Biomedical Engineering Reference
In-Depth Information
from the previous level. We perform the registration from resolution
k
c
until res-
olution
k
f
(in general
k
f
=
0). This is done using cost function (8.2) but with
∇
f
k
(
s
,
t
)
=
∇
f
k
(
s
+
ˆ
w
s
,
t
2
) and
f
t
(
s
,
t
)
=
f
k
(
s
+
ˆ
w
s
,
t
2
)
−
f
k
(
s
,
t
1
) instead of
∇
f
k
(
s
,
t
) and
f
t
(
s
,
t
).
To compute the spatial and temporal gradients, we construct the warped
volume
f
k
(
s
+
ˆ
w
s
,
t
2
) from volume
f
k
(
s
,
t
2
) and the deformation field ˆ
w
s
, using
trilinear interpolation. The spatial gradient is hence calculated using the recur-
sive implementation of the derivatives of the Gaussian [45]. At each voxel, we
calculate the difference between the source volume and the reconstructed vol-
ume, and the result is filtered with a Gaussian to construct the temporal gradient.
As previously, these quantities come from the linearization of the constancy as-
sumption expressed for the whole displacement ˆ
w
s
+
d
w
s
. The regularization
term becomes
<
s
,
r
>
∈
C
ρ
2
(
||
ˆ
w
s
+
d
w
s
−
ˆ
w
r
−
d
w
r
||
).
8.3.2.6
Multigrid Minimization Scheme
Motivations.
The direct minimization of Eq. (8.3) is intractable. Some iter-
ative procedure has to be designed. Unfortunately, the propagation of infor-
mation through local interaction is often very slow, leading to an extremely
time-consuming algorithm. To overcome this difficulty (which is classical in
computer vision when minimizing a cost function involving a large number of
variables), multigrid approaches have been designed and used in the field of
computer vision [48, 98, 133]. Multigrid minimization consists in performing the
estimation through a set of nested subspaces. As the algorithm goes further,
the dimension of these subspaces increases, thus leading to a more accurate
estimation. In practice, the multigrid minimization usually consists in choosing
a set of basis functions and estimating the projection of the “real” solution on
the space spanned by these basis functions.
Description.
At each level of resolution, we use a multigrid minimization
(see Fig. 8.2) based on successive partitions of the initial volume [98]. At each
resolution level
k
, and at each grid level
, corresponding to a partition of cubes,
we estimate an incremental deformation field
d
w
k
,
that refines the estimate
ˆ
w
k
, obtained from the previous resolution levels. This minimization strategy,
where the starting point is provided by the previous result—which we hope
to be a rough estimate of the desired solution—improves the quality and the
