Biomedical Engineering Reference
In-Depth Information
by a finite number of parameters
c
j
:
g
(
x
)
=
x
+
c
j
ϕ
j
(
x
)
(9.9)
j
∈
J
where
J
is a set of parameter indexes and
ϕ
j
are the corresponding basis
functions. This transforms a variational problem into a much easier finite-
dimensional minimization problem, for which numerous algorithms exist [59].
Moreover, the restriction of the family
G
of all possible functions
g
can already
guarantee some useful properties, such as the regularity (smoothness) of the
solution.
9.4.4.1
B-Spline Deformation Model
There are various possibilities for the choice of the basis functions
ϕ
j
for
the deformation model (9.9). These include polynomials [37], harmonic func-
tions [38,39], radial basis functions [1,102], and wavelets [64,65,103], all of which
have been used in registration algorithms before. However, we have again chosen
B-splines, basically for the same reasons that lead us to choose them to interpo-
late our images (see Section 9.4.3): their good approximation properties, com-
putational efficiency [96], scalability, and additionally physical plausibility [95]
(such as minimizing the “strain energy”
g
2
by cubic B-splines [104, 105]) and
low interdependency thanks to short support. Their property of being able to
represent affine transformations, including rigid body motion, is welcome, too.
We have also evaluated the alternative wavelet representation of the same B-
spline space [106], only to find that direct B-spline representation was again
more efficient [96].
The B-spline deformation model is obtained by substituting a scaled version
of the B-spline (or tensor product thereof) in (9.9)
g
(
x
)
=
x
+
c
j
β
n
m
(
x
/
h
−
j
)
(9.10)
j
∈
I
c
⊂Z
N
where
n
m
is the degree of splines used,
h
is the knot spacing, and the division is
taken elementwise. This corresponds to placing the knots on a regular grid over
the image. For efficiency reasons, we require the node spacing
h
to be integer,
which together with the separability of
β
n
m
(
x
) implies that the values of the
B-spline
β
n
m
(
x
) are only needed at a very small number of points (
n
m
+
1)
h
and
that they can be precalculated.
