Wrapped statistical models on manifolds: motivations, the case SE(n), and generalization to symmetric spaces

International audienceWe address here the construction of wrapped probability densities on Lie groups and quotient of Lie groups using the exponential map. The paper starts by briefly reviewing the different approaches to build densities on a manifold and shows the interest of wrapped distributions. We then construct wrapped densities on SE(n) and discuss their statistical estimation. We conclude by an opening to the case of symmetric spaces

A reduced parallel transport equation on Lie Groups with a left-invariant metric

International audienceThis paper presents a derivation of the parallel transport equation expressed in the Lie algebra of a Lie group endowed with a left-invariant metric.The use of this equation is exemplified on the group of rigid body motions SE(3), using basic numerical integration schemes, and compared to the pole ladder algorithm. This results in a stable and efficient implementation of parallel transport. The implementation leverages the python package geomstats and is available online

Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac motion modeling

International audienceTransporting the statistical knowledge regressed in the neighbourhood of a point to a different but related place (transfer learning) is important for many applications. In medical imaging, cardiac motion modelling and structural brain changes are two such examples: for a group-wise statistical analysis, subjectspecific longitudinal deformations need to be transported in a common template anatomy. In geometric statistics, the natural (parallel) transport method is defined by the integration of a Riemannian connection which specifies how tangent vectors are compared at neighbouring points. In this process, the numerical accuracy of the transport method is critical. Discrete methods based on iterated geodesic parallelograms inspired by Schild’s ladder were shown to be very efficient and apparently stable in practice. In this chapter, we show that ladder methods are actually second order schemes, even with numerically approximated geodesics. We also propose a new original algorithm to implement these methods in the context of the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework that endows the space of diffeomorphisms with a right-invariant RKHS metric. When applied to the motion modelling of the cardiac right ventricle under pressure or volume overload, the method however exhibits unexpected effects in the presence of very large volume differences between subjects. We first investigate an intuitive rescaling of the modulus after parallel transport to preserve the ejection fraction. The surprisingly simple scaling/volume relationship that we obtain suggests to decouples the volume change from the deformation directly within the LDMMM metric. The parallel transport of cardiac trajectories with this new metric now reveals statistical insights into the dynamics of each disease. This example shows that parallel transport could become a tool of choice for data-driven metric optimization

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

International audienceParallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes.We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space

Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics

Cardiac Motion Modeling with Parallel Transport and Shape Splines

International audienceIn cases of pressure or volume overload, probing cardiac function may be difficult because of the interactions between shape and deformations.In this work, we use the LDDMM framework and parallel transport to estimate and reorient deformations of the right ventricle. We then propose a normalization procedure for the amplitude of the deformation, and a second-order spline model to represent the full cardiac contraction. The method is applied to 3D meshes of the right ventricle extracted from echocardiographic sequences of 314 patients divided into three disease categories and a control group. We find significant differences between pathologies in the model parameters, revealing insights into the dynamics of each disease

Multi-Spectral Reflection Matrix for Ultra-Fast 3D Label-Free Microscopy

Label-free microscopy exploits light scattering to obtain a three-dimensional image of biological tissues. However, light propagation is affected by aberrations and multiple scattering, which drastically degrade the image quality and limit the penetration depth. Multi-conjugate adaptive optics and time-gated matrix approaches have been developed to compensate for aberrations but the associated frame rate is extremely limited for 3D imaging. Here we develop a multi-spectral matrix approach to solve these fundamental problems. Based on an interferometric measurement of a polychromatic reflection matrix, the focusing process can be optimized in post-processing at any voxel by addressing independently each frequency component of the wave-field. A proof-of-concept experiment demonstrates the three-dimensional image of an opaque human cornea over a 0.1 mm^3-field-of-view at a 290 nm-resolution and a 1 Hz-frame rate. This work paves the way towards a fully-digital microscope allowing real-time, in-vivo, quantitative and deep inspection of tissues.Comment: 27 pages, 4 figure

Mapping Sandy Titano-Uraniferous Deposits in Tabou Region, South-West Cote d’Ivoire: Contribution of Magnetometry and Gamma-Ray Spectrometry

The work of the Geological and Mining Research Office (BRGM) in Côte d'Ivoire, with Tagini and Papon, has revealed, in addition to substances such as gold and manganese, indications of nuclear substances in some areas of the country. Côte d'Ivoire's mining policy is to develop the riches of its subsoil. The lack of information on the reserves of nuclear substances does not currently allow the Ivorian state to exploit these nuclear resources. This work is part of the framework to deepen our knowledge on these substances.Carried out in the south-west of Côte d'Ivoire, along the coast, this study confirmed the reported uraniferous indications and helped to identify their origin. Uranium accumulations were found in beach sediments in significant quantities between Bliéron and Soublaké. Keywords: Uranium, Magnetometry, Gamma-ray spectrometry, Tabou, Côte d’Ivoire. DOI: 10.7176/JEES/12-11-02 Publication date: November 30th 2022

Attempts, Successes, and Failures of Distance Learning in the Time of COVID-19

Over 1.7 billion students around the world have had their education disrupted by the spread of the Coronavirus disease worldwide. Schools and universities have not faced this level of disruption since World War II. The COVID-19 pandemic presented a colossal challenge for teachers to urgently and massively adapt all their classes to distance learning in order to maintain educational continuity with the same quality. Even if some teachers and certain classes were ready to face the situation, a large majority had to adapt their teaching and learning in a very short time without training, with insufficient bandwidth, and with little preparation. This unexpected and rapid transition to online learning has led to a multiplication of teachers’ strategies for distance learning in lectures, tutorials, project groups, lab works, and assessments. The purpose of this paper is to present the feedback from students and teachers who participated in the lockdown semester of two different groups of a 5-year program in Chemistry, Environment and Chemical Engineering (100 students) at INSA Toulouse (France). The analysis has highlighted some great successes and some failures in the solutions proposed. Consequently, some guidelines can be given to help us all to learn the lessons of such a singular experience in order to face the unexpected future with more knowledge and more successful distance learning. Teachers have shown very strong resilience during this crisis, at the cost of significant personal commitment. They admit that they have learned more about distance education in two months than in the last 10 years