2,059 research outputs found
RAAGs in Ham
We prove that every RAAG (a Right-Angled Artin Group) embeds in the group of
Hamiltonian symplectomorphisms of the 2-sphere.Comment: 23 pages, 2 figure
Geodesic Warps by Conformal Mappings
In recent years there has been considerable interest in methods for
diffeomorphic warping of images, with applications e.g.\ in medical imaging and
evolutionary biology. The original work generally cited is that of the
evolutionary biologist D'Arcy Wentworth Thompson, who demonstrated warps to
deform images of one species into another. However, unlike the deformations in
modern methods, which are drawn from the full set of diffeomorphism, he
deliberately chose lower-dimensional sets of transformations, such as planar
conformal mappings.
In this paper we study warps of such conformal mappings. The approach is to
equip the infinite dimensional manifold of conformal embeddings with a
Riemannian metric, and then use the corresponding geodesic equation in order to
obtain diffeomorphic warps. After deriving the geodesic equation, a numerical
discretisation method is developed. Several examples of geodesic warps are then
given. We also show that the equation admits totally geodesic solutions
corresponding to scaling and translation, but not to affine transformations
Accurate Optical Flow via Direct Cost Volume Processing
We present an optical flow estimation approach that operates on the full
four-dimensional cost volume. This direct approach shares the structural
benefits of leading stereo matching pipelines, which are known to yield high
accuracy. To this day, such approaches have been considered impractical due to
the size of the cost volume. We show that the full four-dimensional cost volume
can be constructed in a fraction of a second due to its regularity. We then
exploit this regularity further by adapting semi-global matching to the
four-dimensional setting. This yields a pipeline that achieves significantly
higher accuracy than state-of-the-art optical flow methods while being faster
than most. Our approach outperforms all published general-purpose optical flow
methods on both Sintel and KITTI 2015 benchmarks.Comment: Published at the Conference on Computer Vision and Pattern
Recognition (CVPR 2017
Exploring complex networks via topological embedding on surfaces
We demonstrate that graphs embedded on surfaces are a powerful and practical
tool to generate, characterize and simulate networks with a broad range of
properties. Remarkably, the study of topologically embedded graphs is
non-restrictive because any network can be embedded on a surface with
sufficiently high genus. The local properties of the network are affected by
the surface genus which, for example, produces significant changes in the
degree distribution and in the clustering coefficient. The global properties of
the graph are also strongly affected by the surface genus which is constraining
the degree of interwoveness, changing the scaling properties from
large-world-kind (small genus) to small- and ultra-small-world-kind (large
genus). Two elementary moves allow the exploration of all networks embeddable
on a given surface and naturally introduce a tool to develop a statistical
mechanics description. Within such a framework, we study the properties of
topologically-embedded graphs at high and low `temperatures' observing the
formation of increasingly regular structures by cooling the system. We show
that the cooling dynamics is strongly affected by the surface genus with the
manifestation of a glassy-like freezing transitions occurring when the amount
of topological disorder is low.Comment: 18 pages, 7 figure
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