64 research outputs found
Degenerate neckpinches in Ricci flow
In earlier work, we derived formal matched asymptotic profiles for families
of Ricci flow solutions developing Type-II degenerate neckpinches. In the
present work, we prove that there do exist Ricci flow solutions that develop
singularities modeled on each such profile. In particular, we show that for
each positive integer , there exist compact solutions in all dimensions
that become singular at the rate (T-t)^{-2+2/k}$
Finsler Active Contours
©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TPAMI.2007.70713In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery
Dynamics of Convex Mean Curvature Flow
There is an extensive and growing body of work analyzing convex ancient
solutions to Mean Curvature Flow (MCF), or equivalently of Rescaled Mean
Curvature Flow (RMCF). The goal of this paper is to complement the existing
literature, which analyzes ancient solutions one at a time, by considering the
space X of all convex hypersurfaces M, regard RMCF as a semiflow on this space,
and study the dynamics of this semiflow. To this end, we first extend the well
known existence and uniqueness of solutions to MCF with smooth compact convex
initial data to include the case of arbitrary non compact and non smooth
initial convex hypersurfaces. We identify a suitable weak topology with good
compactness properties on the space X of convex hypersurfaces and show that
RMCF defines a continuous local semiflow on X whose fixed points are the
shrinking cylinder solitons, and for which the Huisken energy is a Lyapunov
function. Ancient solutions to MCF are then complete orbits of the RMCF
semiflow on X. We consider the set of all hypersurfaces that lie on an ancient
solution that in backward time is asymptotic to one of the shrinking cylinder
solitons and prove various topological properties of this set. We show that
this space is a path connected, compact subset of X, and, considering only
point symmetric hypersurfaces, that it is topologically trivial in the sense of
Cech cohomology. We also give a strong evidence in support of the conjecture
that the space of all convex ancient solutions with a point symmetry is
homeomorphic to an n-1 dimensional simplex
Uniqueness of two-convex closed ancient solutions to the mean curvature flow
In this paper we consider closed non-collapsed ancient solutions to the mean
curvature flow () which are uniformly two-convex.
We prove that any two such ancient solutions are the same up to translations
and scaling. In particular, they must coincide up to translations and scaling
with the rotationally symmetric closed ancient non-collapsed solution
constructed by Brian White in (2000), and by Robert Haslhofer and Or
Hershkovits in (2016).Comment: 74 pages, 5 figure
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