On the hexagonal Shepard method

Dell’Accio, Francesco; Tommaso, Filomena Di

doi:10.1016/j.apnum.2019.09.005

Cited by 18 publications

(12 citation statements)

References 18 publications

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“…In line with [1, Equation (1-5)] we represent f (x) and f (x i ) = y i in truncated Taylor series of order d centered at x (10) with integral remainder (11) and we obtain…”

Section: Remark 1 Denoting Bymentioning

confidence: 99%

“…The previous results are useful to estimate the error of approximation in several processes of scattered data interpolation which use polynomials as local interpolants, like for example triangular Shepard [4,11], hexagonal Shepard [10] and tetrahedral Shepard methods [5]. They are also crucial to realize extensions of those methods to higher dimensions [9].…”

Section: Remarkmentioning

confidence: 99%

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Numerical differentiation on scattered data through multivariate polynomial interpolation

et al. 2021

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We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

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“…In line with [1, Equation (1-5)] we represent f (x) and f (x i ) = y i in truncated Taylor series of order d centered at x (10) with integral remainder (11) and we obtain…”

Section: Remark 1 Denoting Bymentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Numerical differentiation on scattered data through multivariate polynomial interpolation

et al. 2021

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“…For example, error estimates, direct and converse theorems, saturation results, rational approximation, simultaneous approximation may be found in [14][15][16]31,[34][35][36]. There are also several modifications of the original Shepard operators in order to increase the accuracy of approximation or to solve specific interpolation problems in CAGD such as scattered data and image compression [2,6,7,9,10,[17][18][19][20][21][22]24]. However, to the best of our knowledge, there is a limited information in the literature about a complex version of Shepard operators.…”

Section: Introductionmentioning

confidence: 99%

Complex Shepard Operators and Their Summability

Duman

Vecchia

2021

Results Math

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In this paper, we construct the complex Shepard operators to approximate continuous and complex-valued functions on the unit square. We also examine the effects of regular summability methods on the approximation by these operators. Some applications verifying our results are provided. To illustrate the approximation theorems graphically we consider the real or imaginary part of the complex function being approximated and also use the contour lines of the modulus of the function.

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“…Shepard triangular can also be used to produce a smooth surface from regular and irregular scattered data. For instance, Cavoretto et al [2], Dell'Accio and Di Tommaso [5] and Dell' Accio et al [6], [7] have discussed the application of Shepard triangular in surface reconstruction. But the main drawback is that their schemes require more computation time in order to produce the final interpolating surface (Karim et al [21]).…”

Section: Introductionmentioning

confidence: 99%

Scattered Data Interpolation Using Rational Quartic Triangular Patches With Three Parameters

2020

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This paper discusses the construction of scattered data interpolation scheme based on rational quartic triangular patches with C 1 continuity. The C 1 sufficient condition is derived on each adjacent triangle. We employ rational corrected scheme comprising three local schemes defined on each triangle. The final scheme is then being applied for scattered data interpolation. In this study, we tested the proposed scheme on regular and irregular scattered data. For regular data, we used 36, 65 and 100 data sets, while for irregular data, we used some real data sets. Besides that, we compared the performance between the proposed scheme with cubic Ball and cubic Bézier schemes. We measure the error analysis by calculating the Root Mean Square Error (RMSE), maximum error (Max error), coefficient of determination (R 2) and CPU time (in seconds). Based on the comparisons, our proposed scheme performs better than the other two existing schemes. Furthermore, for large scattered data sets, the proposed scheme requires less computation time than existing schemes. The free parameters in the description of the proposed scheme provide greater flexibility in controlling the quality of the final interpolating surface. This is very useful in the designing processes. INDEX TERMS Rational quartic triangular patches, C 1 continuity, sufficient condition.

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On the hexagonal Shepard method

Cited by 18 publications

References 18 publications

Numerical differentiation on scattered data through multivariate polynomial interpolation

Numerical differentiation on scattered data through multivariate polynomial interpolation

Complex Shepard Operators and Their Summability

Scattered Data Interpolation Using Rational Quartic Triangular Patches With Three Parameters

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