The Stability of Convergence of Curve Evolutions in Vector Fields

TL;DR

This paper develops a theoretical framework to analyze and enhance the stability of curve evolution convergence in vector fields, demonstrating improved convergence through proposed modifications validated by numerical experiments.

Contribution

It introduces a new theory for analyzing and improving the stability of curve evolution convergence, and proposes modifications that enhance convergence stability.

Findings

The convergence of a known curve evolution is marginally stable.

Proposed modifications improve the stability of curve evolution convergence.

Numerical experiments validate the effectiveness of the proposed stability improvements.

Abstract

Curve evolution is often used to solve computer vision problems. If the curve evolution fails to converge, we would not be able to solve the targeted problem in a lifetime. This paper studies the theoretical aspect of the convergence of a type of general curve evolutions. We establish a theory for analyzing and improving the stability of the convergence of the general curve evolutions. Based on this theory, we ascertain that the convergence of a known curve evolution is marginal stable. We propose a way of modifying the original curve evolution equation to improve the stability of the convergence according to our theory. Numerical experiments show that the modification improves the convergence of the curve evolution, which validates our theory.