Multigrid convergence for the MDCA-curvature estimator

TL;DR

This paper introduces an improved curvature estimator for digital object boundaries that is smoother and more accurate, with proven convergence properties and better performance near corners.

Contribution

The paper extends the MDCA curvature estimator to the $eta$-MDCA version, providing convergence proofs and empirical validation showing enhanced accuracy and smoothness.

Findings

Convergence order close to $ ext{O}(h^{1/3})$ for convex sets.

$eta$-MDCA outperforms classical MDCA near corners.

The estimator is effective on objects with known curvature profiles.

Abstract

We consider the problem of estimating the curvature profile along the boundaries of digital objects in segmented black-and-white images. We start with the curvature estimator proposed by Roussillon et al., which is based on the calculation of \emph{maximal digital circular arcs} (MDCA). We extend this estimator to the $\lambda$-MDCA curvature estimator that considers several MDCAs for each boundary pixel and is therefore smoother than the classical MDCA curvature estimator. We prove an explicit order of convergence result for convex subsets in $\mathbb{R}^2$ with positive, continuous curvature profile. In addition, we evaluate the curvature estimator on various objects with known curvature profile. We show that the observed order of convergence is close to the theoretical limit of $\mathcal{O}(h^{\frac13})$. Furthermore, we establish that the $\lambda$-MDCA curvature estimator outperforms the MDCA curvature estimator, especially in the neighborhood of corners.