Biomedical Engineering Reference
In-Depth Information
are given by
k
∈
G
µ
[
k
]
e
j
n
,θ
[
k
]
and
k
∈
G
η
[
k
]
e
j
n
,θ
[
k
]
u
d
[
n
]
=
w
d
[
n
]
=
(6.4)
for
n
∈
G
and
G
={
(
n
1
,
n
2
,
n
3
):0
≤
n
1
<
N
1
,
0
≤
n
2
<
N
2
,
0
≤
n
3
<
N
3
}
. The
displacement fields associated with the inverse of the forward and reverse trans-
formations are given by replacing
u
,
w
,
µ
, and
η
in Eq. (6.4) with
u
,
w
,
µ
, and
η
, respectively. The Fourier series parameterization is periodic and therefore
imposes cyclic boundary conditions on the boundary of
.
6.2.7
Multiresolution Registration
Multiresolution formulation of the registration problem has the benefit of min-
imizing computation time and helps to avoid local registration minima. The
Fourier series parameterization in Eq. (6.4) is an example of a multiresolution
decomposition of the displacement fields in parameter space. Let
G
[
r
]
=
G
\
G
[
r
]
represent a family of subsets of
d
where
G
[
r
]
={
n
∈
G
|
r
1
<
n
1
<
N
1
−
r
1
;
r
2
<
n
2
<
N
2
−
r
2
;
r
3
<
n
3
<
N
3
−
r
3
}
and the set subtraction notation
A
\
B
is defined
as all elements of A not in B. In practice, the low frequency basis coefficients
are estimated before the higher ones allowing the global image features to be
registered before the local features. This is accomplished by replacing Eq. (6.4)
by
k
∈
d
[
r
]
µ
[
k
]
e
j
n
,θ
[
k
]
and
k
∈
d
[
r
]
η
[
k
]
e
j
n
,θ
[
k
]
.
u
d
[
n
,
r
]
=
w
d
[
n
,
r
]
=
(6.5)
where
r
∈
G
determines the number of harmonics used to represent the dis-
placement fields. The components of
r
are initially set small and are periodi-
cally increased throughout the iterative minimization procedure. The set
G
[
r
]
can be replaced by
G
when all of the components of
r
are greater than or equal
to (
N
−
1)
/
2 since the set
G
[
r
] is empty. The constants
r
1
,
r
2
, and
r
3
represent
the largest
x
1
,
x
2
, and
x
3
harmonic components of the displacement fields. Each
displacement field in Eq. (6.5) is efficiently computed using three
N
1
×
N
2
×
N
3
FFTs, i.e., each component of the 3
×
1 vectors
u
d
and
w
d
are computed with a
FFT after zeroing out the coefficients not present in the summations.
The approach of spatial multiresolution is to register two images at a course
spatial resolution initially and then to refine the registration at a higher spatial
