Biomedical Engineering Reference
In-Depth Information
The algorithm can be used for 2D and 3D problems, is reasonably fast, and
is capable of accepting expert hints in the form of soft
landmark constraints
[1, 19-21].
9.4.1
Problem Formulation
The input images are given as two
N
-dimensional discrete signals
f
r
(
i
) and
f
t
(
i
),
where
i
∈
I
⊂ Z
N
, and
I
is an
N
-dimensional discrete interval representing the
set of all pixel coordinates in the image. We call
f
r
and
f
t
reference
and
test
images, respectively. We suppose that the test image is a geometrically deformed
version of the reference image, and vice versa. This is to say that the points with
the same coordinate
x
in the reference image
f
r
(
x
) and in the warped test
image
f
w
(
x
)
=
f
t
g
(
x
)
should correspond. Here,
f
t
is a continuous version
of the test image and
g
(
x
) is a deformation (correspondence) function to be
identified.
9.4.2
Cost Function
The two images
f
r
,
f
w
will not be identical because of noise and also because
the assumption that there is a geometrical mapping between the two images
is not necessarily correct. Therefore, we define the solution to our registration
problem as the result of the minimization
g
=
arg min
g
∈
G
E
(
g
), where
G
is the
space of all admissible deformation functions
g
. We have chosen the SSD (sum
of squared differences) criterion
1
I
1
I
e
i
=
(
f
w
(
i
)
−
f
r
(
i
))
2
E
=
i
∈
I
i
∈
I
1
I
(
f
t
(
g
(
i
))
−
f
r
(
i
))
2
(9.7)
=
i
∈
I
because it is fast to evaluate and yields a smooth criterion surface which lends
itself well to optimization. Minimization of (9.7) yields the optimal solution
g
in
the ML (maximum likelihood) sense under the assumption that
f
r
is a deformed
(warped) version of
f
t
with i.i.d. (independent and identically distributed) Gaus-
sian noise added to each pixel. The SSD criterion proved to be robust enough,
especially if preprocessing was used to equalize the image values—we mostly
applied high-pass filtering and histogram normalization [98]. In principle, there
