Related papers: Shape spaces: From geometry to biological plausibi…

This paper extends parts of the results from [P.W.Michor and D. Mumford, \emph{Appl. Comput. Harmon. Anal.,} 23 (2007), pp. 74--113] for plane curves to the case of hypersurfaces in $\mathbb R^n$. Let $M$ be a compact connected oriented…

Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article…

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that…

We present a framework for shape matching in computational anatomy allowing users control of the degree to which the matching is diffeomorphic. This control is given as a function defined over the image and parameterises the template…

In this paper we study pseudo-Riemannian spaces with a degenerate curvature structure i.e. there exists a continuous family of metrics having identical polynomial curvature invariants. We approach this problem by utilising an idea coming…

Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given…

Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing…

Some geometric structures with associated Riemannian metrics have been considered in the book.

We propose a solution to the problem of estimating a Riemannian metric associated with a given differentiable manifold. The metric learning problem is based on minimizing the relative volume of a given set of points. We derive the details…

We introduce the shape module of the Python package Geomstats to analyze shapes of objects represented as landmarks, curves and surfaces across fields of natural sciences and engineering. The shape module first implements widely used shape…

The projective shape of a configuration of k points or "landmarks" in RP(d) consists of the information that is invariant under projective transformations and hence is reconstructable from uncalibrated camera views. Mathematically, the…

Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order…

The space of all Riemannian metrics on a smooth second countable finite dimensional manifold is itself a smooth manifold modeled on the space of symmetric (0,2)-tensor fields with compact support. It carries a canonical Riemannian metric…

We generalize the classical Fisher information metric on statistical models to $L^p$-metrics on various spaces of differential forms or group of diffeomorphisms. Using this new interpretation from information geometry, we derive several new…

We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well-known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of…

Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework…

While classical data analysis has addressed observations that are real numbers or elements of a real vector space, at present many statistical problems of high interest in the sciences address the analysis of data that consist of more…

We present a Riemannian framework for linear and quadratic discriminant classification on the tangent plane of the shape space of curves. The shape space is infinite dimensional and is constructed out of square root velocity functions of…

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…

Deformational structures, in many aspects generalizing standard elasticity theory, are investigated in abstract form. Within free deformational structures we define algebra of deformations, classify them by its special properties, define…