Weighted quasi-interpolant spline approximations: Properties and applications
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Weighted quasi-interpolant spline approximations: Properties and applications Andrea Raffo1,2,3 · Silvia Biasotti3 Received: 13 December 2019 / Accepted: 20 July 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 non-parametric 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. Keywords Spline methods · Quasi-interpolation · Non-parametric regression · Point clouds · Raw data · Noise Mathematics Subject Classification 2010 41A25 · 65D25 · 68U07
1 Introduction Modeling sampled data with a continuous representation is essential in many applications such as, for instance, image resampling [8], geometric modeling [19], Andrea Raffo
[email protected] Silvia Biasotti
[email protected] 1
Department of Mathematics and Cybernetics, SINTEF, Forskningsveien 1, 0373 Oslo, Norway
2
Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851 Oslo, Norway
3
Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes”, Consiglio Nazionale delle Ricerche, Via de Marini 6, 16149 Genoa, Italy
Numerical Algorithms
isogeometric analysis (IgA) [30], and the numerical solution of PDE boundary problems [4]. Spline interpolation is largely adopted to approximate data from a function or a physical object because of the simplicity of its construction, its ease and accuracy of evaluation, and its capacity to approximate complex shapes through mathematical element fitting and interactive design [48]. It is often preferred to polynomial interpolation because it yields visually effective results even when using low-degree polynomials, while avoiding the Runge’s phenomenon for higher degrees [28]. B-splines represent a popular way for dealing with spline interpolation and are nowadays the most powerful tool in CAGD [9]. Several generalizations to non-polynomial splines are possible, such as generalized splines [7], which admit also trigonometric or exponential bases, or non-uniform rational B-splines (NURBS) [45]. The B-spline extension to higher dimensions consists of multivariate spline functions based on a tensor product approach. Unfortunately, classical tensor product splines lack local refinement, which is often fundame
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