It has been known that the curvature of data spaces plays a role in data analysis. For example, the Frechet mean (intrinsic mean) always exists uniquely for a probability measure on a non-positively curved metric space. In this paper, we use the curvature of data spaces in a novel manner. A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. The p...

We present a variant of the Fréchet distance (as well as geodesic and homotopic Fréchet distance) which forces the motion between the input objects to follow an ambient isotopy. This provides a measure of how much you need to continuously deform one shape into another while maintaining topologically equivalently shapes throughout the deformation.

Glioma is one of the most challenging types of brain tumors to treat or control locally. One of the main problems is to determine which areas of the apparently normal brain contain glioma cells, as gliomas are known to infiltrate several centimeters beyond the clinically apparent lesion that is visualized on standard Computed Tomography scans (CT) or Magnetic Resonance Images (MRIs). To ensure ...

1. A comparison theorem for complete Riemannian manifolds with sectional curvatures ≥ k says that distance functions in such manifolds are more concave than in the model space Sk of constant curvature k. In other words, the restriction of any distance function distp to any geodesic γ (always parametrised by the arclength) satisfies a certain concavity condition (∗)k. For example, the condition ...

This paper focuses on the study of open curves in a Riemannian manifold M , and proposes a reparametrization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [13] to define a Riemannian metric on the space of immersions M = Imm([0, 1],M) by pullback of a natural metric on the tangent bundle TM. This induces a first-o...

We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S1 to the plane modulo the group of diffeomorphisms of S1, acting as reparametrizations. In particular we investigate the metric, for a constant A > 0, Gc (h, k) := ∫ S1 (1+ Aκc(θ) 2)〈h(θ), k(θ)〉|c(θ)| dθ where κc is the curvature of the curve c and h, k are normal vector ...

In this paper we propose a novel semantic label transfer method using supervised geodesic propagation (SGP). We use supervised learning to guide the seed selection and the label propagation. Given an input image, we first retrieve its similar image set from annotated databases. A Joint Boost model is learned on the similar image set of the input image. Then the recognition proposal map of the i...