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Bivariate interpolation based on univariate subdivision schemes

Nir Sharon, Nira Dyn
2012 Journal of Approximation Theory  
The paper presents a bivariate subdivision scheme interpolating data consisting of univariate functions along equidistant parallel lines by repeated refinements.  ...  The bivariate subdivision scheme can be reduced to a countable number of univariate, interpolatory, non-stationary subdivision schemes.  ...  Acknowledgments The authors thank the anonymous referees for valuable comments and suggestions which helped improve the manuscript.  ... 
doi:10.1016/j.jat.2012.02.002 fatcat:psipaq2xz5bbdj3cdo5ez7gjku

Improved accuracy of multiquadric interpolation using variable shape parameters

E.J. Kansa, R.E. Carlson
1992 Computers and Mathematics with Applications  
An important unsolved problem is finding a method for computing the optimal value of R 2 for CMQ.  ...  They developed an empirical relation for computing R 2, and no theory exists for computing the optimal R ~ for CMQ.  ...  BIVARIATE FUNCTIONS Franke [9] used a collection of six bivariate functions and three scattered data sets to compare the performance of a number of scattered data interpolation schemes.  ... 
doi:10.1016/0898-1221(92)90174-g fatcat:ichc4mvvkzhghh6s4z4vi4dpve

Page 2289 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews  
Taylor, Monotone rational approximation on finite sets (265-272); Theodore Kilgore, Polynomial and function inequalities for derivatives (273-280); Jiirgen Koch, Algorithm for geometric Hermite interpolation  ...  Jean Savoie and Frangois Dubeau, On optimal error bounds for spline interpolation on a uniform partition (485-490); Robert Schaback, Multivariate interpolation and approximation by trans- lates of a basis  ... 

Page 2966 of Mathematical Reviews Vol. , Issue 88f [page]

1988 Mathematical Reviews  
Interpolation by bivariate splines in S}(A) is considered for the case in which A; is the triangulation of type one of [0, N]?. The regularity of a Hermite-type interpolation scheme is shown.  ...  P. (6-RANI); Rana, S.S. (6-RANI) Local behaviour of the derivative of a mid point cubic spline interpolator. Internat. J. Math. Math. Sci. 10 (1987), no. 1, 63-67.  ... 

Page 7039 of Mathematical Reviews Vol. , Issue 92m [page]

1992 Mathematical Reviews  
{For the entire collection see MR 92k:65005.} 92m:65033 65D17 65D05 Costantini, Paolo (I-SIN); Manni, Carla (I-PISA-IF) A local scheme for bivariate co-monotone interpolation. Comput. Aided Geom.  ...  The authors provide a local method for the construction of piece- wise polynomial surfaces interpolating a function and its first Numerical approximation 92m:65037 derivatives on a rectangular grid.  ... 

Positivity Preserving Interpolation Using Rational Bicubic Spline

Samsul Ariffin Abdul Karim, Kong Voon Pang, Azizan Saaban
2015 Journal of Applied Mathematics  
This paper discusses the positivity preserving interpolation for positive surfaces data by extending theC1rational cubic spline interpolant of Karim and Kong to the bivariate cases.  ...  The sufficient conditions for the positivity are derived on every four boundary curves network on the rectangular patch. Numerical comparison with existing schemes also has been done in detail.  ...  Acknowledgment The first author would like to acknowledge Universiti Teknologi PETRONAS (UTP) for the financial support received in the form of a research grant: Short Term Internal Research Funding (STIRF  ... 
doi:10.1155/2015/572768 fatcat:4h2szjff7zdcdahio5p7mbntd4

Continuous Histograms for Anisotropy of 2D Symmetric Piece-wise Linear Tensor Fields [article]

Talha Bin Masood, Ingrid Hotz
2019 arXiv   pre-print
Our derivation is based on a topological sub-division of the mesh in triangles that exhibit a monotonic behavior.  ...  We compare the results to a na\"ive approach based on linear interpolation on the original mesh or the subdivision.  ...  cos θA) 2 + θr 2 2 − r 2 sin θ cos θ 2 − θAr 2 AB 2 − r 2 AB sin θA cos θA 2 = θr 2 − θAr 2 AB 2 + rrAB sin θA cos θ − cos θA sin θ 2 .  ... 
arXiv:1912.00739v1 fatcat:i3m6bogs2fephhgwv3ydst7edm

On a class of polynomial triangular macro-elements

Paolo Costantini, Carla Manni
1996 Journal of Computational and Applied Mathematics  
In this paper \ve present a neM class of polynomial triangular macro-elements of arbitrary degree which are an extension of the classical Clough-Tocher cubic scheme.  ...  Their most important property is that the degree plays the role of a tension parameter. since these macro elements tend to the plane interpolating the vertices data.  ...  Sablonniere for the helpful discussions we had with them at different stages of our work.  ... 
doi:10.1016/0377-0427(96)00033-7 fatcat:svqg4imx6zemrjyrr2o6xypqbm

Construction of C1 Rational Bi-Quartic Spline With Positivity-Preserving Interpolation: Numerical Results and Analysis

Samsul Ariffin Abdul Karim, Azizan Saaban, Van Thien Nguyen
2021 Frontiers in Physics  
This finding is significant in producing a better interpolating surface for computer graphics applications since the proposed scheme has a smaller error compared with existing schemes.  ...  We also show that the proposed scheme is local quadratic reproducing such that it can exactly reproduce the quadratic surface.  ...  [30] described a new bivariate rational quartic spline (quartic/quadratic) for positivity-and monotonicity-preserving interpolation.  ... 
doi:10.3389/fphy.2021.555517 fatcat:yctrux74cbf3nav7gn37xmvcne

Monotone Data Visualization Using Rational Trigonometric Spline Interpolation

Farheen Ibraheem, Maria Hussain, Malik Zawwar Hussain
2014 The Scientific World Journal  
Rational cubic and bicubic trigonometric schemes are developed to conserve monotonicity of curve and surface data, respectively.  ...  The monotonicity of curve and surface data is retained by developing constraints on some of these parameters in description of rational cubic and bicubic trigonometric functions.  ...  Monotone surfaces in Figures 6 and 8 are produced by interpolating the same data by the monotonicity preserving scheme developed in Section 5.  ... 
doi:10.1155/2014/602453 pmid:24955413 pmcid:PMC3998008 fatcat:yfnkm33bmnfq3fgtjrshbxmqii

Nonlinear subdivision through nonlinear averaging

S. Schaefer, E. Vouga, R. Goldman
2008 Computer Aided Geometric Design  
More generally, we show that stationary subdivision schemes for e p(x) , cos(p(x)) and sin(p(x)) for any polynomial or piecewise polynomial p(x) can be generated using only addition, subtraction, multiplication  ...  locally invertible nonlinear map.  ...  Acknowledgements We would like to thank Nira Dyn for her help and advice with this paper as well as for suggesting [11] as a reference.  ... 
doi:10.1016/j.cagd.2007.07.003 fatcat:kdtze4qrnvchpf7txw6pp7do2q

C¹ Positive Surface over Positive Scattered Data Sites

Farheen Ibraheem, Malik Zawwar Hussain, Akhlaq Ahmad Bhatti, Cheng-Yi Xia
2015 PLoS ONE  
The aim of this paper is to develop a local positivity preserving scheme when the data amassed from different sources is positioned at sparse points.  ...  Half of the parameters in the description of the interpolant are constrained to keep up the positive shape of data while the remaining half are set free for users' requirement.  ...  The proposed scheme was local and C 1 continous. Hussain and Hussain [10] arranged the scattered data over a triangular grid to preserve the positivity and monotonicity.  ... 
doi:10.1371/journal.pone.0120658 pmid:26057122 pmcid:PMC4461286 fatcat:abfpklo5a5cbjpm32tmfrig5ci

Page 1383 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
The author provides algorithms which include fast univari- ate spline algorithms, algorithms for bivariate spline wavelets on sparse grids, interpolation algorithms for sparse grids, decomposi- tion algorithms  ...  Summary: “In this paper a strategy is presented to construct a shape-preserving quasi-interpolant function expressed as a linear combination of quadratic splines with local support where the coefficients  ... 

Continuous Histograms for Anisotropy of 2D Symmetric Piece-Wise Linear Tensor Fields [chapter]

Talha Bin Masood, Ingrid Hotz
2021 Mathematics and Visualization  
Our derivation is based on a sub-division of the mesh in triangles that exhibit a monotonic behavior.  ...  We compare the results to a naïve approach based on linear interpolation on the original mesh or the subdivision.  ...  , for fruitful discussions.  ... 
doi:10.1007/978-3-030-56215-1_3 fatcat:3mm5nggvufh2lc5dvczt2bb6sq

Multivariate empirical mode decomposition

N. Rehman, D. P. Mandic
2009 Proceedings of the Royal Society A  
a prerequisite for direct multichannel data analysis.  ...  An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals.  ...  The signal envelopes are produced by using componentwise spline interpolation, and the local maxima and minima are then averaged to obtain the local mean of the bivariate signal.  ... 
doi:10.1098/rspa.2009.0502 fatcat:uuqegpxeb5dgbopzogy2c3wjei
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