Biomedical Engineering Reference
In-Depth Information
Gee, first interested in mechanical models [59], adopted a statistical Bayesian
framework [58]. Let us note
I
R
the reference volume,
I
T
the target volume,
z
={
I
R
,
I
T
}
the data and
u
the deformation field. The problem is then to minimize
the cost functional:
P
(
z
|
u
)
∝
exp
−{
S
(
I
T
(
x
)
,
I
R
(
x
+
u
(
x
)))
dx
}
,
x
∈
T
where
S
is the similarity measure that has been chosen to be cross-correlation.
The regularization follows either a membrane model
P
(
u
)
∝
λ
(
u
x
+
u
y
)
dx
or
a thin-plate model
P
(
u
)
∝
λ
(
u
xx
+
2
u
xy
+
u
yy
)
dx
. Gee also made it possible
to incorporate landmark points in the registration process. If the transforma-
tion
X
matches
p
i
with
p
i
, the associated potential is:
P
(
Z
=
(
p
i
,
p
i
)
|
θ
=
X
)
∝
exp
−
1
2
σ
2
. This probabilistic approach is useful to mix mechani-
cal regularization, photometric similarity and landmark matching. It also make
it possible to experiment and compare different kinds of regularization [58].
Cachier
et al.
[21] have proposed the Pasha algorithm where the lo-
cal correlation coefficient is used. This coefficient can be efficiently com-
puted using convolutions with a Gaussian window function. The regulariza-
tion is a mixture of competitive and incremental regularization using quadratic
energies.
i
||
X
(
p
i
)
−
p
i
||
8.2.3.5
Demons
Thirion has proposed a method well known as the Demon's algorithm [136]. At
each demon's location, force is computed so as to repulse the model toward
the data. The force depends on the polarity of the point (inside or outside the
model), the image difference and gradients. For small displacements, it has been
shown that the demon's method and optical flow are equivalent. The method is
alternated: computation of forces and regularization of the deformation field
by a Gaussian smoothing. The choice of the smoothing parameter is therefore
important. The Demon's algorithm has been successfully used by Dawant
et al.
[40].
Cachier and Pennec [106] have shown that the Demon's method can be
viewed as a second-order gradient descent of the SSD (Sum of Square Differ-
ences). This amounts to a minmax problem: maximization of similarity and
regularization of solution.
