Evolution of curves on a surface driven by the geodesic curvature
and external force
Karol Mikula and Daniel Sevcovic
Abstract
We study a flow of closed curves on a given surface driven by the geodesic
curvature and external force. We use a direct method for solving the
evolution of surface curves based on vertical projection to the plane.
It is shown that this geometric problem can be reduced to a solution of a
fully nonlinear system of parabolic differential equations. We prove short
time existence of classical solutions. Various Lyapunov like functionals
for the flow of surface curves are derived. A special attention is put on
the
analysis of closed stationary surface curves. We give sufficient conditions
for their dynamic stability. We also discuss an important link between
the geodesic flow and the edge detection problem in the theory of image
segmentation. An efficient numerical scheme for solving the governing system
of equations is presented. Several computational examples of evolution of
surface curves driven by the geodesic curvature and external force on
various
surfaces are presented in this paper.
AMS MOS 35K65, 65N40, 53C80
Paper
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Animation of a flow
Lecture slides
Computational and qualitative aspects of evolution of curves
driven by curvature and external force
Karol Mikula and Daniel Sevcovic
Abstract
We propose a direct method for solving the evolution of plane curves
satisfying the geometric equation $v= \beta(x, k, \nu)$ where $v$ is
the normal velocity, $k$ and $\nu$ are the curvature and tangential
angle of a plane curve $\Gamma\subset\R^2$ at a point $x\in \Gamma$.
We derive and analyze the governing system of partial differential equations
for the curvature, tangential angle, local length and position vector of an
evolving family of plane curves. The governing equations include a
nontrivial
tangential velocity functional yielding uniform redistribution of grid
points
along the evolving family of curves preventing thus numerically computed
solutions from forming various instabilities. We also propose a full
space-time discretization of the governing system of equations and study its
experimental order of convergence.
Several computational examples of evolution of plane curves driven by
curvature and external force as well as the geodesic curvatures driven
evolution of curves on various complex surfaces are presented in this paper.
Keywords: curve evolution, image and shape multiscale analysis,
phase interface, nonlinear degenerate parabolic equations, numerical
solution
AMS MOS 35K65, 65N40, 53C80
Paper
Compressed Postscript
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Animation of a flow
Lecture slides
A direct method for solving an anisotropic mean curvature flow
of plane curves with an external force
Karol Mikula and Daniel Sevcovic
Abstract
A new method for solution of
the evolution of plane curves satisfying the geometric equation
$v= \beta(x, k, \nu)$ where $v$ is the normal velocity, $k$ and $\nu$
are the curvature and tangential angle of a plane curve $\Gamma\subset\R^2$
at the point $x\in \Gamma$ is proposed.
We derive a governing system of partial differential equations for the
curvature, tangential angle, local length and position vector of an evolving
family of plane curves and prove local in
time existence of a classical solution. These equations include a
nontrivial
tangential velocity functional governing
a uniform redistribution of grid points and thus preventing numerically
computed solutions from forming various
instabilities. We discretize the governing system of equations
in order to find a numerical solution for 2D anisotropic interface
motions with applications in thermomechanics and medical image segmentation
problems.
Keywords: curve evolution, image and shape multiscale analysis,
phase interface, nonlinear degenerate parabolic equations, numerical
solution
AMS MOS 35K65, 65N40, 53C80
Paper
Compressed Postscript
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Lecture slides
Evolution of plane curves driven by a nonlinear function of curvature
and anisotropy
Karol Mikula and Daniel Sevcovic
Abstract
In this paper we study the evolution of plane curves satisfying the
geometric
equation $v= \beta(k, \nu),$ where $v$ is the normal velocity, $k$ and $\nu$
are the curvature and tangential angle of the plane curve $\Gamma$. We
follow the direct approach and we analyze the so-called intrinsic heat
equation governing the motion of plane curves obeying the geometric
equation.
By contrast to the usual approach, the intrinsic heat equation is modified
to include an appropriate nontrivial tangential velocity functional
$\alpha$.
We also show how the presence of a tangential velocity $\alpha$ can prevent
numerical solutions from forming various instabilities. From
analytical point of view we present some new results on short time
existence
of a regular family of evolving curves in the case when
$\beta(k,\nu)=\gamma(\nu) k^m$, $0\le m\le 2$ and the governing system of
equations includes a nontrivial tangential velocity functional.
Keywords: curve evolution, image and shape multiscale analysis,
phase interface, nonlinear degenerate parabolic equations, numerical
solution
AMS MOS 35K65, 65N40, 53C80
Paper
Compressed Postscript
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Lecture slides
Solution of Nonlinearly Curvature Driven
Evolution of Plane Curves
Karol Mikula and Daniel Sevcovic
Abstract
The evolution of plane curves obeying the equation v= beta(k),
where v is normal velocity and k curvature of the curve is studied.
Morphological image and shape multiscale analysis of Alvarez,
Guichard, Lions and Morel and affine invariant scale space of curves
introduced by Sapiro and Tannenbaum as well as isotropic motions of plane
phase interfaces studied by Angenent and Gurtin are included in the model.
We introduce and analyse a numerical scheme for solving the governing
equation and present numerical experiments.
Keywords: curve evolution, image and shape multiscale analysis,
phase interface, nonlinear degenerate parabolic equations, numerical
solution
AMS MOS 35K65, 65N40, 53C80
Paper
Compressed Postscript
Adobe Acrobat PDF file
Lecture slides