Convex shapes and convergence speed of discrete tangent estimators

TL;DR

This paper investigates the convergence properties of digital tangent estimators for convex shapes, demonstrating that they are multigrid convergent with an average convergence rate of O(h^(2/3)), supported by experiments.

Contribution

It provides a theoretical analysis of the convergence speed of tangent estimators based on digital straight segments for convex shapes, establishing an average rate of O(h^(2/3)).

Findings

Estimators are multigrid convergent for certain convex shapes.

The average convergence speed is O(h^(2/3)).

Experimental results support the theoretical bound.

Abstract

Discrete geometric estimators aim at estimating geometric characteristics of a shape with only its digitization as input data. Such an estimator is multigrid convergent when its estimates tend toward the geometric characteristics of the shape as the digitization step h tends toward 0. This paper studies the multigrid convergence of tangent estimators based on maximal digital straight segment recognition. We show that such estimators are multigrid convergent for some family of convex shapes and that their speed of convergence is on average O(h^(2/3)). Experiments confirm this result and suggest that the bound is tight.