2,055 research outputs found
Covariant coarse-graining of inhomogeneous dust flow in General Relativity
A new definition of coarse-grained quantities describing the dust flow in
General Relativity is proposed. It assigns the coarse--grained expansion, shear
and vorticity to finite-size comoving domains of fluid in a covariant,
coordinate-independent manner. The coarse--grained quantities are all
quasi-local functionals, depending only on the geometry of the boundary of the
considered domain. They can be thought of as relativistic generalizations of
simple volume averages of local quantities in a flat space. The procedure is
based on the isometric embedding theorem for S^2 surfaces and thus requires the
boundary of the domain in question to have spherical topology and positive
scalar curvature. We prove that in the limit of infinitesimally small volume
the proposed quantities reproduce the local expansion, shear and vorticity. In
case of irrotational flow we derive the time evolution for the coarse-grained
quantities and show that its structure is very similar to the evolution
equation for their local counterparts. Additional terms appearing in it may
serve as a measure of the backreacton of small-scale inhomogeneities of the
flow on the large-scale motion of the fluid inside the domain and therefore the
result may be interesting in the context of the cosmological backreaction
problem. We also consider the application of the proposed coarse-graining
procedure to a number of known exact solutions of Einstein equations with dust
and show that it yields reasonable results.Comment: 17 pages, 5 figures. Version accepted in Classical and Quantum
Gravity
A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion
A variational principle is derived for two-dimensional incompressible
rotational fluid flow with a free surface in a moving vessel when both the
vessel and fluid motion are to be determined. The fluid is represented by a
stream function and the vessel motion is represented by a path in the planar
Euclidean group. Novelties in the formulation include how the pressure boundary
condition is treated, the introduction of a stream function into the
Euler-Poincar\'e variations, the derivation of free surface variations, and how
the equations for the vessel path in the Euclidean group, coupled to the fluid
motion, are generated automatically.Comment: 19 pages, 3 figure
Unsupervised Monocular Depth Reconstruction of Non-Rigid Scenes
Monocular depth reconstruction of complex and dynamic scenes is a highly
challenging problem. While for rigid scenes learning-based methods have been
offering promising results even in unsupervised cases, there exists little to
no literature addressing the same for dynamic and deformable scenes. In this
work, we present an unsupervised monocular framework for dense depth estimation
of dynamic scenes, which jointly reconstructs rigid and non-rigid parts without
explicitly modelling the camera motion. Using dense correspondences, we derive
a training objective that aims to opportunistically preserve pairwise distances
between reconstructed 3D points. In this process, the dense depth map is
learned implicitly using the as-rigid-as-possible hypothesis. Our method
provides promising results, demonstrating its capability of reconstructing 3D
from challenging videos of non-rigid scenes. Furthermore, the proposed method
also provides unsupervised motion segmentation results as an auxiliary output
Variational Autoencoders for Deforming 3D Mesh Models
3D geometric contents are becoming increasingly popular. In this paper, we
study the problem of analyzing deforming 3D meshes using deep neural networks.
Deforming 3D meshes are flexible to represent 3D animation sequences as well as
collections of objects of the same category, allowing diverse shapes with
large-scale non-linear deformations. We propose a novel framework which we call
mesh variational autoencoders (mesh VAE), to explore the probabilistic latent
space of 3D surfaces. The framework is easy to train, and requires very few
training examples. We also propose an extended model which allows flexibly
adjusting the significance of different latent variables by altering the prior
distribution. Extensive experiments demonstrate that our general framework is
able to learn a reasonable representation for a collection of deformable
shapes, and produce competitive results for a variety of applications,
including shape generation, shape interpolation, shape space embedding and
shape exploration, outperforming state-of-the-art methods.Comment: CVPR 201
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