Computation of Hermite interpolation in terms of B-spline basis using polar forms

Lamnii, A.; Lamnii, M.; Oumellal, Fatima

doi:10.1016/j.matcom.2016.09.009

Cited by 9 publications

(4 citation statements)

References 15 publications

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“…As in Lamnii et al, 17 one further knot is inserted in each interval [ x i , x i + 1 ] to produce a

{scriptC}^{1}

quadratic spline interpolant which interpolates the values of

f^{false(r false)} false(x_{i} false), i = 0, \dots, n_{k} - 1, r = 0,1

. More precisely, let

\overset{scriptX}{true}

be the new partition obtained by inserting new knots in

scriptX

, such that

\tilde{X} = \{\begin{array}{ll} {\overset{x}{true}}_{i} = - \frac{π}{2} + i h_{k}, & i = - 2, - 1, 2 n_{k} - 1, 2 n_{k}, \\ {\overset{x}{true}}_{2 i + 1} = - \frac{π}{2} + false(2 i + 1 false) h_{k}, & i = 0, \dots, n_{k} - 2, \\ {\overset{x}{true}}_{2 i} = x_{i}, & i = 0, \dots, n_{k} - 1, \end{array}

and let

S_{2}^{1} (I, \tilde{X}) = {s \in C^{1} (I) : s_{|_{[{\tilde{x}}_{i},}}

…”

Section: Quadratic Spline Interpolantmentioning

confidence: 78%

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A C1 composite spline Hermite interpolant on the sphere

Bouhiri

Lamnii

et al. 2021

Math Methods in App Sciences

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In this paper, we identified the sphere‐like surface by a rectangular domain. We constructed a new interpolant on the sphere using the tensor product of the quadratic composite‐spline interpolant and the third‐order trigonometric composite‐spline interpolant. This construction on the rectangular domain is described in detail, with the study and the proof of the error bounds of each interpolant. Furthermore, we present numerical examples to show the efficiency of this method.

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“…As in Lamnii et al, 17 one further knot is inserted in each interval [ x i , x i + 1 ] to produce a

{scriptC}^{1}

quadratic spline interpolant which interpolates the values of

f^{false(r false)} false(x_{i} false), i = 0, \dots, n_{k} - 1, r = 0,1

. More precisely, let

\overset{scriptX}{true}

be the new partition obtained by inserting new knots in

scriptX

, such that

\tilde{X} = \{\begin{array}{ll} {\overset{x}{true}}_{i} = - \frac{π}{2} + i h_{k}, & i = - 2, - 1, 2 n_{k} - 1, 2 n_{k}, \\ {\overset{x}{true}}_{2 i + 1} = - \frac{π}{2} + false(2 i + 1 false) h_{k}, & i = 0, \dots, n_{k} - 2, \\ {\overset{x}{true}}_{2 i} = x_{i}, & i = 0, \dots, n_{k} - 1, \end{array}

and let

S_{2}^{1} (I, \tilde{X}) = {s \in C^{1} (I) : s_{|_{[{\tilde{x}}_{i},}}

…”

Section: Quadratic Spline Interpolantmentioning

confidence: 78%

“…As in Lamnii et al, 17 one further knot is inserted in each interval [x i , x i + 1 ] to produce a  1 quadratic spline interpolant which interpolates the values of 𝑓 (r) (x i ), i = 0, … , n k − 1, r = 0, 1. More precisely, let  be the new partition obtained by inserting new knots in , such that…”

Section: Quadratic Spline Interpolantmentioning

confidence: 99%

A C1 composite spline Hermite interpolant on the sphere

Bouhiri

Lamnii

et al. 2021

Math Methods in App Sciences

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show abstract

“…It is common to fit the data collected with geometric forms or mathematical functions [13][14][15][16]. Because the point clouds from TLS measurement contain gaps, leaps, cusps, and so on, it is necessary to construct a fitting model to investigate the 3D deformation of a structure.…”

Section: Curve Fitting With Mathematical Functionsmentioning

confidence: 99%

TLS-Based Feature Extraction and 3D Modeling for Arch Structures

Zhao

Yang

et al. 2017

Journal of Sensors

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Terrestrial laser scanning (TLS) technology is one of the most efficient and accurate tools for 3D measurement which can reveal surface-based characteristics of objects with the aid of computer vision and programming. Thus, it plays an increasingly important role in deformation monitoring and analysis. Automatic data extraction and high efficiency and accuracy modeling from scattered point clouds are challenging issues during the TLS data processing. This paper presents a data extraction method considering the partial and statistical distribution of the point clouds scanned, called the window-neighborhood method. Based on the point clouds extracted, 3D modeling of the boundary of an arched structure was carried out. The ideal modeling strategy should be fast, accurate, and less complex regarding its application to large amounts of data. The paper discusses the accuracy of fittings in four cases between whole curve, segmentation, polynomial, and B-spline. A similar number of parameters was set for polynomial and B-spline because the number of unknown parameters is essential for the accuracy of the fittings. The uncertainties of the scanned raw point clouds and the modeling are discussed. This process is considered a prerequisite step for 3D deformation analysis with TLS.

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“…The Hermite spline interpolants' construction by adding some additional knots to the initial partition and increasing the polynomial pieces' number is not new. This method has been recently studied in the literature (see [17,32]) also by Lamnii et al [11]. The added knots can be chosen to preserve certain geometric shape such as monotonicity and convexity.…”

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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Computation of Hermite interpolation in terms of B-spline basis using polar forms

Cited by 9 publications

References 15 publications

A C1 composite spline Hermite interpolant on the sphere

A C1 composite spline Hermite interpolant on the sphere

TLS-Based Feature Extraction and 3D Modeling for Arch Structures

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

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