Biomedical Engineering Reference
In-Depth Information
is no difficulty in extending this method for more sophisticated pixel-based
similarity measures, such as information-based measures [99], especially mu-
tual information [45], or weighted
p
norms. Only the evaluation of the criterion
and its derivatives (gradient) needs to be changed.
9.4.3
B-Splines and Image Interpolation
We have chosen to interpolate the image using uniform
B-splines
:
f
t
(
x
)
=
b
i
β
n
(
x
−
i
)
(9.8)
i
∈
I
b
⊂Z
N
where
β
n
(
x
) is a tensor product of B-splines
β
n
(
x
) of degree
n
, i.e.,
β
n
(
x
)
=
k
=
1
β
n
(
x
k
), with
x
=
(
x
1
,...,
x
N
). Mirror boundary conditions were used, to
ensure continuity.
Let us recall some basic facts about B-splines. Uniform symmetric B-
splines [100] of degree
n
are piecewise polynomials of degree
n
. The polyno-
mial pieces are delimited by uniformly placed knots. B-splines of degree
n
have
continuous derivatives up to order
n
−
1 everywhere. Their integer shifts form
a basis. The first (degree zero) symmetric B-spline is defined as
β
0
(
x
)
=
1 for
x
∈
(
−
1
2
,
1
2
) and 0 otherwise. Higher order B-splines are defined recursively as
β
n
+
1
=
β
n
∗
β
0
and their support is (
−
n
+
1
n
+
1
2
).
Using B-splines as interpolation functions has many advantages: B-splines
have
good approximation properties
—for example, the error of a cubic B-spline
(
β
3
) approximation decreases asymptotically as
h
4
(measured by any
L
p
or
l
p
norm,
p
∈{
1
,
2
,...,
∞}
). B-splines perform well in comparison with other
bases [11, 101]. B-splines are
fast
—they have a short support (length 4 for
β
3
),
are symmetric, piecewise cubic, and separable in multiple dimensions. They
are simple to compute and
scalable
—the transition from a coarse spline space
with step size (knot distance)
hq
to a finer space with step size
h
is exact for
integer
q
.
2
,
+
9.4.4
Deformation Model Structure
So far, we have considered the deformation function
g
to be an arbitrary admissi-
ble function
R
N
N
. We will restrict it now to a family of functions described
→ R
