Biomedical Engineering Reference
In-Depth Information
of Navier-Stokes can be written as
∂
u
∂
t
−
ν
u
+
(
u
· ∇
)
u
+ ∇
p
=
0
where
ν
is the fluid viscosity,
u
its speed and
p
its pressure. This equation is
highly non-linear (cross-terms) and its resolution is complex, leading to large
computation times. Christensen imposes the constraint that the Jacobian be
positive [27], leading to an homeomorphic transformation.
Christensen and Johnson [28] have extended the registration approach to
introduce the reversibility constraint. Given two subjects
A
and
B
, the method
jointly estimates transformation from
A
to
B
and from
B
to
A
. The inverse
consistency error is zero when the forward and reverse transformations are
inverses of one another. Furthermore, the transformations obey the rules of
continuum mechanics and are parameterized by Fourier series.
Bro-Nielsen [17] has proposed an improvement to solve the following partial
differential equation:
Lv
=
µ
∇
v
(
x
)
+
(
λ
+
µ
)
di
v
(
v
)
=
f
(
x
,
u
(
x
))
where
u
is the displacement and
v
the instantaneous speed. For a small time
change, internal forces are constant and the equation is linear. While Christensen
uses a finite element scheme, Bro-Nielsen considers the impulse response asso-
ciated with operator
L
. The solution is then expressed as linear combinations of
eigenvectors of operator
L
. This significantly decreases the computation time.
Wang and Staib [146] have also proposed two methods that obey the rule of
continuum mechanics. The methods respect the properties of elastic solids or
viscous fluids. A statistical shape information (sparse set of forces) is mixed with
a luminance information (dense set of forces within a Bayesian framework).
8.2.3.4
Correlation
Cross-correlation is a widespread similarity measure. It has been used by popular
methods such as ANIMAL [35] and Gee
et al.
[59]. ANIMAL uses a multireso-
lution strategy to estimate local linear transformations that maximizes cross-
correlation. At a resolution level
σ
, the regularization is based on the statement
that the norm of displacement vectors should not exceed
σ
. Colllins
et al.
[36]
has extended ANIMAL so that sulcal constraints can be taken into account in
the registration process.
