Biomedical Engineering Reference
In-Depth Information
8.2.3.7
Joint estimation of Intensity and
Geometric Transformations
Many artifacts can modify the luminance of an MR image. One of them is the
inhomogeneity of the magnetic field for instance [80]. As a consequence, the
hypothesis of luminance conservation might not be valid anywhere. One solu-
tion consists in using robust estimators to get rid of inconsistent data. Another
solution consists in estimating jointly an intensity correction and a spatial trans-
formation [53, 55, 65].
Gupta and Prince [65] propose an affine correction model for tagged MR:
f
(
r
+
dr
,
t
+
dt
)
=
m
(
r
,
dr
,
t
,
dt
)
f
(
r
,
t
)
+
c
(
r
,
dr
,
t
,
dt
). The optical flow equa-
tion then becomes:
f
(
r
,
t
)
+∇
f
(
r
,
t
)
·
U
(
r
,
t
)
−
f
(
r
,
t
)
∂
m
(
r
,
t
)
∂
t
−
∂
c
(
r
,
t
)
∂
t
=
0
.
The equation is solved in a variational framework using a first-order regular-
ization.
Friston [55] and Feldmar [53] propose to embed the intensity correction and
the spatial transformation in the same cost functional:
(
I
2
(
f
(
M
i
))
−
g
(
I
1
(
M
i
)
,
M
i
))
2
C
(
f
,
g
)
=
,
M
i
∈
i
1
where
f
is the 3D transformation and
g
is the intensity correction. Feldmar
generalizes this approach and considers 3D images as 4D surfaces. The criterion
becomes:
d
((
f
(
x
j
)
,
g
(
x
j
,
i
j
))
,
CP
4
D
(
f
(
x
j
)
,
g
(
x
j
,
i
j
)))
2
C
(
f
,
g
)
=
,
(
x
j
,
i
j
)
where
x
j
is the point of intensity
i
j
and
CP
4
D
is the function that renders the
closest point. In this sense, this method is a generalization of the ICP (iterative
closest point) algorithm. Functions
f
and
g
can be modeled according to the
application. For instance, for a intra-subject monomodal registration,
f
is rigid
and
g
is the identity. For inter-subject registration,
f
can be a combination of
radial basis functions and
f
should correct acquisition artifacts.
