Related papers: Manifolds of Projective Shapes
The program of understanding Shape Theory layer by layer topologically and geometrically -- proposed in Part I -- is now addressed for 4 points in 1-$d$. Topological shape space graphs are far more complex here, whereas metric shape spaces…
The group PGL(2) of linear transformations of the projective line acts naturally on the d-dimensional projective space P^d parametrizing configurations (`d-tuples') of points on the line. In this note we are concerned with the orbits of…
Generating 3D shapes from single RGB images is essential in various applications such as robotics. Current approaches typically target images containing clear and complete visual descriptions of the object, without considering common…
Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural…
Template 3D shapes are useful for many tasks in graphics and vision, including fitting observation data, analyzing shape collections, and transferring shape attributes. Because of the variety of geometry and topology of real-world shapes,…
The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the…
The main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non-compact K\"ahler manifold of finite volume with reasonable geometric assumptions to its universal covering. Applicable examples…
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $\phi$ in the form $K_D^\phi(H,K)=\sum_{i,j}\phi(\lambda_i,\lambda_j)^{-1} Tr P_iHP_jK$ when $\sum_i\lambda_iP_i$ is the spectral…
A complete representation of 3D objects requires characterizing the space of deformations in an interpretable manner, from articulations of a single instance to changes in shape across categories. In this work, we improve on a prior…
We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse…
Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back…
The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis…
We classify projective manifolds with flat holomorphic conformal structures.
In this paper we present an intrinsic characterisation of projective special K\"ahler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the…
Quantum states defined over a parameter space form a Grassmann manifold. To capture the geometry of the associated gauge structure, gauge-invariant quantities are essential. We employ the projector of a multilevel system to quantify the…
This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to…
We hypothesise that properties of space could underly some patterns observed in nature. We explore the possibility that the observed variety of matter particles and the pattern of their properties arise due to the non-orientability of the…
Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps…
We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP^n as a set of flat tori parametrized by the positive octant of…
Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the group of diffeomorphisms of $S^1$. We investige several…