Biomedical Engineering Reference
In-Depth Information
derivatives of
f
t
in (9.12) and (9.13) can be calculated from (9.8) as a ten-
∂
x
m
(
x
)
=
k
∈
I
b
k
β
n
(
x
m
−
k
m
)
l
=
1
sor product
∂
f
t
l
=
m
β
n
(
x
l
−
k
l
). Second-order partial
derivatives of
f
t
are obtained in a similar fashion.
9.4.8.2
Hessian Approximation
Because the evaluation of the Hessian matrix from (9.13) is costly, several modi-
fications have been devised. The Marquardt-Levenberg approximation assumes
that the term
∂
e
i
∂
f
w
(
i
)
is negligibly small or that it sums to zero on the average. This
reduces (9.13) to
2
E
∂
c
j
,
m
∂
c
k
,
n
=
2
I
∂
x
m
∂
f
t
∂
f
t
∂
∂
x
n
∂
g
m
∂
c
j
,
m
∂
g
n
(9.14)
∂
c
j
,
n
i
∈
I
b
Another simplification is to consider only diagonal terms
∂
2
E
/∂
c
j
,
m
. Obviously,
this diagonal Hessian approximation only makes sense if the basis functions
ϕ
j
do not overlap too much. This is another argument for the B-spline model. Each
such approximation makes the evaluation faster at the expense of precision
which may result in slower convergence. Whether it is advantageous to use some
approximation depends on many factors, including the size and the character
of the data.
9.4.8.3
Gradient Calculation
Similarly to the case of evaluating the deformation
g
, the use of an inte-
ger step size
h
leads to computational savings here too. The expanded ex-
pression for
∂
E
∂
c
j
,
m
can be transformed into a discrete separable
convolution
∂
E
∂
c
j
,
m
j
=
i
w
(
i
)
b
(
j
·
h
−
i
)
=
(
w
∗
b
)
↓
h
, where we have substituted
w
for the
first two factors in (9.12),
b
(
q
)
=
β
n
m
(
−
q
/
h
), and
↓
h
indicates downsampling as
defined by the formula, with elementwise multiplication
j
·
h
. The convolution
kernel
b
is separable and the convolution can be calculated as a sequence of
N
unidimensional convolutions
(
w
∗
b
1
)
↓
h
1
∗···
b
n
↓
h
N
. Because of the downsam-
pling, calculating one output value at step
k
consists of a scalar product with
a filter
b
k
of length (
n
m
+
1)
h
k
and shifting this filter by
h
k
.
