Abstract
We present a data reduction scheme for efficient surface storage, by introducing a coefficient–based least squares spline operator that does not require any pointwise evaluation to approximate (in a lower dimension spline space) a given bivariate B–spline function. In order to define an accurate approximation of the target spline with a significant reduction of the space dimension, this operator is subsequently combined with the hierarchical spline framework to design an adaptive method that exploits the capabilities of truncated hierarchical B–splines (THB–splines). The resulting THB–spline simplification approach is validated by several numerical tests. The target B–spline surfaces include approximations of functions whose analytical expression is available, reconstructions of geographic data and parametric surfaces.
Keywords
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Berdinsky, D., Kim, T.-W., Bracco, C., Cho, D., Mourrain, B., Oh, M.-J., Kiatpanichgij, S.: Dimensions and bases of hierarchical tensor-product splines. J. Comput. Appl. Math. 257, 86–104 (2014)
Bracco, C., Giannelli, C., Mazzia, F., Sestini, A.: Bivariate hierarchical Hermite spline quasi-interpolation. BIT 56, 1165–1188 (2016)
Conti, C., Morandi, R., Rabut, C., Sestini, A.: Cubic spline data reduction choosing the knots from a third derivative criterion. Numer. Algorithms 28, 45–61 (2001)
de Boor, C.: A Practical Guide to Splines. Springer, New York (2001). Revised ed
Deng, J., Chen, F., Feng, Y.: Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math. 194, 267–283 (2006)
Deng, J., Chen, F., Li, X., Hu, C., Tong, W., Yang, Z., Feng, Y.: Polynomial splines over hierarchical T-meshes. Graph. Models 70, 76–86 (2008)
Dokken, T., Lyche, T., Pettersen, K.F.: Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Des. 30, 331–356 (2013)
Forsey, D.R., Bartels, R.H.: Hierarchical B-spline refinement. Comput. Graphics 22, 205–212 (1988)
Forsey, D.R., Bartels, R.H.: Surface fitting with hierarchical splines. ACM Trans. Graphics 14, 134–161 (1995)
Forsey, D.R., Wong, D.: Multiresolution surface reconstruction for hierarchical B-splines. In: Graphics, Interface, pp. 57–64 (1998)
Garau, E.M., Vázquez, R.: Algorithms for the implementation of adaptive isogeometric methods using hierarchical splines. Appl. Numer. Math. 123, 58–87 (2018)
Giannelli, C., Jüttler, B., Speleers, H.: THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Des. 29, 485–498 (2012)
Giannelli, C., Jüttler, B., Speleers, H.: Strongly stable bases for adaptively refined multilevel spline spaces. Adv. Comput. Math. 40, 459–490 (2014)
Greiner, G., Hormann, K.: Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 163–172. Vanderbilt University Press, Nashville (1997)
Kiss, G., Giannelli, C., Zore, U., Jüttler, B., Großmann, D., Barner, J.: Adaptive CAD model (re-)construction with THB-splines. Graph. Models 76, 273–288 (2014)
Kraft, R.: Adaptive and linearly independent multilevel B-splines. In: Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.) Surface Fitting and Multiresolution Methods, pp. 209–218. Vanderbilt University Press, Nashville (1997)
Lyche, T., Mørken, K.: A data-reduction strategy for splines with applications to the approximation of functions and data. IMA J. Numer. Anal. 8, 185–208 (1988)
Mazzia, F., Sestini, A.: The BS class of Hermite spline quasi-interpolants on nonuniform knot distribution. BIT 49, 611–628 (2009)
Mokriš, D., Jüttler, B., Giannelli, C.: On the completeness of hierarchical tensor-product B-splines. J. Comput. Appl. Math. 271, 53–70 (2014)
Morandi, R., Sestini, A.: Data reduction in surface approximation. In: Lyche, T., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces: Oslo 2000, pp. 315–324. Vanderbilt University Press, Nashville (2001)
Sederberg, T.W., Cardon, D.L., Finnigan, G.T., North, N.S., Zheng, J., Lyche, T.: T-spline simplification and local refinement. ACM Trans. Graphics 23, 276–283 (2004)
Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCS. ACM Trans. Graphics 22, 477–484 (2003)
Skytt, V., Barrowclough, O., Dokken, T.: Locally refined spline surfaces for representation of terrain data. Comput. Graphics 49, 58–68 (2015)
Speleers, H.: Hierarchical spline spaces: quasi-interpolants and local approximation estimates. Adv. Comput. Math. 43, 235–255 (2017)
Speleers, H., Manni, C.: Effortless quasi-interpolation in hierarchical spaces. Numer. Math. 132, 155–184 (2016)
U.S. Geological Survey. https://www.usgs.gov/, http://dds.cr.usgs.gov/pub/data/nationalatlas/el_usa_hawaii.bil_nt00924.tar.gz
Vázquez, R.: A new design for the implementation of isogeometric analysis in Octave and Matlab: GeoPDEs 3.0. Comput. Math. Appl. 72, 523–554 (2016)
Vuong, A.-V., Giannelli, C., Jüttler, B., Simeon, B.: A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200, 3554–3567 (2011)
Wever, U.A.: Global and local data reduction strategies for cubic splines. Comput. Aided Des. 23, 127–132 (1991)
Acknowledgements
The support by MIUR “Futuro in Ricerca” programme through the project DREAMS (RBFR13FBI3) and by the Istituto Nazionale di Alta Matematica (INdAM) through Gruppo Nazionale per il Calcolo Scientifico (GNCS)—“Finanziamento Giovani Ricercatori” and “Progetti di ricerca” programmes—and Finanziamenti Premiali SUNRISE are gratefully acknowledged.
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Bracco, C., Giannelli, C., Sestini, A. (2017). Coefficient–Based Spline Data Reduction by Hierarchical Spaces. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science(), vol 10521. Springer, Cham. https://doi.org/10.1007/978-3-319-67885-6_2
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