Relationship between RBFs and Implicit Functions

[Olumide]

Hi -

I'm sorry if this question has an obvious answer, but what's the
relationship between Radial Basis Functions and Implicit functions?

Thanks,

- Olumide

[Olumide]

Olumide wrote:
> I'm sorry if this question has an obvious answer, but what's the
> relationship between Radial Basis Functions and Implicit functions?

I now know that implicit surfaces *can* be created from scattered
surface data. References:

1. Variational Implicit Surfaces by Grek Turk and James F. O'brien
http://graphics.cs.uiuc.edu/~jch/cs497jch/variational-is.pdf

2. Interpolating Implicit Surfaces From Scattered Surface Data Using
Compactly Supported Radial Basis Functions
http://www.cs.umbc.edu/~rheingan/pubs/smi2001.pdf

Unfortunately, although both papers treat radial basis functions
(RBFs) in sufficient depth, neither seems to have taken care to
explain how the maths of RBFs produce a(n implicit) function that has
the form, F(x) = 0. My primary concerns are that:

1. The RBF interpolation functions (three of them for the 3D
interpolation problem) interpolate the "height field", which does not
do not equate to zero.

2. Only one implicit function is required, the RBF interpolation
problem produces 2 or 3 equations.

Is the single implicit function produced by making each of the
interpolation equations to zero and summing them? i.e:

X = F1(x) -> G1(x) = X - F1(x) = 0
Y = F2(y) -> G2(y) = Y - F2(y) = 0

and G(x,y) = X + Y - F1(x) - F2(y) = 0 .