Biomedical Engineering Reference
In-Depth Information
so-called DFD (displaced frame difference):
f
(
s
+
w
s
,
t
1
)
−
f
(
s
,
t
2
)
=
0
,
where
s
is a voxel of the volume,
t
1
and
t
2
are the indexes of the volumes (tem-
poral indexes for a dynamic acquisition, indexes in a database for multisubject
registration),
f
is the luminance function and
w
the expected 3D displacement
field. The DFD may not be valid everywhere, because of noise and intensity
inhomogeneities of MR acquisition. The robustness of the registration process
with respect to acquisition artifacts will be discussed later on, the sections 8.3.2
and 8.3.3.
Generally, a linear expansion of this equation is preferred :
∇
f
(
s
,
t
)
·
w
s
+
f
t
(
s
,
t
)
=
0 where
∇
f
(
s
,
t
) stands for the spatial gradient of luminance and
f
t
(
s
,
t
) is the voxelwise difference between the two volumes. The resulting set
of undetermined equations has to be complemented with some prior on the de-
formation field. Using an energy-based framework (which can be viewed either
from the Bayesian point of view, or from the one of the regularization theory),
the registration problem may be formulated as the minimization of the following
cost function:
[
∇
f
(
s
,
t
)
·
w
s
+
f
t
(
s
,
t
)]
2
2
U
(
w
;
f
)
=
+
α
||
w
s
−
w
r
||
,
(8.1)
s
∈
S
<
s
,
r
>
∈
C
where
S
is the voxel lattice,
C
is the set of neighboring pairs w.r.t. a given neigh-
borhood system
V
on
S
(<
s
,
r
>
∈
C
⇔
s
∈
V
(
r
)), and
α
controls the balance
between the two energy terms. The first term captures the brightness con-
stancy constraint, thus modeling the interaction between the field (unknown
variables) and the data (given variables), whereas the second term captures a
simple smoothness prior. The weaknesses of this formulation are known:
(a) Due to the linearization, the optical flow constraint (OFC) is not valid in
case of large displacements.
(b) The OFC might not be valid in all the regions of the volume, be-
cause of the acquisition noise, intensity non-uniformity in MRI data, and
occlusions.
(c) The “real” field is not globally smooth and it probably contains dis-
continuities that might not be preserved because of the quadratic
smoothing.
