A non-uniform corner-cutting subdivision scheme with an improved accuracy

TL;DR

This paper introduces a new non-uniform corner-cutting subdivision scheme that leverages exponential polynomials to enhance approximation accuracy from second to third order, maintaining smoothness and demonstrating improved performance through numerical examples.

Contribution

It proposes a novel shape parameter selection method for NUCC schemes, achieving higher approximation order while preserving smoothness.

Findings

Achieves third-order approximation accuracy.

Maintains C^1 smoothness similar to classical algorithms.

Numerical examples confirm improved approximation performance.

Abstract

The aim of this paper is to construct a new non-uniform corner-cutting (NUCC) subdivision scheme that improves the accuracy of the classical (stationary and nonstationary) methods. The refinement rules are formulated via the reproducing property of exponential polynomials. An exponential polynomial has a shape parameter so that it may be adapted to the characteristic of the given data. In this study, we propose a method of selecting the shape parameter, so that it enables the associated scheme to achieve an improved approximation order (that is, {\em three}), while the classical methods attain the second order accuracy. An analysis of convergence and smoothness of the proposed scheme is conducted. The proposed scheme is shown to have the same smoothness as the classical Chaikin's corner-cutting algorithm, that is, $C^1$. Finally, some numerical examples are presented to demonstrate the advantages of the new corner-cutting algorithm.