Efficient Algorithms for the Implementation of General B-Splines

Nonuniform B-splines are usually computed using the traditional recurrence relation $\text{B}{}_{\mathrm{i,r}}\text{(}\text{u}\text{) =}\text{u}{}_{\mathrm{i}}\text{−}\text{u}\text{u}{}_{\mathrm{i+r-1}}\text{−}\text{u}{}_{\mathrm{i}}\text{Bi,}{}_{\mathrm{r-1}}\text{(}\text{u}\text{) +}\text{u}{}_{\mathrm{i+r}}\text{−}\text{u}\text{u}{}_{\mathrm{i+r}}\text{−}\text{u}{}_{\mathrm{i+1}}\text{B}{}_{\mathrm{i+1,r-1}}\text{(}\text{u}\text{).}$We derive a recurrence relation which relates the rth derivative of B_i,r(ū) to the (r − 1)th derivatives of B_i,r−1($\text{u}$) and B_{i + 1, r − 1}$\text{u}$[formula]B^(r)_{i, r}($\text{u}$) is comprised of r + 1 impulses (Dirac functions) at the knots [ū_i, ū_{i + 1}, . . . , ū_{i + r}]. The amplitudes of the impulses are found from the recurrence. We show that equally spaced samples of the continuous B-spline function B_{i, r}(ū) can be computed exactly using recursive summation.