Bicubic Spline

A bicubic spline is a special case of bicubic interpolation which uses an interpolation function of the form

$\displaystyle y(x_1,x_2)$	$\textstyle =$	$\displaystyle \sum_{i=1}^4\sum_{j=1}^4 c_{ij}t^{i-1}u^{j-1}$
$\displaystyle y_{x_1}(x_1,x_2)$	$\textstyle =$	$\displaystyle \sum_{i=1}^4\sum_{j=1}^4 (i-1)c_{ij}t^{i-2}u^{j-1}$
$\displaystyle y_{x_2}(x_1,x_2)$	$\textstyle =$	$\displaystyle \sum_{i=1}^4\sum_{j=1}^4 (j-1)c_{ij}t^{i-1}u^{j-2}$
$\displaystyle y_{x_1x_2}$	$\textstyle =$	$\displaystyle \sum_{i=1}^4\sum_{j=1}^4 (i-1)(j-1)c_{ij}t^{i-2}u^{j-2},$

where $c_{ij}$ are constants and

and

are parameters ranging from 0 to 1. For a bicubic spline, however, the partial derivatives at the grid points are determined globally by 1-D Splines.

See also B-Spline, Spline

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 118-122, 1992.