Computation of multi-degree B-splines

TL;DR

This paper introduces a simple algorithm to construct basis functions for multi-degree splines, called MDB-splines, enabling efficient evaluation and refinement similar to standard B-splines.

Contribution

It presents a novel construction method for MDB-splines using an extraction operator, facilitating their practical computation and integration with existing B-spline algorithms.

Findings

MDB-splines can be represented as linear combinations of local B-splines.

The proposed method allows efficient evaluation and refinement of multi-degree splines.

A MATLAB implementation demonstrates the practicality of the approach.

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.