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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 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).

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