Multivariate and Rational Splines

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

$f (x, y, z) = \sum_{u = 1}^{U} \sum_{v = 1}^{V} \sum_{w = 1}^{W} B_{u, k} (x) B_{v, l} (y) B_{w, m} (z) a_{u, v, w}$

with B_u,k,B_v,l,B_w,m univariate B-splines. Correspondingly, this spline is of order k in x, of order l in y, and of order m in z. 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

$f (x) = \sum_{j = 1}^{n - 3} Ψ (x - c_{j}) a_{j} + x (1) a_{n - 2} + x (2) a_{n - 1} + a_{n}$

with ψ(x)=|x|²log|x|² the thin-plate spline basis function, and |x| denoting the Euclidean length of the vector x. Here, for convenience, denote the independent variable by x, but x is now a vector whose two components, x(1) and x(2), play the role of the two independent variables earlier denoted x and y. Correspondingly, the sites c_j are points in ℜ².

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

$p \sum_{i = 1}^{n - 3} | y_{i} - f c_{i}^{2} | + (1 - p) \int ({| D_{1} D_{1} f |}^{2} + 2 {| D_{1} D_{2} f |}^{2} + {| D_{2} D_{2} f |}^{2})$

over all sufficiently smooth functions f. Here, the y_i are data values given at the data sites c_i, p is the smoothing parameter, and D_jf denotes the partial derivative of f with respect to x(j). The integral is taken over the entire ℜ². The upper summation limit, n–3, 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 Ψ(· -c) of one fixed function, Ψ. This so-called basis function for the thin-plate spline is special in that it is radially symmetric, meaning that Ψ(x) only depends on the Euclidean length, |x|, of x. For that reason, thin-plate splines are also known as RBFs or radial basis functions. See Constructing and Working with stform Splines for more information.

Rational Splines

A rational spline is any function of the form r(x) = s(x)/w(x), with both s and w splines and, in particular, w a scalar-valued spline, while s often is vector-valued.

Rational splines are attractive because it is possible to describe various basic geometric shapes, like conic sections, exactly as the range of a rational spline. For example, a circle can so be described by a quadratic rational spline with just two pieces.

In this toolbox, there is the additional requirement that both s and w be of the same form and even of the same order, and with the same knot or break sequence. This makes it possible to store the rational spline r as the ordinary spline R whose value at x is the vector [s(x);w(x)]. Depending on whether the two splines are in B-form or ppform, such a representation is called here the rBform or the rpform of such a rational spline.

It is easy to obtain r from R. For example, if v is the value of R at x, then v(1:end-1)/v(end) is the value of r at x. As another example, consider getting derivatives of r from those of R. Because s = wr, Leibniz' rule tells us that

$D^{m} s = \sum_{j = 0}^{m} (\begin{matrix} m \\ j \end{matrix}) D^{j} w D^{m - j} r$

where D^ms the mth derivative of s.

Hence, if v(:,j) contains D^j–1R(x), j = 1...m + 1, then

$(((v (1 : end - 1, m + 1) - \sum_{j = 1}^{m} (\begin{matrix} m \\ j \end{matrix}) v (end, j + 1) v (1 : end - 1, j + 1)) / v (end, 1))$

provides the value of D^mR(x).