Weighted least squares subdivision schemes for noisy data on triangular meshes

TL;DR

This paper introduces a new family of linear subdivision schemes based on weighted least squares for refining noisy data on triangular meshes, capable of handling various grid types and providing denoising and approximation properties.

Contribution

The paper proposes a novel subdivision scheme using weighted least squares for noisy data on triangular meshes, with proven properties like reproduction, approximation order, and convergence.

Findings

Performance comparable to advanced local linear regression methods

Suitable for multiresolution analysis of noisy geometric data

Effective denoising capabilities demonstrated through numerical experiments

Abstract

This paper presents and analyses a new family of linear subdivision schemes to refine noisy data given on triangular meshes. The subdivision rules consist of locally fitting and evaluating a weighted least squares approximating first-degree polynomial. This type of rules, applicable to any type of triangular grid, including finite grids or grids containing extraordinary vertices, are geometry-dependent which may result in non-uniform schemes. For these new subdivision schemes, we are able to prove reproduction, approximation order, denoising capabilities and, for some special type of grids, convergence as well. Several numerical experiments demonstrate that their performance is similar to advanced local linear regression methods but their subdivision nature makes them suitable for use within a multiresolution context as well as to deal with noisy geometric data as shown with an example.