517 research outputs found
Stability of Reeb graphs under function perturbations: the case of closed curves
Reeb graphs provide a method for studying the shape of a manifold by encoding
the evolution and arrangement of level sets of a simple Morse function defined
on the manifold. Since their introduction in computer graphics they have been
gaining popularity as an effective tool for shape analysis and matching. In
this context one question deserving attention is whether Reeb graphs are robust
against function perturbations. Focusing on 1-dimensional manifolds, we define
an editing distance between Reeb graphs of curves, in terms of the cost
necessary to transform one graph into another. Our main result is that changes
in Morse functions induce smaller changes in the editing distance between Reeb
graphs of curves, implying stability of Reeb graphs under function
perturbations.Comment: 23 pages, 12 figure
Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
We give a classification of generic coadjoint orbits for the groups of
symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic
surface. We also classify simple Morse functions on symplectic surfaces with
respect to actions of those groups. This gives an answer to V.Arnold's problem
on describing all invariants of generic isovorticed fields for the 2D ideal
fluids. For this we introduce a notion of anti-derivatives on a measured Reeb
graph and describe their properties.Comment: 38 pages, 11 figures; to appear in Annales de l'Institut Fourie
A discrete Reeb graph approach for the segmentation of human body scans
Segmentation of 3D human body (HB) scan is a very challenging problem in applications exploiting human scan data. To tackle this problem, we propose a topological approach based on discrete Reeb graph (DRG) which is an extension of the classical Reeb graph to unorganized cloud of 3D points. The essence of the approach is detecting critical nodes in the DRG thus permitting the extraction of branches that represent the body parts. Because the human body shape representation is built upon global topological features that are preserved so long as the whole structure of the human body does not change, our approach is quite robust against noise, holes, irregular sampling, moderate reference change and posture variation. Experimental results performed on real scan data demonstrate the validity of our method
First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds
We calculate the first and the second variation formula for the
sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We
consider general variations that can move the singular set of a C^2 surface and
non-singular variation for C_H^2 surfaces. These formulas enable us to
construct a stability operator for non-singular C^2 surfaces and another one
for C2 (eventually singular) surfaces. Then we can obtain a necessary condition
for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in
term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally
we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly
changed and Remark 9.9 adde
The Weinstein Conjecture for Planar Contact Structures in Dimension Three
In this paper we describe a general strategy for approaching the Weinstein
conjecture in dimension three. We apply this approach to prove the Weinstein
conjecture for a new class of contact manifolds (planar contact manifolds). We
also discuss how the present approach reduces the general Weinstein conjecture
in dimension three to a compactness problem for the solution set of a first
order elliptic PDE.Comment: 19 pages, corrected some typo
An edit distance for Reeb graphs
We consider the problem of assessing the similarity of 3D shapes
using Reeb graphs from the standpoint of robustness under
perturbations. For this purpose, 3D objects are viewed as spaces
endowed with real-valued functions, while the similarity between
the resulting Reeb graphs is addressed through a graph edit
distance. The cases of smooth functions on manifolds and piecewise
linear functions on polyhedra stand out as the most interesting
ones. The main contribution of this paper is the introduction of a
general edit distance suitable for comparing Reeb graphs in these
settings. This edit distance promises to be useful for
applications in 3D object retrieval because of its stability
properties in the presence of noise
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