×

A closed PP form of box splines via Green’s function decomposition. (English) Zbl 1392.41006

A set construction scheme is presented for the class of non-degenerate box splines. This scheme reparably decomposes the Green’s function of a box spline, yielding its explicit piecewise polynomial form. It is possible to use the well-known recursive formulation to obtain these polynomial pieces, but that procedure is quite expensive. The fast evaluation schemes using piecewise polynomial form of box splines are created. The spatial Fourier form of Green’s function and the spatial form of the difference operator are used. It is proved that, under certain conditions, the proposed decomposition procedure is of asymptotically orders of magnitude lower than the recursive procedure. This allows to evaluate box splines with more direction vectors than what would be feasible under the recursive scheme. The exemplary constructions for some known box splines are provided.

MSC:

41A15 Spline approximation
41A10 Approximation by polynomials

Software:

SageMath
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] de Boor, C., On the evaluation of box splines, Numer. Algorithms, 5, 1, 5-23 (1993) · Zbl 0798.65012
[2] de Boor, C.; Höllig, K.; Riemenschneider, S., Box Splines (1993), Springer-Verlag New York, Inc.: Springer-Verlag New York, Inc. New York, NY, USA · Zbl 0814.41012
[3] Entezari, A.; Van De Ville, D.; Moller, T., Practical box splines for reconstruction on the body centered cubic lattice, IEEE Trans. Vis. Comput. Graph., 14, 2, 313-328 (2008)
[4] Kim, M.; Entezari, A.; Peters, J., Box spline reconstruction on the Face-Centered Cubic lattice, IEEE Trans. Vis. Comput. Graph., 14, 6, 1523-1530 (2008)
[5] Kim, M.; Peters, J., Fast and stable evaluation of box-splines via the BB-form, Numer. Algorithms, 50, 4, 381-399 (2009) · Zbl 1162.65005
[6] Kim, M.; Peters, J., Symmetric Box-splines on the A*N Lattice, J. Approx. Theory, 162, 9, 1607-1630 (2010) · Zbl 1205.41009
[7] Kobbelt, L., Stable evaluation of box splines, Numer. Algorithms, 14, 4, 377-382 (1997) · Zbl 0885.65010
[8] Peña, J. M., On the multivariate Horner scheme, SIAM J. Numer. Anal., 37, 4, 1186-1197 (2000) · Zbl 0961.65007
[9] Peña, J. M.; Sauer, T., On the multivariate Horner scheme II: Running error analysis, Computing, 65, 4, 313-322 (2000) · Zbl 0984.65006
[10] A.S. William, et al., Sage Mathematics Software (Version 8.0.0) 2017. http://www.sagemath.org; A.S. William, et al., Sage Mathematics Software (Version 8.0.0) 2017. http://www.sagemath.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.