De Boor's algorithm

In the mathematical subfield of numerical analysis, de Boor's algorithm^[1] is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de Casteljau's algorithm for Bézier curves. The algorithm was devised by German-American mathematician Carl R. de Boor. Simplified, potentially faster variants of the de Boor algorithm have been created but they suffer from comparatively lower stability.^[2][3]

Introduction

A general introduction to B-splines is given in the main article. Here we discuss de Boor's algorithm, an efficient and numerically stable scheme to evaluate a spline curve ${\displaystyle \mathbf {S} (x)}$ at position ${\displaystyle x}$. The curve is built from a sum of B-spline functions ${\displaystyle B_{i,p}(x)}$ multiplied with potentially vector-valued constants ${\displaystyle \mathbf {c} _{i}}$, called control points,

${\displaystyle \mathbf {S} (x)=\sum _{i}\mathbf {c} _{i}B_{i,p}(x).}$

B-splines of order ${\displaystyle p+1}$ are connected piece-wise polynomial functions of degree ${\displaystyle p}$ defined over a grid of knots ${\displaystyle {t_{0},\dots ,t_{i},\dots ,t_{m}}}$ (we always use zero-based indices in the following). De Boor's algorithm uses O(p²) + O(p) operations to evaluate the spline curve. Note: the main article about B-splines and the classic publications^[1] use a different notation: the B-spline is indexed as ${\displaystyle B_{i,n}(x)}$ with ${\displaystyle n=p+1}$.

Local support

B-splines have local support, meaning that the polynomials are positive only in a finite domain and zero elsewhere. The Cox-de Boor recursion formula^[4] shows this:

${\displaystyle B_{i,0}(x):={\begin{cases}1&{\text{if }}\quad t_{i}\leq x<t_{i+1}\\0&{\text{otherwise}}\end{cases}}}$

${\displaystyle B_{i,p}(x):={\frac {x-t_{i}}{t_{i+p}-t_{i}}}B_{i,p-1}(x)+{\frac {t_{i+p+1}-x}{t_{i+p+1}-t_{i+1}}}B_{i+1,p-1}(x).}$

Let the index ${\displaystyle k}$ define the knot interval that contains the position, ${\displaystyle x\in [t_{k},t_{k+1})}$. We can see in the recursion formula that only B-splines with ${\displaystyle i=k-p,\dots ,k}$ are non-zero for this knot interval. Thus, the sum is reduced to:

${\displaystyle \mathbf {S} (x)=\sum _{i=k-p}^{k}\mathbf {c} _{i}B_{i,p}(x).}$

It follows from ${\displaystyle i\geq 0}$ that ${\displaystyle k\geq p}$. Similarly, we see in the recursion that the highest queried knot location is at index ${\displaystyle k+1+p}$. This means that any knot interval ${\displaystyle [t_{k},t_{k+1})}$ which is actually used must have at least ${\displaystyle p}$ additional knots before and after. In a computer program, this is typically achieved by repeating the first and last used knot location ${\displaystyle p}$ times. For example, for ${\displaystyle p=3}$ and real knot locations ${\displaystyle (0,1,2)}$, one would pad the knot vector to ${\displaystyle (0,0,0,0,1,2,2,2,2)}$.

The algorithm

With these definitions, we can now describe de Boor's algorithm. The algorithm does not compute the B-spline functions ${\displaystyle B_{i,p}(x)}$ directly. Instead it evaluates ${\displaystyle \mathbf {S} (x)}$ through an equivalent recursion formula.

Let ${\displaystyle \mathbf {d} _{i,r}}$ be new control points with ${\displaystyle \mathbf {d} _{i,0}:=\mathbf {c} _{i}}$ for ${\displaystyle i=k-p,\dots ,k}$. For ${\displaystyle r=1,\dots ,p}$ the following recursion is applied:

${\displaystyle \mathbf {d} _{i,r}=(1-\alpha _{i,r})\mathbf {d} _{i-1,r-1}+\alpha _{i,r}\mathbf {d} _{i,r-1};\quad i=k-p+r,\dots ,k}$

${\displaystyle \alpha _{i,r}={\frac {x-t_{i}}{t_{i+1+p-r}-t_{i}}}.}$

Once the iterations are complete, we have ${\displaystyle \mathbf {S} (x)=\mathbf {d} _{k,p}}$, meaning that ${\displaystyle \mathbf {d} _{k,p}}$ is the desired result.

De Boor's algorithm is more efficient than an explicit calculation of B-splines ${\displaystyle B_{i,p}(x)}$ with the Cox-de Boor recursion formula, because it does not compute terms which are guaranteed to be multiplied by zero.

Optimizations

The algorithm above is not optimized for the implementation in a computer. It requires memory for ${\displaystyle (p+1)+p+\dots +1=(p+1)(p+2)/2}$ temporary control points ${\displaystyle \mathbf {d} _{i,r}}$. Each temporary control point is written exactly once and read twice. By reversing the iteration over ${\displaystyle i}$ (counting down instead of up), we can run the algorithm with memory for only ${\displaystyle p+1}$ temporary control points, by letting ${\displaystyle \mathbf {d} _{i,r}}$ reuse the memory for ${\displaystyle \mathbf {d} _{i,r-1}}$. Similarly, there is only one value of ${\displaystyle \alpha }$ used in each step, so we can reuse the memory as well.

Furthermore, it is more convenient to use a zero-based index ${\displaystyle j=0,\dots ,p}$ for the temporary control points. The relation to the previous index is ${\displaystyle i=j+k-p}$. Thus we obtain the improved algorithm:

Let ${\displaystyle \mathbf {d} _{j}:=\mathbf {c} _{j+k-p}}$ for ${\displaystyle j=0,\dots ,p}$. Iterate for ${\displaystyle r=1,\dots ,p}$:

${\displaystyle \mathbf {d} _{j}:=(1-\alpha _{j})\mathbf {d} _{j-1}+\alpha _{j}\mathbf {d} _{j};\quad j=p,\dots ,r\quad }$ (${\displaystyle j}$ must be counted down)

${\displaystyle \alpha _{j}:={\frac {x-t_{j+k-p}}{t_{j+1+k-r}-t_{j+k-p}}}.}$

After the iterations are complete, the result is ${\displaystyle \mathbf {S} (x)=\mathbf {d} _{p}}$.

Example implementation

The following code in the Python programming language is a naive implementation of the optimized algorithm.

def deBoor(k: int, x: int, t, c, p: int):
    """Evaluates S(x).

    Arguments
    ---------
    k: Index of knot interval that contains x.
    x: Position.
    t: Array of knot positions, needs to be padded as described above.
    c: Array of control points.
    p: Degree of B-spline.
    """
    d = [c[j + k - p] for j in range(0, p + 1)] 

    for r in range(1, p + 1):
        for j in range(p, r - 1, -1):
            alpha = (x - t[j + k - p]) / (t[j + 1 + k - r] - t[j + k - p]) 
            d[j] = (1.0 - alpha) * d[j - 1] + alpha * d[j]

    return d[p]

See also

• De Casteljau's algorithm
• Bézier curve
• Non-uniform rational B-spline

External links

• De Boor's Algorithm
• The DeBoor-Cox Calculation

Computer code

• PPPACK: contains many spline algorithms in Fortran
• GNU Scientific Library: C-library, contains a sub-library for splines ported from PPPACK
• SciPy: Python-library, contains a sub-library scipy.interpolate with spline functions based on FITPACK
• TinySpline: C-library for splines with a C++ wrapper and bindings for C#, Java, Lua, PHP, Python, and Ruby
• Einspline: C-library for splines in 1, 2, and 3 dimensions with Fortran wrappers

References

Works cited

• Carl de Boor (2003). A Practical Guide to Splines, Revised Edition. Springer-Verlag. ISBN 0-387-95366-3.