A Family of 5-Point Nonlinear Ternary Interpolating Subdivision Schemes with C2 Smoothness

Aslam, Muhammad

doi:10.3390/mca23020018

Cited by 3 publications

(7 citation statements)

References 9 publications

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“…In this article, we proposed a general formula for (2n−1)-point nonlinear ternary interpolating subdivision schemes. It is shown that 3-point and 5-point nonlinear ternary subdivision schemes developed in ( [4], [5]) are special cases to our proposed schemes. Convergence of another special case (5-point nonlinear ternary interpolating subdivision scheme) is proved and it is shown that it is at least C 2 continuous.…”

Section: Resultsmentioning

confidence: 98%

“…The above equation can be simplified to the the following subdivision scheme of [5] which is C 2 for 1 324 < w < 1 162 .…”

Section: )mentioning

confidence: 99%

“…To avoid Gibbs phenomenon while keeping all initial data points in the limiting curve, nonlinear subdivision Schemes like ENO, WENO, PPH and PCHIP ( [1], [2], [3], [4], [5], [6], [8], [9], [10]) were introduced during last several years. Recently we have introduced two families of nonlinear ternary interpolating subdivision schemes to reduce undesirable oscillations.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], we introduced a family of 3-point nonlinear ternary interpolating subdivision schemes and proved that it is C 1 continuous. In [5], we developed a family of 5-point nonlinear ternary interpolating subdivision schemes and showed that this family is C 2 continuous. In this article, we introduce a general form of (2n−1)-point nonlinear ternary interpolating subdivision schemes and show that several of existing subdivision schemes including ( [4], [5]) become special cases of our proposed family of schemes.…”

Section: Introductionmentioning

confidence: 99%

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$(2n-1)$-POINT NONLINEAR TERNARY INTERPOLATING SUBDIVISION SCHEMES

Aslam¹

2018

Int. J. of Appl. Math.

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Nonlinear interpolating subdivision schemes have been introduced in recent years to reduce Gibbs phenomenon near irregular initial data points. In this article, we presented a class of (2n − 1)-point nonlinear ternary interpolating subdivision schemes. It is shown that several of the existing nonlinear ternary interpolating subdivision schemes become special cases for our proposed class of schemes. Convergence for one special case of 5-point nonlinear interpolating subdivision schemes is proved.

show abstract

Section: Resultsmentioning

confidence: 98%

“…The above equation can be simplified to the the following subdivision scheme of [5] which is C 2 for 1 324 < w < 1 162 .…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

$(2n-1)$-POINT NONLINEAR TERNARY INTERPOLATING SUBDIVISION SCHEMES

Aslam¹

2018

Int. J. of Appl. Math.

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“…The nonlinear subdivision schemes that have been applied in practical situations until now (see for example [1,2,3,4,5,9,10]) are usually compatible with either the point-value framework or the cell-average framework. Since our interest for the present work lays outside of these frameworks, we will focus only on the linear case.…”

Section: Introductionmentioning

confidence: 99%

A Harten's Multiresolution Framework for Subdivision Schemes

López-Ureña¹

2019

Preprint

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Harten's Multiresolution framework has been applied in different contexts, such as in the numerical simulation of PDE with conservation laws or in image compression, showing its flexibility to describe and manipulate the data in a multilevel fashion. Two basic operators form the basis of this theory: the decimation and the prediction. The decimation is chosen first and determines the type of data that is being manipulated. For instance, the data could be the point evaluations or the cell-averages of a function, which are the two classical environments. Next, the prediction is chosen, and it must be compatible with the decimation.Subdivision schemes can be used as prediction operators, but sometimes they not fit into one of the two typical environments. In this paper we show how to invert this order so we can choose a prediction first and then define a compatible decimation from that prediction. Moreover, we also prove that any possible decimation can be obtained in this way.

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A Computational Method for Subdivision Depth of Ternary Schemes

et al. 2020

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Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

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A Family of 5-Point Nonlinear Ternary Interpolating Subdivision Schemes with C2 Smoothness

Cited by 3 publications

References 9 publications

$(2n-1)$-POINT NONLINEAR TERNARY INTERPOLATING SUBDIVISION SCHEMES

$(2n-1)$-POINT NONLINEAR TERNARY INTERPOLATING SUBDIVISION SCHEMES

A Harten's Multiresolution Framework for Subdivision Schemes

A Computational Method for Subdivision Depth of Ternary Schemes

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