Biomedical Engineering Reference
In-Depth Information
9.4.5
Existence and Unicity
Since the useful range of the parameters
c
is naturally compactly constrained
(for example, by excluding displacements larger than the image size) and since
the criterion
E
is non-negative and continuous with respect to
c
(if at least
linear interpolation for
f
t
is used), it is clear that
E
as a function of
c
has
a minimum; i.e., the proposed problem has a solution. However, depending on
the images at hand, the solution does not have to be unique and there can
be local minima. Fortunately, this does not pose problems in practice, thanks
to a multiresolution approach (section 9.4.6.2) which smoothes out images at
coarse levels and brings us sufficiently close to the solution at fine resolution
levels. The algorithm will find a solution if started within the attraction basin
of that solution. The virtual springs (section 9.4.7) play a role of an a prioiri
information and a regularization term; extra regularization can be applied, if
desired [88, 107] (section 9.4.9).
9.4.6
Optimization Strategy
9.4.6.1
Optimization Algorithm
Recall from (9.7) and (9.10) that we need to minimize a criterion
E
with respect
to a finite number of parameters
c
. To determine which of the many available
algorithms performs best in our context, we tested four local iterative algorithms
which can be cast into a following common framework: At each step
i
we take
the actual estimate
c
(
i
)
and calculate a proposed update
c
(
i
)
. If the step is
successful, i.e., the criterion decreases, then the proposed point is accepted,
c
(
i
+
1)
=
c
(
i
)
+
c
(
i
)
. Otherwise, a more conservative update
c
(
i
)
is calculated,
and the test is repeated.
1.
Gradient descent with feedback step size adjustment.
The update rule is:
c
(
i
)
=−
µ
∇
c
E
(
c
(
i
)
). After a successful step,
µ
is multiplied by
µ
f
, other-
wise it is divided by
µ
f
.
6
2.
Gradient descent with quadratic step size estimation.
We choose a step
size
µ
∗
minimizing the following approximation of the criterion around
c
(
i
)
:
E
(
c
(
i
)
+
x
)
=
E
(
c
(
i
)
)
+
x
T
∇
c
E
(
c
(
i
)
)
+
α
x
2
, where
α
is identified from the
6
We used
µ
f
=
10 and
µ
f
=
15.
