A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

Abstract

SUMMARYA new method for solution of the evolution of plane curves satisfying the geometric equation v=ÿ(x; k; ), where v is the normal velocity, k and are the curvature and tangential angle of a plane curve ⊂ R 2 at the point x ∈ , is proposed. We derive a governing system of partial di erential 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 non-trivial 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 ÿnd a numerical solution for 2D anisotropic interface motions and image segmentation problems.

Evolution of plane curves with a curvature adjusted tangential velocity

“…This result can be considered as an improvement of that of [14] in which satisfactory results were obtained only for the case when β = β(k) is linear or sublinear function with respect to k. Next, in the series of papers [16][17][18], Mikula and the first author proposed a method of asymptotically uniform redistribution. In terms of our notation, they derived (3.3) with ϕ ≡ 1 and nontrivial relaxation function ω(t) for a general class of normal velocities of the form β = β(x, ν, k).…”

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

“…In [26,27,28] the system (2.6)-(2.8) was solved numerically for the curvature k, tangent angle ν and the local length g. Knowing these quantities one can reconstruct the curve evolution. Asymptotically uniform redistributions for the second order flows with driving force (see [26]) and for the fourth order flows (see [28]) were also proposed.…”

“…Asymptotically uniform redistributions for the second order flows with driving force (see [26]) and for the fourth order flows (see [28]) were also proposed.…”

“…In this paper we follow a different approach when compared to [26,28]. It is much faster and simpler from computational point of view.…”

“…[35,34]). In [26] the authors proposed and analyzed a new type of tangential redistribution referred to as the asymptotically uniform grid points redistribution. Here we follow a similar idea but adopt it in a rather different way.…”