488,645 research outputs found
Shape-from-intrinsic operator
Shape-from-X is an important class of problems in the fields of geometry
processing, computer graphics, and vision, attempting to recover the structure
of a shape from some observations. In this paper, we formulate the problem of
shape-from-operator (SfO), recovering an embedding of a mesh from intrinsic
differential operators defined on the mesh. Particularly interesting instances
of our SfO problem include synthesis of shape analogies, shape-from-Laplacian
reconstruction, and shape exaggeration. Numerically, we approach the SfO
problem by splitting it into two optimization sub-problems that are applied in
an alternating scheme: metric-from-operator (reconstruction of the discrete
metric from the intrinsic operator) and embedding-from-metric (finding a shape
embedding that would realize a given metric, a setting of the multidimensional
scaling problem)
A Shape Theorem for Riemannian First-Passage Percolation
Riemannian first-passage percolation (FPP) is a continuum model, with a
distance function arising from a random Riemannian metric in . Our main
result is a shape theorem for this model, which says that large balls under
this metric converge to a deterministic shape under rescaling. As a
consequence, we show that smooth random Riemannian metrics are geodesically
complete with probability one
An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
Here shape space is either the manifold of simple closed smooth
unparameterized curves in or is the orbifold of immersions from
to modulo the group of diffeomorphisms of . We
investige several Riemannian metrics on shape space: -metrics weighted by
expressions in length and curvature. These include a scale invariant metric and
a Wasserstein type metric which is sandwiched between two length-weighted
metrics. Sobolev metrics of order on curves are described. Here the
horizontal projection of a tangent field is given by a pseudo-differential
operator. Finally the metric induced from the Sobolev metric on the group of
diffeomorphisms on is treated. Although the quotient metrics are
all given by pseudo-differential operators, their inverses are given by
convolution with smooth kernels. We are able to prove local existence and
uniqueness of solution to the geodesic equation for both kinds of Sobolev
metrics.
We are interested in all conserved quantities, so the paper starts with the
Hamiltonian setting and computes conserved momenta and geodesics in general on
the space of immersions. For each metric we compute the geodesic equation on
shape space. In the end we sketch in some examples the differences between
these metrics.Comment: 46 pages, some misprints correcte
Gauge Invariant Framework for Shape Analysis of Surfaces
This paper describes a novel framework for computing geodesic paths in shape
spaces of spherical surfaces under an elastic Riemannian metric. The novelty
lies in defining this Riemannian metric directly on the quotient (shape) space,
rather than inheriting it from pre-shape space, and using it to formulate a
path energy that measures only the normal components of velocities along the
path. In other words, this paper defines and solves for geodesics directly on
the shape space and avoids complications resulting from the quotient operation.
This comprehensive framework is invariant to arbitrary parameterizations of
surfaces along paths, a phenomenon termed as gauge invariance. Additionally,
this paper makes a link between different elastic metrics used in the computer
science literature on one hand, and the mathematical literature on the other
hand, and provides a geometrical interpretation of the terms involved. Examples
using real and simulated 3D objects are provided to help illustrate the main
ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern
Analysis and Machine Intelligence in a better resolutio
Curvature weighted metrics on shape space of hypersurfaces in -space
Let be a compact connected oriented dimensional manifold without
boundary. In this work, shape space is the orbifold of unparametrized
immersions from to . The results of \cite{Michor118}, where
mean curvature weighted metrics were studied, suggest incorporating Gau{\ss}
curvature weights in the definition of the metric. This leads us to study
metrics on shape space that are induced by metrics on the space of immersions
of the form G_f(h,k) = \int_{M} \Phi . \bar g(h, k) \vol(f^*\bar{g}). Here
f \in \Imm(M,\R^n) is an immersion of into and are tangent vectors at . is the standard
metric on , is the induced metric on ,
\vol(f^*\bar g) is the induced volume density and is a suitable smooth
function depending on the mean curvature and Gau{\ss} curvature. For these
metrics we compute the geodesic equations both on the space of immersions and
on shape space and the conserved momenta arising from the obvious symmetries.
Numerical experiments illustrate the behavior of these metrics.Comment: 12 pages 3 figure
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