Introduction
A multivariate function form quite different from the tensor-product
construct is the scattered translates form, or stform for short. As the
name suggests, it uses arbitrary or scattered translates
of one fixed function ψ,
in addition to some polynomial terms. Explicitly, such a form describes
a function
in terms of the basis function
, a sequence
of sites called centers and a corresponding
sequence
of
coefficients, with the final
coefficients,
, involved in the polynomial
part, p.
When the basis function is radially symmetric, meaning that
depends only on the Euclidean length
of its argument,
, then
is called a radial basis function,
and, correspondingly,
is then often called an RBF.
At present, the toolbox works with just one kind of stform,
namely a bivariate thin-plate spline and its first partial derivatives.
For the thin-plate spline, the basis function is
, with
, i.e., a radial basis function.
Its polynomial part is a linear polynomial, i.e.,
. The first partial derivative with
respect to its first argument uses, correspondingly, the basis function
, with
and
, and
.
 | The stform | | Properties of the stform |  |
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