Computational and qualitative aspects of evolution of curves driven by curvature and external force

Mikula, Karol; Ševčovič, Daniel

doi:10.1007/s00791-004-0131-6

Cited by 53 publications

(119 citation statements)

References 30 publications

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“…This has been reported in a number of places, for example [4] [9] [15]. During evolution the curve points bunch together in some places while spreading out at other places along the curve.…”

Section: Problems With Parametric Contoursmentioning

confidence: 64%

On Stabilisation of Parametric Active Contours

Srikrishnan

Chaudhuri

Roy

et al. 2007

2007 IEEE Conference on Computer Vision and Pattern Recognition

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Section: Problems With Parametric Contoursmentioning

confidence: 64%

On Stabilisation of Parametric Active Contours

Srikrishnan

Chaudhuri

Roy

et al. 2007

2007 IEEE Conference on Computer Vision and Pattern Recognition

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“…In these experiments we used coupling τ ≈ h 2 , cf. [9,10,11,27,28], the exact area at any time moment is A e = 3π and the area error is computed as…”

Section: Discussion On Numerical Experimentsmentioning

confidence: 99%

“…The corresponding numerical scheme is efficiently stabilized by an appropriate choice of the tangential velocity term α in (2.4). Let us note that other known tangential velocities, like the one preserving relative local length [18,21,25], locally diffusive redistribution [8,27,28], crystalline curvature redistribution [35] or curvature adjusted redistribution [34,4] can be also incorporated straightforwardly to the numerical scheme by corresponding change of the term α entering equation (2.4).…”

Section: Governing Equationsmentioning

confidence: 99%

“…The case when b = − 1 2 k 3 arises from the model of the Euler-Bernoulli elastic rod -an important problem in structural mechanics [6,12,11]. The evolutionary models having the normal velocity of the form (1.1) are often adopted in image segmentation where elastic and geodesic curves are used in order to find image objects in an automatic way [19,7,20,27,29]. By our method we are able to handle a regularized backward mean curvature flow in which b is a decreasing function of the curvature k like e.g.…”

Section: Introductionmentioning

confidence: 99%

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A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

Mikula¹,

Ševčovič

Balažovjech³

2010

CiCP

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Abstract.A new simple Lagrangian method with favorable stability and efficiency properties for computing general plane curve evolutions is presented. The method is based on the flowing finite volume discretization of the intrinsic partial differential equation for updating the position vector of evolving family of plane curves. A curve can be evolved in the normal direction by a combination of fourth order terms related to the intrinsic Laplacian of the curvature, second order terms related to the curvature, first order terms related to anisotropy and by a given external velocity field. The evolution is numerically stabilized by an asymptotically uniform tangential redistribution of grid points yielding the first order intrinsic advective terms in the governing system of equations. By using a semi-implicit in time discretization it can be numerically approximated by a solution to linear penta-diagonal systems of equations (in presence of the fourth order terms) or tri-diagonal systems (in the case of the second order terms). Various numerical experiments of plane curve evolutions, including, in particular, nonlinear, anisotropic and regularized backward curvature flows, surface diffusion and Willmore flows, are presented and discussed.

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“…in [13,21]; for further Lagrangean methods we refer e.g. to [12,25,26]. Other than the level set, but also Eulerian, approach is represented by the phase field method where the convergence of numerical approximation to the solution of the so-called Allen-Cahn equation (modelling diffused interface evolution) is studied, see e.g.…”

mentioning

confidence: 99%

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

Handlovičová

Mikula

2008

Appl Math

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Abstract. We show stability and consistency of the linear semi-implicit complementary volume numerical scheme for solving the regularized, in the sense of Evans and Spruck, mean curvature flow equation in the level set formulation. The numerical method is based on the finite volume methodology using so-called complementary volumes to a finite element triangulation. The scheme gives solution in efficient and unconditionaly stable way.

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Computational and qualitative aspects of evolution of curves driven by curvature and external force

Cited by 53 publications

References 30 publications

On Stabilisation of Parametric Active Contours

On Stabilisation of Parametric Active Contours

A Simple, Fast and Stabilized Flowing Finite Volume Method for Solving General Curve Evolution Equations

Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation

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