Multivariate Splines :: Splines: An Overview (Spline Toolbox™)

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Multivariate Splines

Multivariate splines can be obtained from univariate splines by the tensor product construct. For example, a trivariate spline in B-form is given by

with univariate B-splines. Correspondingly, this spline is of order in , of order in , and of order in . Similarly, the ppform of a tensor-product spline is specified by break sequences in each of the variables and, for each hyper-rectangle thereby specified, a coefficient array. Further, as in the univariate case, the coefficients may be vectors, typically 2-vectors or 3-vectors, making it possible to represent, e.g., certain surfaces in R³.

A very different bivariate spline is the thin-plate spline. This is a function of the form

with the thin-plate spline basis function, and denoting the Euclidean length of the vector . Here, for convenience, we denote the independent variable by , but is now a vector whose two components, and , play the role of the two independent variables earlier denoted and . Correspondingly, the sites are points in .

Thin-plate splines arise as bivariate smoothing splines, meaning a thin-plate spline minimizes

over all sufficiently smooth functions . Here, the are data values given at the data sites , p is the smoothing parameter, and denotes the partial derivative of with respect to . The integral is taken over the entire . The upper summation limit, , reflects the fact that 3 degrees of freedom of the thin-plate spline are associated with its polynomial part.

Thin-plate splines are functions in stform, meaning that, up to certain polynomial terms, they are a weighted sum of arbitrary or scattered translates of one fixed function, . This so-called basis function for the thin-plate spline is special in that it is radially symmetric, meaning that only depends on the Euclidean length, , of . For that reason, thin-plate splines are also known as RBFs or radial basis functions. See The stform for more information.