# Timetable (GFSW03)

## Shape analysis and computational anatomy

Monday 13th November 2017 to Friday 17th November 2017

 09:00 to 09:30 Registration 09:35 to 09:45 Welcome from Christie Marr (INI Deputy Director) 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. INI 1 10:30 to 11:00 Morning Coffee 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. INI 1 11:30 to 12:15 tba INI 1 12:30 to 13:30 Lunch @ Wolfson Court 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. INI 1 14:45 to 15:30 Alain Goriely Morphoelasticity and the Geometry of Growth INI 1 15:30 to 16:00 Afternoon Tea 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. INI 1 16:45 to 17:45 Welcome Wine Reception at INI
 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. INI 1 09:45 to 10:30 Ganesh Sundaramoorthi Accelerated optimization on manifolds INI 1 10:30 to 11:00 Morning Coffee 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. INI 1 11:30 to 12:15 Krastan Blagoev On loss of form in cancer growth INI 1 12:30 to 13:30 Lunch @ Wolfson Court 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. INI 1 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. INI 1 15:30 to 16:00 Afternoon Tea 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. INI 1