Subdivision schemes based on weighted local polynomial regression. A new technique for the convergence analysis

TL;DR

This paper introduces a family of binary WLPR-based subdivision schemes for curve construction from noisy data. By embedding weighted local polynomial regression into the refinement rules, it achieves polynomial reproduction up to degree $d$, controlled denoising via a bandwidth $\lambda$ and weight function $\phi$, and efficient convergence analysis. For $d=0,1$ the schemes are proven convergent with positive masks, monotonicity preservation, and Gibbs-avoidance; for $d=2,3$ new asymptotic convergence tools are developed to establish convergence and to study approximation versus denoising trade-offs with explicit results for several weight choices. Numerical experiments on noisy geometric data confirm convergence to $C^1$ limit curves, illustrate denoising benefits, and demonstrate practical guidance for selecting weight functions to balance accuracy and smoothing.

Abstract

The generation of curves and surfaces from given data is a well-known problem in Computer-Aided Design that can be approached using subdivision schemes. They are powerful tools that allow obtaining new data from the initial one by means of simple calculations. However, in some applications, the collected data are given with noise and most of schemes are not adequate to process them. In this paper, we present some new families of binary univariate linear subdivision schemes using weighted local polynomial regression. We study their properties, such as convergence, monotonicity, polynomial reproduction and approximation and denoising capabilities. For the convergence study, we develop some new theoretical results. Finally, some examples are presented to confirm the proven properties.

Figures (6)

• Figure 1: Limit functions of the subdivision schemes $S_{1,\mathbf{w}^\lambda}$ for some weight functions (see Table \ref{['tabla1nucleos']}) and $\lambda=3.2$ (blue), $\lambda=3.4$ (orange), $\lambda=3.6$ (yellow) and $\lambda=3.8$ (purple).
• Figure 2: Left, the pair of values $(2I_4(H),\|H\|_2^2)$ for several choices of $\phi$. Thus, the lower $x$ and $y$ axis values, the better approximation and denoising capabilities, respectively. Blue, $\phi(x) = (1-x^p)^q$ for several values $(p,q)$ pairs such that $1\leq p \leq 20$, $\frac{1}{2}\leq q \leq 20$; red, the Pareto front of the previous pairs; green, $\phi(x) = \exp(-\xi x)$ for $\frac{1}{2}\leq \xi\leq 10$. Right, the red line represents the pairs of values ($p,q$) for which $\phi(x) = (1-x^p)^q$ is Pareto-optimal.
• Figure 3: Several subdivision schemes (by columns) applied to the star-shaped data in \ref{['eqnumexpej2']}. In the first row, they are applied to the original data. In the second and third row, the data is contaminated by normal noise with $\sigma=0.5$ and $\sigma=1$, respectively.
• Figure 4: Five iterations of $S_{3,\texttt{rect}^{\lambda_k}}$ applied to $\widetilde{\mathbf{g}}^{0,h_k}$, for $k=1,2,3,4$. The blue circles are the initial data, the red line is the smooth function $G$ and the black line represents the limit function. The parameters for each graphic are (by rows): $h_1=10^{-1}$, $\lambda_1 = 4.5$; $h_2=10^{-2}$, $\lambda_2 = 13.5$; $h_3=10^{-3}$, $\lambda_3 = 103.5$; $h_4=10^{-4}$, $\lambda_4 = 1003.5$.
• Figure 5: Limit curves for discontinuous data using subdivision schemes with rect (red line) and trwt (black line) weight functions, $d=0,1$.

Theorems & Definitions (44)

• Remark 2.1
• Definition 1
• Definition 2
• Lemma 2.1
• Definition 3
• Definition 4
• Lemma 2.2
• Lemma 2.3
• proof
• Theorem 2.4