10,636 research outputs found
Matching Structures by Computing Minimal Paths on a Manifold
International audienceThe general problem of matching structures is very pervasive in computer vision and image processing. The research presented here tackles the problem of object matching in a very general perspective. It is formulated for the matching of surfaces. It applies to objects having small or large deformation and arbitrary topological changes. The process described hinges on a geodesic distance equation for a family of curves or surfaces embedded in the graph of a cost function. This geometrical approach to object matching has the advantage that the similarity criterion can be used to define the shape of the cost function. Matching paths are computed on the cost manifolds using distance maps. These distance maps are generated by solving a general partial differential equation which is a generalization of the geodesic dis- tance evolution scheme introduced by R. Kimmel, A. Amir, and A. F. Bruckstein (1995, IEEE Trans. Pattern Anal. Mach. Intell. 17, 635-640). An Eulerian level-set formulation is also introduced, leading to a numerical scheme used for solving par- tial differential equations originating from hyperbolic conservation laws, which has proven to be very robust and stable
Distinguishing between exotic symplectic structures
We investigate the uniqueness of so-called exotic structures on certain exact
symplectic manifolds by looking at how their symplectic properties change under
small nonexact deformations of the symplectic form. This allows us to
distinguish between two examples based on those found in
\cite{maydanskiy,maydanskiyseidel}, even though their classical symplectic
invariants such as symplectic cohomology vanish. We also exhibit, for any ,
an exact symplectic manifold with distinct but exotic symplectic
structures, which again cannot be distinguished by symplectic cohomology.Comment: 33 pages, 6 figures. Final version, accepted by Journal of Topolog
Computing Invariants of Simplicial Manifolds
This is a survey of known algorithms in algebraic topology with a focus on
finite simplicial complexes and, in particular, simplicial manifolds. Wherever
possible an elementary approach is chosen. This way the text may also serve as
a condensed but very basic introduction to the algebraic topology of simplicial
manifolds.
This text will appear as a chapter in the forthcoming book "Triangulated
Manifolds with Few Vertices" by Frank H. Lutz.Comment: 13 pages, 3 figure
Diffeomorphic density matching by optimal information transport
We address the following problem: given two smooth densities on a manifold,
find an optimal diffeomorphism that transforms one density into the other. Our
framework builds on connections between the Fisher-Rao information metric on
the space of probability densities and right-invariant metrics on the
infinite-dimensional manifold of diffeomorphisms. This optimal information
transport, and modifications thereof, allows us to construct numerical
algorithms for density matching. The algorithms are inherently more efficient
than those based on optimal mass transport or diffeomorphic registration. Our
methods have applications in medical image registration, texture mapping, image
morphing, non-uniform random sampling, and mesh adaptivity. Some of these
applications are illustrated in examples.Comment: 35 page
Multilevel Artificial Neural Network Training for Spatially Correlated Learning
Multigrid modeling algorithms are a technique used to accelerate relaxation
models running on a hierarchy of similar graphlike structures. We introduce and
demonstrate a new method for training neural networks which uses multilevel
methods. Using an objective function derived from a graph-distance metric, we
perform orthogonally-constrained optimization to find optimal prolongation and
restriction maps between graphs. We compare and contrast several methods for
performing this numerical optimization, and additionally present some new
theoretical results on upper bounds of this type of objective function. Once
calculated, these optimal maps between graphs form the core of Multiscale
Artificial Neural Network (MsANN) training, a new procedure we present which
simultaneously trains a hierarchy of neural network models of varying spatial
resolution. Parameter information is passed between members of this hierarchy
according to standard coarsening and refinement schedules from the multiscale
modelling literature. In our machine learning experiments, these models are
able to learn faster than default training, achieving a comparable level of
error in an order of magnitude fewer training examples.Comment: Manuscript (24 pages) and Supplementary Material (4 pages). Updated
January 2019 to reflect new formulation of MsANN structure and new training
procedur
Degenerations of ideal hyperbolic triangulations
Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The
deformation variety D(T), a subset of which parameterises (incomplete)
hyperbolic structures obtained on M using T, is defined and compactified by
adding certain projective classes of transversely measured singular
codimension-one foliations of M. This leads to a combinatorial and geometric
variant of well-known constructions by Culler, Morgan and Shalen concerning the
character variety of a 3-manifold.Comment: 31 pages, 11 figures; minor changes; to appear in Mathematische
Zeitschrif
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Elastic Registration of Geodesic Vascular Graphs
Vascular graphs can embed a number of high-level features, from morphological
parameters, to functional biomarkers, and represent an invaluable tool for
longitudinal and cross-sectional clinical inference. This, however, is only
feasible when graphs are co-registered together, allowing coherent multiple
comparisons. The robust registration of vascular topologies stands therefore as
key enabling technology for group-wise analyses. In this work, we present an
end-to-end vascular graph registration approach, that aligns networks with
non-linear geometries and topological deformations, by introducing a novel
overconnected geodesic vascular graph formulation, and without enforcing any
anatomical prior constraint. The 3D elastic graph registration is then
performed with state-of-the-art graph matching methods used in computer vision.
Promising results of vascular matching are found using graphs from synthetic
and real angiographies. Observations and future designs are discussed towards
potential clinical applications
- …