443 research outputs found
Diskrete Spin-Geometrie für Flächen
This thesis proposes a discrete framework for spin geometry of surfaces. Specifically, we discretize the basic notions in spin geometry, such as the spin structure, spin connection and Dirac operator. In this framework, two types of Dirac operators are closely related as in smooth case. Moreover, they both induce the discrete conformal immersion with prescribed mean curvature half-density.In dieser Arbeit wird ein diskreter Zugang zur Spin-Geometrie vorgestellt. Insbesondere diskretisieren wir die grundlegende Begriffe, wie zum Beispiel die Spin-Struktur, den Spin-Zusammenhang und den Dirac Operator. In diesem Rahmen sind zwei Varianten fĂĽr den Dirac Operator eng verwandt wie in der glatten Theorie. DarĂĽber hinaus induzieren beide die diskret-konforme Immersion mit vorgeschriebener Halbdichte der mittleren KrĂĽmmung
Disentangling Geometric Deformation Spaces in Generative Latent Shape Models
A complete representation of 3D objects requires characterizing the space of
deformations in an interpretable manner, from articulations of a single
instance to changes in shape across categories. In this work, we improve on a
prior generative model of geometric disentanglement for 3D shapes, wherein the
space of object geometry is factorized into rigid orientation, non-rigid pose,
and intrinsic shape. The resulting model can be trained from raw 3D shapes,
without correspondences, labels, or even rigid alignment, using a combination
of classical spectral geometry and probabilistic disentanglement of a
structured latent representation space. Our improvements include more
sophisticated handling of rotational invariance and the use of a diffeomorphic
flow network to bridge latent and spectral space. The geometric structuring of
the latent space imparts an interpretable characterization of the deformation
space of an object. Furthermore, it enables tasks like pose transfer and
pose-aware retrieval without requiring supervision. We evaluate our model on
its generative modelling, representation learning, and disentanglement
performance, showing improved rotation invariance and intrinsic-extrinsic
factorization quality over the prior model.Comment: 22 page
Simulating water-entry/exit problems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library
In this paper we employ two implementations of the fictitious domain (FD)
method to simulate water-entry and water-exit problems and demonstrate their
ability to simulate practical marine engineering problems. In FD methods, the
fluid momentum equation is extended within the solid domain using an additional
body force that constrains the structure velocity to be that of a rigid body.
Using this formulation, a single set of equations is solved over the entire
computational domain. The constraint force is calculated in two distinct ways:
one using an Eulerian-Lagrangian framework of the immersed boundary (IB) method
and another using a fully-Eulerian approach of the Brinkman penalization (BP)
method. Both FSI strategies use the same multiphase flow algorithm that solves
the discrete incompressible Navier-Stokes system in conservative form. A
consistent transport scheme is employed to advect mass and momentum in the
domain, which ensures numerical stability of high density ratio multiphase
flows involved in practical marine engineering applications. Example cases of a
free falling wedge (straight and inclined) and cylinder are simulated, and the
numerical results are compared against benchmark cases in literature.Comment: The current paper builds on arXiv:1901.07892 and re-explains some
parts of it for the reader's convenienc
Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds
Sparsity-based representations have recently led to notable results in
various visual recognition tasks. In a separate line of research, Riemannian
manifolds have been shown useful for dealing with features and models that do
not lie in Euclidean spaces. With the aim of building a bridge between the two
realms, we address the problem of sparse coding and dictionary learning over
the space of linear subspaces, which form Riemannian structures known as
Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into
the space of symmetric matrices by an isometric mapping. This in turn enables
us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we
propose closed-form solutions for learning a Grassmann dictionary, atom by
atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann
sparse coding and dictionary learning algorithms through embedding into Hilbert
spaces.
Experiments on several classification tasks (gender recognition, gesture
classification, scene analysis, face recognition, action recognition and
dynamic texture classification) show that the proposed approaches achieve
considerable improvements in discrimination accuracy, in comparison to
state-of-the-art methods such as kernelized Affine Hull Method and
graph-embedding Grassmann discriminant analysis.Comment: Appearing in International Journal of Computer Visio
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