Superconvergent local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation

Sbibih, D.; Serghini, A.; Tijini, A.

doi:10.1016/j.amc.2014.10.090

Cited by 6 publications

(3 citation statements)

References 26 publications

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“…A rather simple generalization, known as Variation Diminishing Spline Approximation (VDSA), generalizes this construction to B-splines (see, for example [5,6]). Since its inception, quasi-interpolation has been studied to obtain methods that apply to different domains and with the aim of increasing the order of convergence: recent developments include univariate and tensorproduct spaces [7][8][9], triangular meshes [10][11][12][13], quadrangulations [14] and tetrahedra partitions [15], among others.…”

Section: Introductionmentioning

confidence: 99%

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

Raffo

Biasotti

2021

Mathematics

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The approximation of curvilinear profiles is very popular for processing digital images and leads to numerous applications such as image segmentation, compression and recognition. In this paper, we develop a novel semi-automatic method based on quasi-interpolation. The method consists of three steps: a preprocessing step exploiting an edge detection algorithm; a splitting procedure to break the just-obtained set of edge points into smaller subsets; and a final step involving the use of a local curve approximation, the Weighted Quasi Interpolant Spline Approximation (wQISA), chosen for its robustness to data perturbation. The proposed method builds a sequence of polynomial spline curves, connected C0 in correspondence of cusps, G1 otherwise. To curb underfitting and overfitting, the computation of local approximations exploits the supervised learning paradigm. The effectiveness of the method is shown with simulation on real images from various application domains.

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Section: Introductionmentioning

confidence: 99%

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

Raffo

Biasotti

2021

Mathematics

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“…In numerical analysis, the superconvergence is a phenomenon where the order of convergence of the approximant error at certain special points is higher than the order of convergence of the approximant error over the definition's domain (see [6,26,27,28,29]). Then by considering a local linear polynomial operator in the neighborhood of the support of the B-splines that reproduces the space of polynomials of degree at most m ≥ 2, we propose a method to bluid superconvergent discrete quasi-interpolants of a function f. It satisfies an interesting property that these quasi-interpolants are globally of order 3 and of order m + 1 at the knots of the initial partition τ .…”

Section: Introductionmentioning

confidence: 99%

Construction of a normalized basis of a univariate quadratic $C^1$ spline space and application to the quasi-interpolation

Rahouti

Serghini

Tijini

2021

bspm

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In this paper, we use the finite element method to construct a new normalized basis of a univariate quadratic $C^1$ spline space. We give a new representation of Hermite interpolant of any piecewise polynomial of class at least $C^1$ in terms of its polar form. We use this representation for constructing several superconvergent and super-superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones.

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“…Quasi-interpolation is a well known technique [18,19] that does not require to solve any linear system, unlike the traditional spline approaches, and therefore it allows to define more efficient algorithms. Whilst there are works on the use of quasi-interpolant spline methods for function approximation [18,20,21,22,23], to the best of our knowledge, only recently, few efforts have been devoted to define quasi-interpolant schemes for point clouds [24,25,26].…”

Section: Introductionmentioning

confidence: 99%

Weighted Quasi Interpolant Spline Approximations: Properties and Applications

Raffo,

Biasotti

2019

Preprint

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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.

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Superconvergent local quasi-interpolants based on special multivariate quadratic spline space over a refined quadrangulation

Cited by 6 publications

References 26 publications

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

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

Construction of a normalized basis of a univariate quadratic $C^1$ spline space and application to the quasi-interpolation

Weighted Quasi Interpolant Spline Approximations: Properties and Applications

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