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Multilevel refinable triangular PSP-splines (Tri-PSPS)

Li, Qingde; Tian, Jie

Authors

Jie Tian



Abstract

A multi-level spline technique known as partial shape preserving splines (PSPS) (Li and Tian, 2011) has recently been developed for the design of piecewise polynomial freeform geometric surfaces, where the basis functions of the PSPS can be directly built from an arbitrary set of polygons that partitions a giving parametric domain. This paper addresses a special type of PSPS, the triangular PSPS (Tri-PSPS), where all spline basis functions are constructed from a set of triangles. Compared with other triangular spline techniques, Tri-PSPS have several distinctive features. Firstly, for each given triangle, the corresponding spline basis function for any required degree of smoothness can be expressed in closed-form and directly written out in full explicitly as piecewise bivariate polynomials. Secondly, Tri-PSPS are an additive triangular spline technique, where the spline function built from a given triangle can be replaced with a set of refined spline functions built on a set of smaller triangles that partition the initial given triangle. In addition, Tri-PSPS are a multilevel spline technique, Tri-PSPS surfaces can be designed to have a continuously varying levels of detail, achieved simply by specifying a proper value for the smoothing parameter introduced in the spline functions. In terms of practical implementation, Tri-PSPS are a parallel computing friendly spline scheme, which can be easily implemented on modern programmable GPUs or on high performance computer clusters, since each of the basis functions of Tri-PSPS can be directly computed independent of each other in parallel.

Publication Date 2015-10
Journal Computers and mathematics with applications
Print ISSN 0898-1221
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 70
Issue 8
Pages 1781-1798
Institution Citation Li, Q., & Tian, J. (2015). Multilevel refinable triangular PSP-splines (Tri-PSPS). Computers & mathematics with applications, 70(8), 1781-1798. https://doi.org/10.1016/j.camwa.2015.07.017
DOI https://doi.org/10.1016/j.camwa.2015.07.017
Keywords Triangular splines; Refinable splines; Spline basis functions; Multilevel splines; Partition of unity; Spline approximation
Publisher URL http://www.sciencedirect.com/science/article/pii/S0898122115003521
Copyright Statement © 2016, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/
Additional Information Authors' accepted manuscript of an article published in: Computers and mathematics with applications, 2015, v.70, issue 8.

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Copyright Statement
© 2016, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/





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