[Paper Review] Evolution of plane curves with a curvature adjusted tangential velocity

Daniel Ševčovič, Shigetoshi Yazaki|arXiv (Cornell University)|Sep 14, 2010

Geometric Analysis and Curvature Flows18 references4 citations

TL;DR

This paper introduces a curvature-adjusted tangential velocity method for evolving plane curves with curvature-dependent normal velocity, enabling dynamic grid point redistribution that concentrates or disperses points based on local curvature. The approach improves numerical stability and accuracy in solving geometric evolution equations, particularly for area and length preservation in polygonal approximations of evolving curves.

ABSTRACT

We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted control in the tangential motion of evolving curves. The tangential velocity distributes grid points along the curve not only uniform but also lead to a suitable concentration and/or dispersion depending on the curvature. Our study is based on solutions to the governing system of nonlinear parabolic equations for the position vector, tangent angle and curvature of a curve. We furthermore present a semi-implicit numerical discretization scheme based on the flowing finite volume method. Several numerical examples illustrating capability of the new tangential redistribution method are also presented in this paper.

Motivation & Objective

• Address numerical instabilities in curve evolution simulations caused by uneven grid point distribution due to curvature-driven motion.
• Develop a systematic method for tangential velocity that controls grid point concentration and dispersion based on curvature.
• Ensure better approximation of geometric quantities (area, length) in polygonal discretizations of evolving curves.
• Provide a semi-implicit numerical scheme using the flowing finite volume method for efficient and stable computation.
• Generalize existing tangential velocity strategies to handle arbitrary curvature, angle, and position-dependent normal velocities.

Proposed method

• Formulate the curve evolution as a system of nonlinear parabolic PDEs for position vector, tangent angle, and curvature.
• Introduce a curvature-adjusted tangential velocity component that dynamically redistributes grid points based on local curvature.
• Use a semi-implicit time discretization combined with the flowing finite volume method for numerical stability and accuracy.
• Derive the tangential velocity from a variational principle that minimizes error in length and area approximation of polygonal curves.
• Implement the method for various curvature-driven flows, including anisotropic mean curvature flow and image segmentation flows.
• Validate the method through numerical experiments with closed, embedded plane curves under diverse normal velocity laws.

Experimental results

Research questions

• RQ1 How can tangential velocity be designed to achieve optimal grid point distribution along evolving curves with curvature-dependent normal motion?
• RQ2 What is the impact of curvature-adjusted tangential velocity on the accuracy of area and length approximation in polygonal curve representations?
• RQ3 Can the proposed method stabilize numerical schemes that otherwise suffer from grid clustering or dispersion in curve evolution?
• RQ4 How does the curvature-adjusted tangential velocity compare to uniform redistribution or fixed tangential velocity strategies in preserving geometric properties?
• RQ5 What is the theoretical and numerical performance of the semi-implicit flowing finite volume scheme in handling general curvature-driven flows?

Key findings

• The curvature-adjusted tangential velocity significantly improves the accuracy of polygonal approximations by minimizing errors in length and area preservation.
• Numerical experiments confirm that the method effectively prevents grid point clustering and dispersion, especially in regions of high curvature.
• The proposed semi-implicit scheme based on the flowing finite volume method demonstrates stability and convergence in simulating complex curve evolutions.
• The method successfully captures phenomena such as convexification and loss of convexity in anisotropic and forced curvature flows.
• For image segmentation, the method enables robust edge detection and curve evolution toward high-gradient regions using a gradient flow formulation.
• The curvature-adjusted tangential velocity outperforms uniform redistribution and fixed tangential velocity strategies in maintaining geometric fidelity during evolution.