Biomedical Engineering Reference
In-Depth Information
When we move to curves and surfaces, we are dealing with the locally singular
locus of points in certain directions only: curves and surfaces are continuous along
their tangent. The extension of the notion of distributions that allows to take that
information into account is the geometric integration theory. The basic idea is
to integrate differential forms, but for objects like smooth surfaces in 3D, we
shall simply define currents by their action on vector fields, similarly to the way
distributions are defined by their action on scalar functions. In fact, one could see
currents as the characterization of surfaces by the flux of vector fields through it.
5.2.2.1
Surfaces Seen as Currents
Let
W
be a Hilbert space of test vector fields. This is a possibly infinite complete
vector space provided with a scalar product. For a given surface
S
, we can measure
the flux of any vector field
ω
∈
W
through this surface:
x∈S
S
(
ω
)=
ω
(
x
)
|
n
(
x
)
dσ
(
x
)
,
where
n
(
x
)
is the normal to the smooth surface at point
x
∈
S
and
dσ
(
x
)
the surface
element around point
x
. The shape of the surface
S
is characterized by the variation
of the flux as the test vector field
ω
varies in
W
. Thus, the surface actually defines
a continuous linear form on
W
which can be identified as an element of the dual
space
W
∗
, which is the space of linear functionals from
W
to
R
(currents). The
nice property of currents is that it is a vector space: we can add or subtract currents
from each other, or multiply them by a scalar. However, we should keep in mind
that the space of currents is larger than the space of smooth surfaces: one can for
instance add many pieces of surfaces together in a non-continuous way to create a
non-continuous object.
5.2.2.2
The Kernel Trick for Currents
Now that we have identified surfaces to currents, we need to define more carefully
what is the space
W
that we consider. The core element proposed by Glaunes [
7
]is
to consider a kernel metric (typically the Gaussian kernel
K
W
(
x, z
)=exp(
−
x
−
2
/λ
2
W
)
z
) in order to turn
W
into a Reproducible Kernel Hilbert Space (RKHS).
The reproducibility property implies that
W
is the dense span of basis vector fields
of the form
ω
x
(
α
. This means that any vector field of
W
can be
written as an infinite linear combination of such basis vectors. The kernel induces
.
)=
K
W
(
x, .
)
a scalar product which is easily computed on two basis vectors:
ω
x
ω
y
W
=
α
T
K
W
(
x, y
)
β
.
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