Biomedical Engineering Reference
In-Depth Information
Since W is a Hilbert space, by the Rietz representation theorem, there exists
an isometric linear mapping
from W to W which maps to each ω
L W
W the
ω )=
ω W
linear form that reproduces the scalar product:
.This
canonical isomorphism allows to define the dual of an element, or conversely to
map to each surface the vector field that optimizes the flux. For instance, the dual
of the above basis vectors are the Dirac delta currents δ x = L W (
( L W (
ω
))(
ω
|
ω x )
as we have
ω x |
. These can be seen as vector fields whose spatial support
is concentrated at one point only. The space of currents W is the dense span of
these basis elements. For instance, the surface S is represented by the current
ω
W =
α T ω
(
x
)
S =
x∈S
δ n ( x )
(
x
)
. The usual way to define a norm on the dual space is to take the
x
S W =sup ω∈W,ω =1 |S (
ω
) |
operator norm:
. The distance induced by this norm
might seem difficult to use, but thanks to the RKHS properties, we have a closed
form for basis vectors:
δ x δ y W = α
δ x − δ y 2 W = δ x 2 W + δ y 2 W 2 δ x δ y W .
T
K W ( x, y ) β
and
5.2.2.3
Numerical Issues
It is interesting to notice that this distance can be approximated by
O
4 W 4 ,
δ x
δ y 2 W =
2 +2
2 2 W α T β
α
β
x
y
+
x
y
when the points x and y are within a fraction of λ W , while the distance is essentially
constant (and equal to
L 2 ) when the points x and y are more than a few
λ W apart. This behavior is typical of a robust distance in statistics, meaning that
outliers (over a few λ W in distance) will have (almost) no effect on the optimization
of the distance between the surfaces as it is an almost constant penalty.
In practice, surfaces are often represented by discrete triangulated meshes.
Assuming that each face has a support which is smaller than ε times the scale
λ W , we can approximate the surface by the current
L 2 +
α
β
S = k
δ α k
x k
,where x k is
the barycenter and α k the normal weighted by the area of the face. From the above
Taylor expansion of the distance, we can see that the approximation error is less
than ε 2 for each face in the W norm.
The scalar product between two discrete currents is obtained by linearity:
δ α k
x k
δ β j
y j
=
α T
k
K W (
x k ,y j )
β j ,
k
j
k,j
W
and the distance between two discrete surfaces is simply dist 2 (
S, S )= S−S 2 W .
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