A proof of equivalence between level lines shortening and curvature motion in image processing

TL;DR

This paper establishes a rigorous mathematical equivalence between level lines shortening and curvature motion in image processing, providing a solid analytical foundation for their numerical implementation.

Contribution

It proves that the continuous Level Lines Shortening evolution computes a viscosity solution for mean curvature motion, linking geometric image processing to PDE theory.

Findings

Provides an exact analytical framework for numerical implementation

Shows equivalence between level lines shortening and curvature motion

Enables online image processing using the proposed method

Abstract

In this paper we define the continuous Level Lines Shortening evolution of a two-dimensional image as the Curve Shortening operator acting simultaneously and independently on all the level lines of the initial data, and show that it computes a viscosity solution for the mean curvature motion. This provides an exact analytical framework for its numerical implementation, which runs online on any image at http://www.ipol.im/. Analogous results hold for its affine variant version, the Level Lines Affine Shortening.