Biomedical Engineering Reference
In-Depth Information
and from
orthogonality conditions
i
=
1
λ
i
=
0
N
i
=
1
λ
i
x
i
=
0
N
i
=
1
λ
i
y
i
=
0
N
(9.4)
The method we have just briefly described is called
thin-plate spline
interpola-
tion [1, 85, 86, 88].
9.3.3
Fractional Landmark Interpolation
Although the thin-plate splines have been known to work well, in many appli-
cations we might benefit from a wider choice of interpolation functions, while
keeping the general spirit and the invariance properties (affine geometrical trans-
formations including scaling) we are interested in. The straightforward way to
do it is to consider minimizing different criteria, namely fractional derivatives
(in 1D) and fractional Laplacian (in multiple dimensions). In some sense, these
are the
only
reasonable criteria guaranteeing the useful properties described
above (see [85, 86] for a more precise statement).
9.3.3.1
The Criterion and The Interpolation Formula
2
f
∂
x
2
2
f
2
f
=
∂
+
∂
The Laplacian is defined in the space domain by
∇
∂
y
2
. In the Fourier
domain we have
∇
2
f
, provided that all quantities exist.
This can be extended to
fractional orders
as
∇
α
f
=
ω
α
f
, yielding a general-
ized version of the Laplacian based criterion (9.1):
f
+
ω
f
=
ω
x
y
2
f
=
ω
∇
α
g
(
x
)
2
d
x
∝
2
α
2
d
ω
J
(
g
)
=
ω
|
g
(
ω
)
|
(9.5)
To get some intuition, note that in the univariate case we would be measuring
the norm of the
α
-th
fractional derivative
[89, 90] of
g
.
There is an interesting relationship between
fractional Brownian mo-
tion
[91] and fractional derivatives, since fractional derivatives whiten the frac-
tional Brownian motion and thus effectively yield an uncorrelated Gaussian
white noise. The criterion (9.5) can be therefore interpreted as Bayesian
fractal
prior
(see [88] for details and also Poggio [92] for the non-fractal case), assum-
ing that the underlying true function is close to the fractional Brownian motion
model. We then find the solution to our interpolation problem combining this
knowledge with the information given by the constraints.
