Weighted Quasi Interpolant Spline Approximations: Properties and Applications

TL;DR

This paper introduces the Weighted Quasi Interpolant Spline Approximation (wQISA) method, which provides robust, shape-preserving continuous representations for sampled data, especially effective in noisy and outlier-prone scenarios, demonstrated through various real data applications.

Contribution

The paper presents a novel wQISA method that offers shape-preserving, noise-robust spline approximations with proven bounds and broad applicability in data modeling and simulation.

Findings

wQISA preserves shape properties of classical schemes.

The method is robust to noise and outliers.

Effective in applications like image curve fitting and rainfall simulation.

Abstract

Continuous representations are fundamental for modeling sampled data and performing computations and numerical simulations directly on the model or its elements. To effectively and efficiently address the approximation of point clouds we propose the Weighted Quasi Interpolant Spline Approximation method (wQISA). We provide global and local bounds of the method and discuss how it still preserves the shape properties of the classical quasi-interpolation scheme. This approach is particularly useful when the data noise can be represented as a probabilistic distribution: from the point of view of nonparametric regression, the wQISA estimator is robust to random perturbations, such as noise and outliers. Finally, we show the effectiveness of the method with several numerical simulations on real data, including curve fitting on images, surface approximation and simulation of rainfall precipitations.