Algorithm 999

Abstract

Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar properties to standard B-splines. These basis functions are called multi-degree B-splines (or MDB-splines). The construction relies on an extraction operator that represents all MDB-splines as linear combinations of local B-splines of different degrees. This enables the use of existing efficient algorithms for B-spline evaluations and refinements in the context of multi-degree splines. A Matlab implementation is provided to illustrate the computation and use of MDB-splines.An extension of the standard concept of (B-)splines is to consider polynomial pieces of different degrees glued together with certain smoothness. This idea of multi-degree (B-)splines dates back to Nürnberger et al. [1984] but has been rejuvenated several times under different forms (and names). A special case was investigated by Sederberg et al. [2003] for degrees (1, 2, 3) and (1, n) in the context of CAGD. Later, Shen and Wang [2010a,b] constructed multi-degree B-splines with general degrees by means of an integral recurrence relation. Li et al. [2012] observed the latter as being difficult to compute, and instead presented a geometric-oriented recursive procedure. Recently, multi-degree B-splines were revisited by Beccari et al. [2017] illustrating their capabilities in CAGD, but still based on an integral recurrence relation and so-called transition functions identified through Hermite interpolation. Even though the theoretical framework is very elegant, multi-degree B-splines are a burden to implement when dealing with recursive integration or when solving several Hermite interpolation problems. This might explain why they have not been popular in a practical context, despite their (theoretical) potential. Besides CAGD, multi-degree splines were also applied in IGA, in the frame of T-splines [Liu et al. 2016] and polar splines [Toshniwal et al. 2017]. Non-polynomial extensions can be found in [Buchwald and Mühlbach 2003;Nürnberger et al. 1984;Sommer and Strauss 1988] considering local spaces generated by ECT-systems and Descartes systems.The purpose of this paper is to show that the implementation of multi-degree B-splines (in short, MDB-splines) can be painless, and we provide a small Matlab toolbox as illustration. We closely follow the algorithmic construction proposed by Toshniwal et al. [2017] in the context of rational multi-degree polar splines, but reformulate the algorithms in a computationally more efficient way. This further simplifies the construction, and might open the door to the practical use of multi-degree (B-)splines in a wide range of application fields, from CAGD and IGA to data fitting and compression. We remark that some computational aspects of multi-degree splines were also addressed in [Beccari et al. 2017].The remainder of the paper is organized as follows. We start by defining no...