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Efficient evaluation of Bernstein-Bézier coefficients of B-spline basis functions over one knot span

Filip Chudy, Paweł Woźny

TL;DR

This work addresses the efficient evaluation of Bernstein-Bézier coefficients for B-spline basis functions over a single knot span. It introduces new differential-recurrence relations that hold for general knot sequences and, crucially, yields an $O(m^2)$ per-span algorithm to compute all $(m+1)^2$ adjusted Bernstein-Bézier coefficients for a fixed knot span, independent of other spans. Numerical evidence shows comparable digit preservation to the de Boor-Cox approach while offering faster per-span performance and enabling parallelization; the method also accommodates non-clamped knot sequences and integrates with linear-time evaluation when Bernstein-Bézier data are known. Practically, this enables faster rendering and evaluation of B-spline curves and surfaces with shared knot sequences, and it provides a more flexible theoretical framework for Bernstein-Bézier representations in spline design.

Abstract

New differential-recurrence relations for B-spline basis functions are given. Using these relations, a recursive method for finding the Bernstein-Bézier coefficients of B-spline basis functions over a single knot span is proposed. The algorithm works for any knot sequence and has an asymptotically optimal computational complexity. Numerical experiments show that the new method gives results which preserve a high number of digits when compared to an approach which uses the well-known de Boor-Cox formula.

Efficient evaluation of Bernstein-Bézier coefficients of B-spline basis functions over one knot span

TL;DR

This work addresses the efficient evaluation of Bernstein-Bézier coefficients for B-spline basis functions over a single knot span. It introduces new differential-recurrence relations that hold for general knot sequences and, crucially, yields an per-span algorithm to compute all adjusted Bernstein-Bézier coefficients for a fixed knot span, independent of other spans. Numerical evidence shows comparable digit preservation to the de Boor-Cox approach while offering faster per-span performance and enabling parallelization; the method also accommodates non-clamped knot sequences and integrates with linear-time evaluation when Bernstein-Bézier data are known. Practically, this enables faster rendering and evaluation of B-spline curves and surfaces with shared knot sequences, and it provides a more flexible theoretical framework for Bernstein-Bézier representations in spline design.

Abstract

New differential-recurrence relations for B-spline basis functions are given. Using these relations, a recursive method for finding the Bernstein-Bézier coefficients of B-spline basis functions over a single knot span is proposed. The algorithm works for any knot sequence and has an asymptotically optimal computational complexity. Numerical experiments show that the new method gives results which preserve a high number of digits when compared to an approach which uses the well-known de Boor-Cox formula.
Paper Structure (8 sections, 5 theorems, 34 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 8 sections, 5 theorems, 34 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The problem
  3. Some new results for B-spline basis functions
  4. The solution
  5. The main idea
  6. The initial conditions
  7. The algorithm
  8. Conclusions

Key Result

Lemma 3.1

For any knot sequence E:KnotSequence, the B-spline functions $N_{m,i}$ satisfy the following identities: where $-m\leq i\leq n-1$ (cf. E:ConventionI) and we assume that for any quantity $Q$ as well as that $N_{-1,k}\equiv 0$ for all $k$.

Figures (2)

  • Figure 4.1: An illustration of using Eq. \ref{['E:InitialRecurrence']} (cf. \ref{['E:InitCondInitialRecurrenceI']}) for $m=5$.
  • Figure 4.2: An illustration of the full recurrence scheme using Theorem \ref{['T:MainRecurrence']} for $m=5$.

Theorems & Definitions (13)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • Theorem 4.1
  • proof
  • Remark 4.2
  • ...and 3 more