GUIGUI , Nicolas

The study of anatomical shapes and deformations is of major interest in cardiology where diseases such as arrhythmia or pulmonary hypertension cause abnormalities in the way the heart contracts or of its shape. The characterization of these abnormalities allows to evaluate the gravity of the disease or the impact of a treatment. A mathematical...

Parallel 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...

This 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...

We 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...

This paper aims to describe a statistical model of wrapped densities for bi-invariant statistics on the group of rigid motions of a Euclidean space. Probability distributions on the group are constructed from distributions on tangent spaces and pushed to the group by the exponential map. We provide an expression of the Jacobian determinant of...

Transporting 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...

As 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...

Myocardial shape and deformation are two relevant descriptors for the study of cardiac function and can undergo strong interactions depending on diseases. Manifold learning provides low dimensional representations of these high-dimensional descriptors, but the choice of normalization can strongly affect the analysis. Besides, whether the shape...

In 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...

In computational anatomy, the statistical analysis of temporal deformations and inter-subject variability relies on shape registration. However, the numerical integration and optimization required in diffeomorphic registration often lead to important numerical errors. In many cases, it is well known that the error can be drastically reduced in...

Kendall shape spaces are a widely used framework for the statistical analysis of shape data arising from many domains, often requiring the parallel transport as a tool to normalise time series data or transport gradient in optimisation procedures. We present an implementation of the pole ladder, an algorithm to compute parallel transport based...

In this paper, we use tools of information geometry to compare, average and classify histograms. Beta distributions are fitted to the histograms and the corresponding Fisher information geometry is used for comparison. We show that this geometry is negatively curved, which guarantees uniqueness of the notion of mean, and makes it suitable to...

We introduce Geomstats, an open-source Python toolbox for computations and statistics on nonlinear manifolds, such as hyperbolic spaces, spaces of symmetric positive definite matrices, Lie groups of transformations, and many more. We provide object-oriented and extensively unit-tested implementations. Among others, manifolds come equipped with...

There is a growing interest in leveraging differential geometry in the machine learning community. Yet, the adoption of the associated geometric computations has been inhibited by the lack of a reference implementation. Such an implementation should typically allow its users: (i) to get intuition on concepts from differential geometry through a...