Weighted quasi-interpolant spline approximations: Properties and applications

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.

Weighted Quasi-Interpolant Spline Approximations of Planar Curvilinear Profiles in Digital Images

“…This section is meant to list some basic definitions regarding the wQISA family, as well as to introduce the essential terminology and notation. Because in this paper we focus on 1-manifolds, we restrict our attention to a univariate formulation of wQISA; for more details on the more general multivariate setting and an extended analysis of the theoretical properties, we refer the reader to [16].…”

“…Unlike traditional quasi interpolation spline methods, which generally focus on the approximation of functions with strong regularity conditions, Weighted Quasi Interpolant Spline Approximations (wQISAs) [16,17] aims to provide local approximations of point clouds which can be affected by artifacts such as noise and outliers. However, a paradigm to compute global approximations was, so far, missing; indeed, the only two global approximations presented in [16] are obtained by manually processing the set of edge points, thanks to the quite simple geometry of the studied profiles. In this paper, we introduce a novel method that exploits wQISAs to construct global approximations of planar curvilinear profiles, with application to digital image processing.…”

Data-driven quasi-interpolant spline surfaces for point cloud approximation

“…where f : Ω ⊂ R 2 → R is a function being approximated, n x ∈ N ∪ {+∞} n y , (x i , y j ) are given nodes and g i,j : Ω ⊂ R 2 → R are functions at our disposal. Differently from classic Quasi Interpolation methods that focus to function approximation, the Weighted Quasi Intepolant Spline Approximation (wQISA, [37]) aims at point cloud approximation, where the data are assumed to be affected by noise, outliers or partially missing. In this Section we summarise the wQISA concept and list its theoretical properties.…”

“…Similarly to the classical quasi-interpolant schemes [10], the wQISA method satisfies a number of desirable regularity properties, here briefly recalled. For more details and for a probabilistic interpretation of the method, we refer the reader to [37].…”

“…The wQISA implementation proposed in this paper is specifically designed to define powerful prediction methods from low quality points. Starting from the theoretical wQISA definition given in [37] for a single tensor-product mesh, in this work we derive a multi-level approximation algorithm and provide an extended comparative analysis with other methods in the literature. The main contributions of the paper include: a detailed description of the wQISA method when used for surface approximation; a multi-level approximation algorithm based on a data-driven definition of the weight functions; an extensive comparison of the wQISA outcome with other well-known continuous approximation methods.…”