Biomedical Engineering Reference
In-Depth Information
Fig. 5.4
Surfaces seen as integrators of vector fields. In a triangulated mesh, the atomic vector
field integrators, the Dirac delta currents, are the normal vectors of every face, centered at the
face barycenters and weighted by the face surface. Subfigure (
a
) displays the original mesh of the
right ventricle with 1,476 Diracs. In subfigure (
b
), a greedy algorithm reduces the amount of delta
currents needed to represent the shape while preserving the accuracy of the representation to 281
Diracs, i.e., a compression of 80 % with less than 5 % error
to define the action of transforming two objects into the same co-ordinate system.
The basic idea is to rephrase registration into a geodesic problem where the initial
conditions are related to the formalism of currents.
Since we are comparing topologically similar shapes, the transformations are
restricted to those which preserve the topology of the object and give a smooth
one-to-one (invertible) transformation (i.e. a diffeomorphism). The space of dif-
feomorphisms give non-linear deformations that allow local smooth variations to
be captured in the registration. However, diffeomorphic transformations have an
infinite number of degrees of freedom, therefore optimizing over the whole group
of diffeomorphisms may not be possible. So, we use a smaller infinite group of
diffeomorphisms to allow computations with discrete parameterizations using the
Large Deformation Diffeomorphic Metric Mapping (LDDMM) method described
in [
18
,
19
] for images.
The LDDMM framework uses a group of diffeomorphisms constructed through
integration of time-varying vector fields that belong to a RKHS. This gives a
geodesic flow of diffeomorphisms
φ
t
for a continuous parameter
t
within the interval
[0
,
1]
gives
the desired transformation
φ
1
which is required for mapping one image to the other.
The path of any point
x
is defined by
φ
t
(
x
)
. At time
t
=0
, we have the identity mapping
φ
0
. The mapping at time
1
and leads to the final position
φ
(
x
)=
φ
1
(
x
)
. By following the reverse path, we can compute the inverse deformation.
The basic idea of the LDDMM framework is to minimize the squared distance
between objects after transformation (this is the similarity energy or data attachment
term) with a penalization for the energy of the deformation trajectory (regulariza-
tion). A time
t
, the speed at point
y
=
φ
t
(
x
)
is
v
t
(
y
)=
dφ
t
(
y
)
/dt
. This suggests to
define the energy of a velocity field at deformation
φ
t
using a right-invariant metric:
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