arXiv research

We give a combinatorial proof, using the hyperbolicity of the curve graphs, of the bounded geodesic image theorem of Masur and Minsky. Recently it has been shown that curve graphs are uniformly hyperbolic, thus a universal bound can be given for the diameter of the geodesic image. We also generalize the theorem for pro…

A new method tracks retinal vessels more accurately than existing methods.

Uniform bound on geodesic images for surfaces using bicorn curves.

New data-driven Cartan connection tracks complex vascular structures.

Researchers describe the Gromov boundary of a graph related to surfaces.

Generic metrics make geodesic nets dense.

A new snake model improves segmentation of SEM images.

Study geodesics on finite-dimensional manifolds.

Generates samples conditioned on labels using optimal transport.

Quantizes geodesics in Kähler and Sasaki geometry.

Study geodesics in Kähler metrics for all time.

New approach constructs symplectic structure on pseudo-Riemannian geodesics.

This paper develops optimal transport methods on the roto-translation group SE2.

The paper studies hyperbolic phenomena on closed surfaces using bicorn curves.

Study on stability of geodesic maps in non-isotropic manifolds.

This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents…

Localized curvature bounds ensure harmonic maps are constant.

The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the image of a non-closed geodesic has 0 distance from the set of conical points. Dynamical properties for the space of geodesics are also proved.

We consider a geometric property of the closest-points projection to a geodesic in Teichmüller space: the projection is called contracting if arbitrarily large balls away from the geodesic project to sets of bounded diameter. (This property always holds in negatively curved spaces.) It is shown here to hold if and only…

We show that, given any finite dimensional, connected, compact metric space Z, there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry f from X to Y, and a geodesic ray c in X, such that the closure of f(c), instersected with the boundary of Y, is homeomorphic to Z. This…

Differentially private geodesic regression for non-Euclidean data.

This paper addresses the morphing of manifold-valued images based on the time discrete geodesic paths model of Berkels, Effland and Rumpf 2015. Although for our manifold-valued setting such an interpretation of the energy functional is not available so far, the model is interesting on its own. We prove the existence of…

Hierarchical geodesic model for analyzing shapes on manifolds.

New proof shows nonholonomic motions are geodesics, minimizing distance.

It is well-known that if $ξ$ is a smooth vector field on a given Riemannian manifold $M^n$ then $ξ$ naturally defines a submanifold $ξ(M^n)$ transverse to the fibers of the tangent bundle $TM^n$ with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We sh…

Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics betw…

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…

We create a smooth manifold of triangular meshes with a geodesically complete metric.

Characterizes geodesic completeness for landmark spaces.

A new method for fast optimal transport using sliced Wasserstein generalized geodesics.

New constructions show stable geodesics and figure-eights in convex hypersurfaces.

A totally geodesic map $f:\mathcal X_1\to\mathcal X_2$ between Hermitian symmetric spaces is tight if its image contains geodesic triangles of maximal area. Tight maps were first introduced in [BIW09], and were classified in [Ham13, Ham14, HO14] in the case of irreducible domain. We complete the classification by analy…

Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.

We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of the mapping class group may not have the strong contractibility pro…

Let $M$ be the image of a smooth CR embedding of a strictly pseudoconvex CR real hypersurface into a sphere. If the CR second fundamental form of $M$ vanishes, we show that $M$ is a totally geodesic submanifold.

In this paper, we propose a method to learn a minimizing geodesic within a data manifold. Along the learned geodesic, our method can generate high-quality interpolations between two given data samples. Specifically, we use an autoencoder network to map data samples into latent space and perform interpolation via an int…

Study coning totally geodesic boundaries of hyperbolic manifolds.

Study on geodesic distances on SE(3)/SO(2) in machine learning.

Study compact Willmore surfaces without complex structure convergence, computing energy loss and geodesic lengths.

Improved assessment of knee osteoarthritis using geodesic B-score.

Closed geodesics found on specific non-compact manifolds.

We propose Gaussian optimal transport for Image style transfer in an Encoder/Decoder framework. Optimal transport for Gaussian measures has closed forms Monge mappings from source to target distributions. Moreover interpolates between a content and a style image can be seen as geodesics in the Wasserstein Geometry. Usi…

The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples. Surprisingly, we show that there are (topologically embedded) smooth suborbifolds which d…

The performance of the state-of-the-art image segmentation methods heavily relies on the high-quality annotations, which are not easily affordable, particularly for medical data. To alleviate this limitation, in this study, we propose a weakly supervised image segmentation method based on a deep geodesic prior. We hypo…

Cohomology fractals illustrate complex 3-manifold properties.

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally geodesic boundary, answering a question of Y. Minsky. This is proven via a filling the…

Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis (PCA) onto manifolds, its effectiveness is limited to data with a single modality…

The $\mathcal{G}_I^0$ distribution is able to characterize different regions in monopolarized SAR imagery. It is indexed by three parameters: the number of looks (which can be estimated in the whole image), a scale parameter and a texture parameter. This paper presents a new proposal for feature extraction and region d…