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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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TL;DR: In this article, the authors study a model of geometry of vision due to Petitot, Citti, and Sarti, where the primary visual cortex V1 lifts an image from a corrupted image to the bundle of directions of the plane, and then the corrupted image is reconstructed by minimizing the energy necessary for activation of the orientation columns corresponding to regions in which the image is corrupted.
Abstract: In this paper we study a model of geometry of vision due to Petitot, Citti, and Sarti. One of the main features of this model is that the primary visual cortex V1 lifts an image from $\mathbb{R}^2$ to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle. In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process intrinsically defines a hypoelliptic heat equation on the bundle of directions of the plane. In the original model, directions are considered both with and without orientation, giving rise, respectively, to a problem on the group of rototranslations of the plane $SE(2)$ or on the projective tangent bundle of the plane $PT\mathbb{R}^2$. We provide a mathematical proof of several important facts for this model. We first prove that the model is mathematically consis...
61 citations
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60 citations
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60 citations
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TL;DR: For Ricci-flat manifold with Euclidean volume growth, it was shown in this article that local tangent cones are unique if one tangent cone has a smooth cross-section.
Abstract: We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.
60 citations