Multivariate and Rational Splines

Multivariate Splines

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

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

withB_u,k,B_v,l,B_w,munivariate B-splines. Correspondingly, this spline is of orderkinx, of orderliny, and of orderminz. 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 thethin-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|²the thin-plate spline basis function, and |x| denoting the Euclidean length of the vectorx. Here, for convenience, denote the independent variable byx, butxis now avectorwhose two components,x(1) andx(2), play the role of the two independent variables earlier denotedxandy. Correspondingly, the sitesc_jare points in ℜ².

Thin-plate splines arise as bivariatesmoothing 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 functionsf. Here, they_iare data values given at the data sitesc_i,pis the smoothing parameter, andD_jfdenotes the partial derivative offwith respect tox(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-calledbasis 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|, ofx. For that reason, thin-plate splines are also known asRBFs orradial basis functions. SeeConstructing and Working with stform Splinesfor more information.

Rational Splines

A有理样条曲线的is any function of the formr(x) =s(x)/w(x), with bothsandwsplines and, in particular,wa scalar-valued spline, whilesoften 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 bothsandwbe 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 splineras the ordinary splineRwhose value atxis 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 obtainrfromR. For example, ifvis the value ofRatx, thenv(1:end-1)/v(end)is the value ofratx. As another example, consider gettingderivatives ofrfrom those ofR. Becauses=wr, Leibniz' rule tells us that

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

whereD^msthemth derivative ofs.

Hence, ifv(:,j)containsD^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 ofD^mR(x).