Quasi-interpolation and outliers removal

Amir, Anat; Levin, David

doi:10.1007/s11075-017-0401-2

Cited by 6 publications

(5 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have computed the approximation order in the specified different regions A 0 , A 1 , A 2 , A 3 and A 4 . More in concrete, in the case of region A 1 we use the interval [2,3] for the x variable, in the case of region A 2 the interval [4,5], and in the case of region A 3 the intervals [x d k +k , 2π], where k indicates the resolution level and the index d k is such that the inflexion point falls into the interval [x d k −1 , x d k ] for each k. We can observe that in the region A 0 both reconstructions are affected by the jump discontinuity and they lose the approximation order due mainly to the subinterval containing the discontinuity. In the region of type A 1 both reconstructions attain fourth order accuracy as expected.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Linear operators behave improperly in presence of jump discontinuities, so that different nonlinear operators have emerged to deal with this problematic. Recent approaches to deal with similar problems of functions affected by discontinuities can be found for example in [1][2][3][4][5]. And these nonlinear methods also give rise to interesting applications.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Ortiz¹,

Trillo²

2021

Mathematics

View full text Add to dashboard Cite

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.

show abstract

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Ortiz¹,

Trillo²

2021

Mathematics

View full text Add to dashboard Cite

show abstract

“…However, this method may eliminate some of non-outliers, in which case the accuracy of approximation will be significantly affected. 28 The other type of algorithm is to identify outliers first and then weaken the influence of outliers by assigning weights to them. In this method, it is difficult to determine appropriate weights when multiple outliers with different levels exist in the discrete data.…”

Section: Introductionmentioning

confidence: 99%

Surface reconstruction method for measurement data with outlier detection by using improved RANSAC and correction parameter

Luo

Tang

et al. 2022

Proceedings of the Institution of Mechanical Engineers, Part B:

View full text Add to dashboard Cite

The moving least squares (MLS) and moving total least squares (MTLS) are two of the most popular methods used for reconstructing measurement data, on account of their good local approximation accuracy. However, their reconstruction accuracy and robustness will be greatly reduced when there are outliers in measurement data. This article proposes an improved MTLS method (IMTLS), which introduces an improved random sample consensus (RANSAC) algorithm and a correction parameter in the support domain, to deal with the outliers and random errors. Based on the nodes within the support domain, firstly the improved RANSAC is used to generate a model for establishing the group of pre-interpolation and calculating the residual of each node. Subsequently, the abnormal degree of the node with the largest residual is evaluated by the correction parameter associated with the node residual and random errors. The node with certain abnormal degree will be eliminated and the remaining nodes are used to obtain the approximation coefficients. The correction parameter can be used for data reconstruction without insufficient or excessive elimination. The results of numerical simulation and measurement experiment show that the reconstruction accuracy and robustness of the IMTLS method is superior to the MLS and MTLS method.

show abstract

“…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 methods for function approximation [18,20,21,22,23,24], to the best of our knowledge, less efforts have been devoted to define quasi-interpolant schemes for point clouds [25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

Weighted quasi-interpolant spline approximations: Properties and applications

Raffo

Biasotti

2020

Numer Algor

View full text Add to dashboard Cite

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.

show abstract

Quasi-interpolation and outliers removal

Cited by 6 publications

References 7 publications

A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

Surface reconstruction method for measurement data with outlier detection by using improved RANSAC and correction parameter

Weighted quasi-interpolant spline approximations: Properties and applications

Contact Info

Product

Resources

About