Editing B-spline (section)

===Cardinal B-spline===

A cardinal B-spline has a constant separation ''h'' between knots. The cardinal B-splines for a given order ''n'' are just shifted copies of each other. They can be obtained from the simpler definition.<ref>de Boor, p. 322.</ref>

:<math>B_{i,n,t}(x) = \frac{x-t_i}{h} n[0, \dots, n](\cdot - t_i)^{n-1}_+.</math>

The "placeholder" notation is used to indicate that the ''n''-th [[divided difference]] of the function <math>(t-x)^{n-1}_+</math> of the two variables ''t'' and ''x'' is to be taken by fixing ''x'' and considering <math>(t - x)^{n-1}_+</math> as a function of ''t'' alone.

A cardinal B-spline has uniformly spaced knots, therefore interpolation between the knots equals convolution with a smoothing kernel.

Example, if we want to interpolate three values in between B-spline nodes (<math>\textbf{b}</math>), we can write the signal as

: <math>\mathbf{x} = [\mathbf{b}_1, 0, 0, \mathbf{b}_2, 0, 0, \mathbf{b}_3, 0, 0, \dots, \mathbf{b}_n, 0, 0].</math>

Convolution of the signal <math>\mathbf{x}</math> with a rectangle function <math>\mathbf{h} = [1/3, 1/3, 1/3]</math> gives first order interpolated B-spline values. Second-order B-spline interpolation is convolution with a rectangle function twice <math>\mathbf{y} = \mathbf{x} * \mathbf{h} * \mathbf{h}</math>; by iterative filtering with a rectangle function, higher-order interpolation is obtained.

Fast B-spline interpolation on a uniform sample domain can be done by iterative mean-filtering. Alternatively, a rectangle function equals [[sinc]] in [[Fourier domain]]. Therefore, cubic spline interpolation equals multiplying the signal in Fourier domain with sinc<sup>4</sup>.

See [[Irwin–Hall distribution#Special cases]] for algebraic expressions for the cardinal B-splines of degree 1–4.