Expand description
Hermite radial basis function kernels (funcions φ
)
We require that the first derivative of the kernel f
go to zero as x -> 0
.
This ensures that the HRBF fitting matrix is well defined.
I.e. in its taylor series representation, f(x) = ∑ᵢ aᵢxⁱ
, we require that a₁ = 0
.
NOTE: Kernels like x^2
, x^3
, exp(-x*x)
, and (1-x)^4 (4x+1)
all satisfy this criterion.
To ensure that derivativevs at zero are computed accurately, each kernel must provide a formula for the following additional function:
df_l(x) = φʹ(x)/x
where φ
is the kernel function. It must be that df(x)
, ddf(x) - df_l(x) -> 0
as x -> 0
for the HRBF
derivaitves to exist.
Furthermore, if the HRBF is to be used in the optimization framework,
where a 3rd or even 4th derivatives are required, we also need the 3rd derivative to vanish.
I.e. in its taylor series representation, f(x) = ∑ᵢ aᵢxⁱ
, we require that a₁ = a₃ = 0
.
NOTE: The exp(-x*x)
and x^2
kernels satisfy this criteria, but x^3, x^2*log(x)
and
(1-x)^4 (4x+1)
do not!
To ensure that derivatives at zero are computed accurately, each kernel must provide a formula for
the function df_l
(as described above) and the following additional functions:
g(x) = φʺ(x)/x - φʹ(x)/x^2
g_l(x) = φʺ(x)/x^2 - φʹ(x)/x^3
h(x,a) = φ‴(x)/x - a(φʺ(x)/x^2 - φʹ(x)/x^3)
to see how these are used, see mesh_implicit_surface.cpp
It must be that g(x), h(x,3) -> 0
as x -> 0
for the HRBF derivatives to exist.
Structs
- Quintic kernel
(1-x)^4 (4x+1)
. - The
(1-x)^6 (35x^2 + 18x + 3)
kernel. - Gaussian
exp(-x*x)
kernel. - The
x^2
kernel. - The
x^3
kernel. - The
x^4
kernel. - The
x^5
kernel.
Traits
- Global kernel trait defines a constructor for kernels without a radial fallofff.
- Kernel trait declaring all of the necessary derivatives.
- Local kernel trait defines the radial fall-off for appropriate kernels.