Weighted least squares subdivision schemes for noisy data on triangular meshes

TL;DR

For these new subdivision schemes, this paper is able to prove reproduction, approximation order, denoising capabilities and, for some special type of grids, convergence as well.

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.

Figures (15)

• Figure 1: An example of triangular mesh refinement. Left, the initial data. Right, the result of a single refinement step. Projected on the bottom, the triangulations to which the data are attached. It can be seen that each face is divided into four smaller faces, requiring the insertion of new vertices.
• Figure 2: An example of triangulation refinement by mid-point insertion. In red, the original triangulation. In black, the result of a single refinement step (left) and two refinement steps (right). The initial triangulation is irregular (see the extraordinary vertices in blue). No other extraordinary vertices are added by the refinement and each patch defined by the initial faces constitutes a uniform triangulation (see the discussion of Section \ref{['sec:properties']}).
• Figure 3: Example of balls used for the refinement rules. The center of the ball is marked with a cross (a vertex, in the left figure, and a mid-point, in the right figure). Red dots show the vertices inside the balls.
• Figure 4: An example of stencil configuration leading to negative coefficients $\alpha^{k+1,\ell}_i$ (the numbers in the graphic), which has been obtained considering all weights equal to 1 and the vertex coordinates (from top to bottom, from left to right) $(-4,1),(-3,1),(-2,1),(-1,1),(4,1),(0,0),(3,-1)$. The new vertex would be inserted at the origin. See Section Reproducibility to recreate this example.
• Figure 5: Examples of grids with different regularities and uniformities. (a) The triangular-rectangular grid and (b) the equilateral grid are regular and uniform. A regular non-uniform grid (c) and an irregular non-uniform grid (d) are also shown.

Theorems & Definitions (24)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Definition 3.1: 0-homogeneouty
• Lemma 3.1
• proof
• Remark 3.4
• Lemma 4.1
• proof
• Lemma 4.2