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Multilevel refinable triangular PSP-splines (Tri-PSPS)

Li, Qingde; Tian, Jie


Jie Tian


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.


Li, Q., & Tian, J. (2015). Multilevel refinable triangular PSP-splines (Tri-PSPS). Computers & mathematics with applications, 70(8), 1781-1798.

Acceptance Date Jul 26, 2015
Online Publication Date Aug 24, 2015
Publication Date 2015-10
Deposit Date Sep 29, 2015
Publicly Available Date Nov 23, 2017
Journal Computers and mathematics with applications
Print ISSN 0898-1221
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 70
Issue 8
Pages 1781-1798
Keywords Triangular splines; Refinable splines; Spline basis functions; Multilevel splines; Partition of unity; Spline approximation
Public URL
Publisher URL
Additional Information Authors' accepted manuscript of an article published in: Computers and mathematics with applications, 2015, v.70, issue 8.


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