Contour Parametrization via Anisotropic Mean Curvature Flows

TL;DR

This work addresses contour recognition by evolving a planar closed contour under anisotropic mean curvature flow (AMCF) constrained by a Poisson potential generated by point charges. The curve velocity in the normal direction is given by v = ∂u/∂n, where u solves a Poisson problem Δu = ρ in Ω with Dirichlet data on ∂Ω, and ρ can be a Dirac delta inside the contour to attract the curve toward image boundaries; when ρ = 0 this reduces to standard MCF. A Lagrangian, explicit numerical scheme is developed, using Green's function/Poisson representations to compute the field and a boundary-integral discretization to form a 2N × 2N linear system at each step, with tangential redistribution ensuring stable parameterization. Theoretical contributions include short-time existence for the isotropic case, a stability criterion for the anisotropic scheme, and conditioning analysis, together with practical contour parametrization results and public code. The method successfully matches non-convex shapes under suitable charge configurations, indicating potential for image-driven contour recognition and guidance for future extensions to more complex charge distributions.

Abstract

We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as background. We include a stability criteria for our numerical approximation.

Figures (11)

• Figure 1: Numerical instability in a straightforward discretization of MCF for a circle (left), and detail (right).
• Figure 2: Loop formation in numerical discretization by standard Euler method of equation (\ref{['eq:MCF']}) for a cycloid (left), and detail (right).
• Figure 3: Numerical MCF of cycloid. (a) An evolving cycloid (outer curve) evolves by MCF, as time increase, it converges to a circumference. (b) 3D projection of (a), the time parameter $t$ is represented by the vertical axis.
• Figure 4: Numerical MCF of non-convex curve (I). (a) The outer curve is the initial non-convex curve, it deforms to a circumference and shrinks as time elapses. (b) 3D projection of (a), the time parameter $t$ is represented by the vertical axis.
• Figure 5: Numerical MCF of non-convex curve (II). (a) The outer curve is the initial non-convex curve, it shrinks as time elapses. (b) 3D projection of (a), the time parameter $t$ is represented by the vertical axis.

Theorems & Definitions (20)

• Theorem 1
• proof
• Theorem 2
• Lemma 3
• proof
• Theorem 4
• Lemma 5
• Lemma 6
• Definition 7
• Lemma 8