Biomedical Engineering Reference
In-Depth Information
9.4.8.4
Multiresolution Spline Representation
To deploy the multiresolution strategy (see section 9.4.6.2), we need to specify
expansion and reduction operators. We will use the same approach for both the
deformation model and the image model.
Let us consider here a 1D signal represented in a B-spline space
f
(
x
)
=
c
i
β
n
(
x
−
i
)
(9.15)
i
The
expansion operator
E
yields a twice expanded version of
f
which is also
a spline
f
e
=
E
f
,
f
e
(
x
)
=
f
(
x
/
2)
=
d
i
β
n
(
x
−
i
)
(9.16)
i
with coefficients
d
i
given by
d
=
c
↑
2
∗
u
n
(9.17)
where
c
↑
2
denotes upsampled version of
c
and
u
n
is a symmetrical binomial filter
defined in the
z
-domain as
(1
+
z
)
n
+
1
2
n
U
n
(
z
)
=
z
−
(
n
+
1)
/
2
(9.18)
The twice reduced signal
f
(2
x
) cannot be represented as a spline with knots
at integers. We need to resort to approximation and we have chosen the
L
2
optimality as described in [109]. The
reduction operator
R
will yield a projection
(denoted
P
1
) in the original spline space with step size 1.
f
r
=
R
f
,
f
r
(
x
)
=
P
1
f
(2
x
)
,
f
r
(
x
)
=
e
i
β
n
(
x
−
i
)
(9.19)
i
The spline coefficients
e
i
are calculated as
e
=
(
h
∗
c
)
↓
2
∗
b
−
(2
n
+
1)
(9.20)
with prefilter
h
=
b
2
n
+
1
∗
u
n
, where
b
2
n
+
1
is the sequence of sampled values of
a B-spline of degree 2
n
+
1,
b
n
(
i
)
=
β
n
(
i
). Finally,
b
−
(2
n
+
1)
is the inverse filter to
b
2
n
+
1
and the convolution can be handled by
recursive filtering
, as described
in [8, 9].
Because
R
is a projection complementary to
E
, we have the projection iden-
tity
RE
f
=
f
. Extension of both operators to multiple dimensions is trivial
thanks to separability.
