Scattered data interpolation on the 2-dimensional surface through Shepard-like technique

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dc.citation.epage289
dc.citation.issue11
dc.citation.journalTitleМатематичне моделювання та комп'ютинг
dc.citation.spage277
dc.citation.volume1
dc.contributor.affiliationУніверситет Ібн Зор
dc.contributor.affiliationУніверситет Абдельмалека Ессааді
dc.contributor.affiliationIbn Zohr University
dc.contributor.affiliationAbdelmalek Essaadi University
dc.contributor.authorЗерроді, Б.
dc.contributor.authorТайек, Х.
dc.contributor.authorЕль Харрак, А.
dc.contributor.authorZerroudi, B.
dc.contributor.authorTayeq, H.
dc.contributor.authorEl Harrak, A.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2025-10-20T07:44:19Z
dc.date.created2024-02-24
dc.date.issued2024-02-24
dc.description.abstractУ цій статті розглядається проблема інтерполяції розсіяних даних на двовимірних поверхнях шляхом пропозиції розширення методу Шепарда та його модифікованої версії для поверхонь. Кожен запропонований оператор є лінійною комбінацією базисних функцій, коефіцієнти яких є значеннями функції або її розкладів Тейлора першого порядку в точках інтерполяції з використанням як функціональних, так і похідних даних. Наведено числові тести для демонстрації ефективності інтерполяції, де кілька числових результатів показують хорошу точність апроксимації запропонованого оператора.
dc.description.abstractIn the current paper, the problem of interpolation of scattered data on two-dimensional surfaces is considered by proposing an extension to the Shepard method and its modified version to surfaces. Each proposed operator is a linear combination of basis functions whose coefficients are the values of the function or its Taylor of first-order expansions at the interpolation points using both functional and derivative data. Numerical tests are given to show the interpolation performance, where several numerical results show a good approximation accuracy of the proposed operator.
dc.format.extent277-289
dc.format.pages13
dc.identifier.citationZerroudi B. Scattered data interpolation on the 2-dimensional surface through Shepard-like technique / B. Zerroudi, H. Tayeq, A. El Harrak // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2024. — Vol 1. — No 11. — P. 277–289.
dc.identifier.citationenZerroudi B. Scattered data interpolation on the 2-dimensional surface through Shepard-like technique / B. Zerroudi, H. Tayeq, A. El Harrak // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2024. — Vol 1. — No 11. — P. 277–289.
dc.identifier.doi10.23939/mmc2024.01.277
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/113787
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofМатематичне моделювання та комп'ютинг, 11 (1), 2024
dc.relation.ispartofMathematical Modeling and Computing, 11 (1), 2024
dc.relation.references[1] Meijering E. A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proceedings of the IEEE. 90 (3), 319–342 (2002).
dc.relation.references[2] Shepard D. A two-dimensional interpolation function for irregularly-spaced data. Proceedings of the 1968 23rd ACM national conference. 517–524 (1968).
dc.relation.references[3] Liszka T. An interpolation method for an irregular net of nodes. International Journal for Numerical Methods in Engineering. 20 (9), 1599–1612 (1984).
dc.relation.references[4] McLain D. H. Drawing contours from arbitrary data points. The Computer Journal. 17 (4), 318–324 (1974).
dc.relation.references[5] Farwig R. Rate of convergence of Shepard’s global interpolation formula. Mathematics of Computation. 46 (174), 577–590 (1986).
dc.relation.references[6] Renka R. J., Brown R. Algorithm 792: Accuracy Tests of ACM Algorithms for Interpolation of Scattered Data in the Plane. ACM Transactions on Mathematical Software (TOMS). 25 (1), 78–94 (1999).
dc.relation.references[7] Thacker W. I., Zhang J., Watson L. T., Birch J. B., Iyer M. A., Berry M. W. Algorithm 905: SHEPPACK: Modified Shepard algorithm for interpolation of scattered multivariate data. ACM Transactions on Mathematical Software (TOMS). 37 (3), 1–20 (2010).
dc.relation.references[8] Karandashev K., Van´ıˇcek J. A combined on-the-fly/interpolation procedure for evaluating energy values needed in molecular simulations. The Journal of chemical physics. 151 (17), 174116 (2019).
dc.relation.references[9] Farrahi G. H., Faghidian S. A., Smith D. J. An inverse approach to determination of residual stresses induced by shot peening in round bars. International Journal of Mechanical Sciences. 51 (9–10), 726–731 (2009).
dc.relation.references[10] Alfeld P., Neamtu M., Schumaker L. L. Fitting scattered data on sphere-like surfaces using spherical splines. Journal of Computational and Applied Mathematics. 73 (1–2), 5–43 (1996).
dc.relation.references[11] Baramidze V., Lai M. I., Shum C. K. Spherical splines for data interpolation and fitting. SIAM Journal on Scientific Computing. 28 (1), 241–259 (2006).
dc.relation.references[12] Cavoretto R., De Rossi A. A spherical interpolation algorithm using zonal basis functions. Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE09). 1, 258–269 (2009).
dc.relation.references[13] Fasshauer G. E. Adaptive least squares fitting with radial basis functions on the sphere. Mathematical Methods for Curves and Surfaces. 141–150 (1995).
dc.relation.references[14] Fasshauer G. E., Schumaker L. L. Scattered data fitting on the sphere. Mathematical Methods for Curves and Surfaces II. 117–166 (1998).
dc.relation.references[15] Meyling R. H. J. G., Pfluger P. R. B-spline approximation of a closed surface. IMA Journal of Numerical Analysis. 7 (1), 73–96 (1987).
dc.relation.references[16] Pottmann H., Eck M. Modified multiquadric methods for scattered data interpolation over a sphere. Computer Aided Geometric Design. 7 (1–4), 313–321 (1990).
dc.relation.references[17] Sloan I. H., Womersley R. S. Constructive polynomial approximation on the sphere. Journal of Approximation Theory. 103 (1), 91–118 (2000).
dc.relation.references[18] Womersley R. S., Sloan I. H. How good can polynomial interpolation on the sphere be? Advances in Computational Mathematics. 14 (3), 195–226 (2001).
dc.relation.references[19] Allasia G., Cavoretto R., De Rossi A. Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds. Applied Mathematics and Computation. 318, 35–50 (2018).
dc.relation.references[20] Dyn N., Narcowich F. J., Ward J. D. A Framework for Interpolation and Approximation on Riemannian. Approximation Theory and Optimization: Tributes to MJD Powell. 133–144 (1997).
dc.relation.references[21] Dyn N., Narcowich F. J., Ward J. D. Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold. Constructive Approximation. 15 (2), 175–208 (1999).
dc.relation.references[22] Narcowich F. J. Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold. Journal of Mathematical Analysis and Applications. 190 (1), 165–193 (1995).
dc.relation.references[23] Horemuˇz M., Andersson J. V. Polynomial interpolation of GPS satellite coordinates. GPS Solutions. 10 (1), 67–72 (2006).
dc.relation.references[24] Alexander R., Alexander S. Geodesics in Riemannian manifolds-with-boundary. Indiana University Mathematics Journal. 30 (4), 481–488 (1981).
dc.relation.references[25] Corral M. Vector calculus. Independent (2013).
dc.relation.references[26] Renka R. J. Multivariate interpolation of large sets of scattered data. ACM Transactions on Mathematical Software. 14 (2), 139–148 (1988).
dc.relation.references[27] Franke R. Scattered data interpolation: Tests of some methods. Mathematics of Computation. 38 (157), 81–200 (1982).
dc.relation.references[28] Hubbert S., Morton T. M. Lp-error estimates for radial basis function interpolation on the sphere. Journal of Approximation Theory. 129 (1), 58–77 (2004).
dc.relation.references[29] Nouisser O., Zerroudi B. Modified Shepard’s method by six-points local interpolant. Journal of Applied Mathematics and Computing. 65 (1), 651–667 (2021).
dc.relation.references[30] Todhunter I. Spherical trigonometry, for the use of colleges and schools: with numerous examples. Macmillan (1863).
dc.relation.references[31] Boykov Y., Kolmogorov V. Computing geodesics and minimal surfaces via graph cuts. Proceedings Ninth IEEE International Conference on Computer Vision. 3, 26–33 (2003).
dc.relation.references[32] Baek J., Deopurkar A., Redfield K. Finding Geodesics on Surfaces. Dept. Comput. Sci., Stanford Univ., Stanford, CA, USA, Tech. Rep. (2007).
dc.relation.referencesen[1] Meijering E. A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proceedings of the IEEE. 90 (3), 319–342 (2002).
dc.relation.referencesen[2] Shepard D. A two-dimensional interpolation function for irregularly-spaced data. Proceedings of the 1968 23rd ACM national conference. 517–524 (1968).
dc.relation.referencesen[3] Liszka T. An interpolation method for an irregular net of nodes. International Journal for Numerical Methods in Engineering. 20 (9), 1599–1612 (1984).
dc.relation.referencesen[4] McLain D. H. Drawing contours from arbitrary data points. The Computer Journal. 17 (4), 318–324 (1974).
dc.relation.referencesen[5] Farwig R. Rate of convergence of Shepard’s global interpolation formula. Mathematics of Computation. 46 (174), 577–590 (1986).
dc.relation.referencesen[6] Renka R. J., Brown R. Algorithm 792: Accuracy Tests of ACM Algorithms for Interpolation of Scattered Data in the Plane. ACM Transactions on Mathematical Software (TOMS). 25 (1), 78–94 (1999).
dc.relation.referencesen[7] Thacker W. I., Zhang J., Watson L. T., Birch J. B., Iyer M. A., Berry M. W. Algorithm 905: SHEPPACK: Modified Shepard algorithm for interpolation of scattered multivariate data. ACM Transactions on Mathematical Software (TOMS). 37 (3), 1–20 (2010).
dc.relation.referencesen[8] Karandashev K., Van´ıˇcek J. A combined on-the-fly/interpolation procedure for evaluating energy values needed in molecular simulations. The Journal of chemical physics. 151 (17), 174116 (2019).
dc.relation.referencesen[9] Farrahi G. H., Faghidian S. A., Smith D. J. An inverse approach to determination of residual stresses induced by shot peening in round bars. International Journal of Mechanical Sciences. 51 (9–10), 726–731 (2009).
dc.relation.referencesen[10] Alfeld P., Neamtu M., Schumaker L. L. Fitting scattered data on sphere-like surfaces using spherical splines. Journal of Computational and Applied Mathematics. 73 (1–2), 5–43 (1996).
dc.relation.referencesen[11] Baramidze V., Lai M. I., Shum C. K. Spherical splines for data interpolation and fitting. SIAM Journal on Scientific Computing. 28 (1), 241–259 (2006).
dc.relation.referencesen[12] Cavoretto R., De Rossi A. A spherical interpolation algorithm using zonal basis functions. Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE09). 1, 258–269 (2009).
dc.relation.referencesen[13] Fasshauer G. E. Adaptive least squares fitting with radial basis functions on the sphere. Mathematical Methods for Curves and Surfaces. 141–150 (1995).
dc.relation.referencesen[14] Fasshauer G. E., Schumaker L. L. Scattered data fitting on the sphere. Mathematical Methods for Curves and Surfaces II. 117–166 (1998).
dc.relation.referencesen[15] Meyling R. H. J. G., Pfluger P. R. B-spline approximation of a closed surface. IMA Journal of Numerical Analysis. 7 (1), 73–96 (1987).
dc.relation.referencesen[16] Pottmann H., Eck M. Modified multiquadric methods for scattered data interpolation over a sphere. Computer Aided Geometric Design. 7 (1–4), 313–321 (1990).
dc.relation.referencesen[17] Sloan I. H., Womersley R. S. Constructive polynomial approximation on the sphere. Journal of Approximation Theory. 103 (1), 91–118 (2000).
dc.relation.referencesen[18] Womersley R. S., Sloan I. H. How good can polynomial interpolation on the sphere be? Advances in Computational Mathematics. 14 (3), 195–226 (2001).
dc.relation.referencesen[19] Allasia G., Cavoretto R., De Rossi A. Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds. Applied Mathematics and Computation. 318, 35–50 (2018).
dc.relation.referencesen[20] Dyn N., Narcowich F. J., Ward J. D. A Framework for Interpolation and Approximation on Riemannian. Approximation Theory and Optimization: Tributes to MJD Powell. 133–144 (1997).
dc.relation.referencesen[21] Dyn N., Narcowich F. J., Ward J. D. Variational principles and Sobolev-type estimates for generalized interpolation on a Riemannian manifold. Constructive Approximation. 15 (2), 175–208 (1999).
dc.relation.referencesen[22] Narcowich F. J. Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold. Journal of Mathematical Analysis and Applications. 190 (1), 165–193 (1995).
dc.relation.referencesen[23] Horemuˇz M., Andersson J. V. Polynomial interpolation of GPS satellite coordinates. GPS Solutions. 10 (1), 67–72 (2006).
dc.relation.referencesen[24] Alexander R., Alexander S. Geodesics in Riemannian manifolds-with-boundary. Indiana University Mathematics Journal. 30 (4), 481–488 (1981).
dc.relation.referencesen[25] Corral M. Vector calculus. Independent (2013).
dc.relation.referencesen[26] Renka R. J. Multivariate interpolation of large sets of scattered data. ACM Transactions on Mathematical Software. 14 (2), 139–148 (1988).
dc.relation.referencesen[27] Franke R. Scattered data interpolation: Tests of some methods. Mathematics of Computation. 38 (157), 81–200 (1982).
dc.relation.referencesen[28] Hubbert S., Morton T. M. Lp-error estimates for radial basis function interpolation on the sphere. Journal of Approximation Theory. 129 (1), 58–77 (2004).
dc.relation.referencesen[29] Nouisser O., Zerroudi B. Modified Shepard’s method by six-points local interpolant. Journal of Applied Mathematics and Computing. 65 (1), 651–667 (2021).
dc.relation.referencesen[30] Todhunter I. Spherical trigonometry, for the use of colleges and schools: with numerous examples. Macmillan (1863).
dc.relation.referencesen[31] Boykov Y., Kolmogorov V. Computing geodesics and minimal surfaces via graph cuts. Proceedings Ninth IEEE International Conference on Computer Vision. 3, 26–33 (2003).
dc.relation.referencesen[32] Baek J., Deopurkar A., Redfield K. Finding Geodesics on Surfaces. Dept. Comput. Sci., Stanford Univ., Stanford, CA, USA, Tech. Rep. (2007).
dc.rights.holder© Національний університет “Львівська політехніка”, 2024
dc.subjectінтерполяція розсіяних даних
dc.subjectалгоритми інтерполяції
dc.subjectметоди Шепарда
dc.subjectапроксимація многовидів
dc.subjectповерхневий трикутник
dc.subjectscattered data interpolation
dc.subjectinterpolation algorithms
dc.subjectShepard methods
dc.subjectmanifolds approximation
dc.subjectsurface triangle
dc.titleScattered data interpolation on the 2-dimensional surface through Shepard-like technique
dc.title.alternativeІнтерполяція розсіяних даних на двовимірній поверхні за допомогою методу Шепарда
dc.typeArticle

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Mathematical Modeling And Computing. – 2024. – Vol. 11, No. 1