturbations to cellular physiology are readily detected based on optical imaging alone.
GFSW03 17th November 2017
11:00 to 11:30
Jenny Larsson Shell Shape of Snails
The beautiful, intricate, and widely diverse shapes of snail shells have fascinated people for centuries. In particular, shells have been analysed by mathematicians wanting to understand their geometric properties. One of the greatest contributors to the field is D'Arcy Thompson with his book On Growth and Form. This book celebrates one hundred years this year, and has played a major role in the field of morphometric analysis.
One well established and simple way of visualising snail shells is built on Raup's growth model, using logarithmic equations with three growth parameters. However, analysing a shell to find the correct values for the growth parameters is not always a straightforward task.

The goal of my research is to develop a method for obtaining the growth parameters from 2D images of shells. I will test my methods with a set of images of the species Littorina saxatilis. This species is biologically interesting because strong natural selection maintains different shell shapes in distinct environments. It would therefore be an advantage to have a way of describing this natural shape variation, and the shapes of laboratory hybrids, in terms of meaningful growth parameters. This is expected to give a better understanding of the operation of selection and of the underlying genetic basis of shape variation, compared to more classical PCA analysis of landmarks.
GFSW03 17th November 2017
11:30 to 12:15
Tilak Ratnanather 3D normal coordinate systems for the cortex: applications in the deafened cortices in babies, adults and cats
We describe a surface-based diffeomorphic algorithm to generate 3D coordinate grids in the cortical ribbon. In the grid, normal coordinate lines from the grey/white (inner) surface to the grey/csf (outer) surface are constrained to be normal at the surfaces. Specifically, the cortical ribbon is described by two triangulated surfaces with open boundaries. Conceptually, the inner surface sits on top of the white matter structure and the outer 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 in the embryo. 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. Applications will be described for the deafened auditory cortices in babies, adults and cats. The approach offers potential for quantitative functional and histological analysis of cortical activity and anatomy.

(Joint work with Laurent Younes, Sylvain Arguillère, Kwame Kutten, Andrej Kral, Peter Hubka and Michael Miller).
GFSW03 17th November 2017
14:00 to 14:45
François-Xavier Vialard Around unbalanced optimal transport: fluid dynamic, growth model, applications.
In this talk, we present the so-called Wasserstein-Fisher-Rao metric (also called Hellinger-Kantorovich) by its dynamical and static formulation. The link between these two formulations is made clear by generalizing the Riemannian submersion of Otto to this new setting. Then the link with the Camassa-Holm equation can be made with this metric, in the same way Brenier made it between optimal transport and incompressible Euler. Passing by, we prove that the Camassa-Holm equation is actually an incompressible Euler equation on a bigger space. We also show the use of this metric to interpret a particular Hele-Shaw model as a gradient flow. We then finish with some examples of use of this new metric as a similarity measure on diffeomorphic registration of shapes.