B-spline

In the mathematical subfield of numerical analysis, a B-spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. A fundamental theorem states that every spline function of a given degree, smoothness, and domain partition, can be represented as a linear combination of B-splines of that same degree and smoothness, and over that same partition. [cite book | author = Carl de Boor | title = A Practical Guide to Splines | publisher = Springer-Verlag | pages = 113-114 | year = 1978] The term "B-spline" was coined by Isaac Jacob Schoenberg and is short for basis spline. [cite book | author = Carl de Boor | title = A Practical Guide to Splines | publisher = Springer-Verlag | pages = 114-115 | year = 1978] B-splines can be evaluated in a numerically stable way by the de Boor algorithm.

In the computer science subfields of computer-aided design and computer graphics the term B-spline frequently refers to a spline curve parametrized by spline functions that are expressed as linear combinations of B-splines (in the mathematical sense above). A B-spline is simply a generalisation of a Bézier curve, and it can avoid the Runge phenomenon without increasing the degree of the B-spline.

Definition

Given "m"+1 values t_i in [0,1] , called "knots", with:t_0 le t_1 le cdots le t_m

a B-spline of degree "n" is a parametric curve

:mathbf{S}: [t_0, t_m] o mathbb{R}^2

composed of basis B-splines of degree "n"

:mathbf{S}(t)= sum_{i=0}^{m-n-1} mathbf{P}_{i} b_{i,n}(t) mbox{ , } t in [t_n,t_{m-n}] .

The P"i" are called control points or de Boor points. (There are "m−n" control points.) A polygon can be constructed by connecting the de Boor points with lines, starting with P0 and finishing with P"m−n−1". This polygon is called the de Boor polygon.

The "m"−"n" basis B-splines of degree "n" can be defined using the Cox-de Boor recursion formula

:b_{j,0}(t) := left{egin{matrix} 1 & mathrm{if} quad t_j leq t < t_{j+1} \0 & mathrm{otherwise} end{matrix} ight.

:b_{j,n}(t) := frac{t - t_j}{t_{j+n} - t_j} b_{j,n-1}(t) + frac{t_{j+n+1} - t}{t_{j+n+1} - t_{j+1 b_{j+1,n-1}(t).

When the knots are equidistant we say the B-spline is uniform otherwise we call it non-uniform. If two knots t_j are identical, we deem frac{0}{0} stackrel{mathrm{def{=} 0.

Note that j+n+1 can not exceed m, which limits both j and n.

Uniform B-spline

When the B-spline is uniform, the basis B-splines for a given degree "n" are just shifted copies of each other. An alternative non-recursive definition for the "m"−"n" basis B-splines is

:b_{j,n}(t) = b_n(t - t_j), qquad; j = 0, ldots, m-n-1

with

:b_{n}(t) := frac{n+1}{n} sum_{i=0}^{n+1} omega_{i,n}(t - t_i)_+^{n} ,;

and

:omega_{i,n} := prod_{j=0, j eq i}^{n+1} frac{1}{t_j - t_i} ,;

where

:(t - t_i)_+^n ,;

is the truncated power function.

Cardinal B-spline

Define B"0" as the characteristic function of [- frac{1}{2}, frac{1}{2}] , and B"k" recursively as the convolution product

:B_k := B_{k-1} * B_0, ~k =1, 2, dots

then B"k" are called (centered) "cardinal B-splines". This definition goes back to Schoenberg.

B"k" has compact support [- frac{k+1}{2}, frac{k+1}{2}] and is an even function. As k ightarrow infty the normalized cardinal B-splines tend to the Gaussian function. [Brinks R: On the convergence of derivatives of B-splines to derivatives of the Gaussian function, Comp. Appl. Math., 27, 1, 2008]

Notes

When the number of knots is the same as the degree, the B-Spline degenerates into a Bézier curve. The shape of the basis functions is determined by the position of the knots. Scaling or translating the knot vector does not alter the basis functions.

The spline is contained in the convex hull of its control points.

A basis B-spline of degree "n":b_{i,n}(t),;is non-zero only in the interval ["t""i", "t""i+n+1"] that is :b_{i,n}(t) = left{egin{matrix} >0 & mathrm{if} quad t_{i} le t < t_{i+n+1} \0 & mathrm{otherwise} end{matrix} ight.In other words if we manipulate one control point we only change the local behaviour of the curve and not the global behaviour as with Bézier curves.

The basis function can be derived from the Bernstein polynomial.

Examples

Constant B-spline

The constant B-spline is the most simple spline. It is defined on only one knot span and is not even continuous on the knots. It is just the indicator function for the different knot spans.

:b_{j,0}(t) = 1_{ [t_j,t_{j+1})} =left{egin{matrix} 1 & mathrm{if} quad t_j le t < t_{j+1} \0 & mathrm{otherwise} end{matrix} ight.

Linear B-spline

The linear B-spline is defined on two consecutive knot spans and is continuous on the knots, but not differentiable.

:b_{j,1}(t) = left{egin{matrix} frac{t - t_j}{t_{j+1} - t_j} & mathrm{if} quad t_j le t < t_{j+1} \frac{t_{j+2} - t}{t_{j+2} - t_{j+1 & mathrm{if} quad t_{j+1} le t < t_{j+2} \0 & mathrm{otherwise} end{matrix} ight.

Uniform quadratic B-spline

Quadratic B-splines with uniform "knot-vector" is a commonly used form of B-spline. The blending function can easily be precalculated, and is equal for each segment in this case. :b_{j,2}(t) = egin{cases} frac{1}{2}t^2 \ -t^2 + t + frac{1}{2} \ frac{1}{2}(1-t)^2 end{cases}

Put in matrix-form, it is: [http://graphics.idav.ucdavis.edu/education/CAGDNotes/Quadratic-Uniform-B-Spline-Curve-Splitting/Quadratic-Uniform-B-Spline-Curve-Splitting.html] : mathbf{S}_i(t) = egin{bmatrix} t^2 & t & 1 end{bmatrix} frac{1}{2} egin{bmatrix}1 & -2 & 1 \-2 & 2 & 0 \1 & 1 & 0 end{bmatrix}egin{bmatrix} mathbf{p}_{i-1} \ mathbf{p}_{i} \ mathbf{p}_{i+1} end{bmatrix} for t in [0,1] , i = 1,2 ldots m-1

Cubic B-Spline

A B-spline formulation for a single segment can be written as:

:mathbf{S}_{i} (t) = sum_{k=0}^3 mathbf{P}_{i-3+k} b_{i-3+k,3} (t) mbox{ ; } t in [0,1]

where S"i" is the "i"th B-spline segment and P is the set of control points, segment "i" and "k" is the local control point index. A set of control points would be P_i^w = ( w_i x_i, w_i y_i, w_i z_i, w_i) where the w_i is weight, pulling the curve towards control point P_i as it increases or moving the curve away as it decreases.

An entire set of segments, "m"-2 curves (S_3,S_4,...,S_m) defined by "m"+1 control points (P_0,P_1,...,P_m, m ge 3), as one B-spline in "t" would be defined as:

:mathbf{S}(t) = sum_{i=0}^m mathbf{P}_{i} b_{i,3} (t)

where "i" is the control point number and "t" is a global parameter giving knot values. This formulation expresses a B-spline curve as a linear combination of B-spline basis functions, hence the name.

There are two types of B-spline - uniform and non-uniform. A non-uniform B-spline is a curve where the intervals between successive control points is not, or not necessarily, equal (the knot vector of interior knot spans are not equal). A common form is where intervals are successively reduced to zero, interpolating control points.

Uniform cubic B-splines

Cubic B-splines with uniform "knot-vector" is the most commonly used form of B-spline. The blending function can easily be precalculated, and is equal for each segment in this case. Put in matrix-form, it is:: mathbf{S}_i(t) = egin{bmatrix} t^3 & t^2 & t & 1 end{bmatrix} frac{1}{6} egin{bmatrix}-1 & 3 & -3 & 1 \ 3 & -6 & 3 & 0 \-3 & 0 & 3 & 0 \ 1 & 4 & 1 & 0 end{bmatrix}egin{bmatrix} mathbf{p}_{i-1} \ mathbf{p}_{i} \ mathbf{p}_{i+1} \ mathbf{p}_{i+2} end{bmatrix} for t in [0,1] .

ee also

*Spline (mathematics)
*Nonuniform rational B-splines (NURBS)
*De Boor algorithm

External links

* [http://www.ibiblio.org/e-notes/Splines/Basis.htm Interactive java applets for B-splines]
* [http://mathworld.wolfram.com/B-Spline.html B-spline on MathWorld]
* [http://math.fullerton.edu/mathews/n2003/B-SplinesMod.html Module B-Splines by John H. Mathews]
* [http://www.qwerkop.de/qwerkop-projects-bspline.php BSpline Java Applet by Stefan Beck (with C++ Source)]
* [http://www.stat.columbia.edu/~ruf/ruf_bspline.pdf B-splines of third order on a non-uniform grid by Johannes Ruf]

References


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