Topic
Unit tangent bundle
About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.
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13 Jun 2010TL;DR: This paper argues that a suitable abstraction is the unit tangent bundle R2 × S1 and shows that a basic principle of “minimum energy consumption” in this space, namely a minimum length completion, entails desired perceptual properties for the completion in the image plane.
Abstract: The phenomenon of visual curve completion, where the visual system completes the missing part (e.g., due to occlusion) between two contour fragments, is a major problem in perceptual organization research. Previous computational approaches for the shape of the completed curve typically follow formal descriptions of desired, image-based perceptual properties (e.g, minimum total curvature, roundedness, etc.). Unfortunately, however, it is difficult to determine such desired properties psychophysically and indeed there is no consensus in the literature for what they should be. Instead, in this paper we suggest to exploit the fact that curve completion occurs in early vision in order to formalize the problem in a space that explicitly abstracts the primary visual cortex. We first argue that a suitable abstraction is the unit tangent bundle R2 × S1 and then we show that a basic principle of “minimum energy consumption” in this space, namely a minimum length completion, entails desired perceptual properties for the completion in the image plane. We present formal theoretical analysis and numerical solution methods, we show results on natural images and their advantage over existing popular approaches, and we discuss how our theory explains recent findings from the perceptual literature using basic principles only.
18 citations
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TL;DR: In this article, a class of g -natural metrics on the tangent bundle of a Finsler manifold which is a generalized version of Sasaki-Matsumoto metric and Miron metric is studied.
Abstract: In this article, we study a class of g -natural metrics on the tangent bundle of a Finsler manifold which is a generalized version of Sasaki–Matsumoto metric and Miron metric. Then, we consider on compatible almost complex structure with together the metric confers to the slit tangent bundle of Finsler manifold and structure of locally conformal almost Kahlerian manifold. We find some conditions under which the slit tangent bundle is locally conformal Kahlerian, Kahlerian, locally Euclidean or the Finsler manifold has scalar or constant flag curvature.
18 citations
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18 citations
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TL;DR: In this paper, it was shown that any flow invariant, isometry invariant C0-function on the unit tangent bundle SH of a 1-connected, irreducible homogeneous space H of nonpositive curvature is necessarily constant, unless H is symmetric of higher rank.
Abstract: Consider the geodesic flow on the unit tangent bundle SH of a 1-connected, irreducible homogeneous space H of nonpositive curvature. We prove that any flow invariant, isometry invariant C0-function on SH is necessarily constant, unless H is symmetric of higher rank. As the main applications, we obtain rigidity and partial classification results for spaces H whose geodesic symmetries are (asymptotically) volume-preserving.
18 citations
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TL;DR: In this article, the authors show that the foliation of a compact manifold is ergodic with respect to the maximal entropy of the associated geodesic flow on the manifold.
Abstract: Let $M$ be the unit tangent bundle of a compact manifold
with negative sectional
curvatures and let $\hat M$ be a $\mathbb Z^d$ cover for
$M$. Let $\mu$ be the measure of maximal entropy for
the associated geodesic
flow on $M$ and let $\hat\mu$ be the lift of $\mu$ to
$\hat M$.
We show that the foliation $\hat{M^{s s}}$ is ergodic with
respect to $\hat\mu$.
(This was proved in the
special case of surfaces by Babillot and Ledrappier by
a different method.)
Our method
extends to certain Anosov and hyperbolic flows.
18 citations