| Spline Toolbox | |
The stform
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 
, 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, 
.
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 
.
This section treats these aspects of the stform:
| | Use | stform | |