Biomedical Engineering Reference
In-Depth Information
Then, the results are propagated to the next finer level and used as a starting
guess for solving the task at that level. This procedure is iterated until the finest
level is reached.
In our algorithm, multiresolution is used twice. First, we build an
image
pyramid
: a set of gradually reduced versions of the original image [108, 109].
This pyramid is compatible with our image representation (9.8) and is optimal
in the
L
2
-sense (i.e., compatible with the SSD criterion (9.7)), which ensures
that the approximation made by substituting the lower resolution image is the
best possible. We reduce images up to the size of 16
∼
32 pixels, which works
well in most cases. The coarse versions of images (half size) are generated using
a reduction operator (see Section 9.4.8.4) and coarse level solutions are extrap-
olated to finer levels using an expansion operator (cubic spline interpolation).
Second, we use multiresolution for the warping function as well. We start
with a deformation
g
described with very few parameters
c
k
, and with a large
distance
h
between knots. After the optimization of
c
k
is complete, we halve
the distance between knots. This approximately corresponds to doubling the
number of knots in each direction, i.e., quadrupling (in 2D) the number of co-
efficients
c
k
. Because of the two-scale spline relation, we can
exactly
represent
the warping function from the old, coarse space, in the new, finer space. The
sequence obeys
h
j
+
1
=
h
j
/
2. The process starts with
g
being identity.
The global strategy combines the two multiresolutions by alternatively de-
creasing the scales for the image and for the model.
The consequence of using multiresolution is that the algorithm works best
for images and deformations that follow the multiresolution model; i.e., when
a low resolution version is a good approximation of the finer resolution version.
9.4.7
Semi-Automatic Registration
We realize that although the multiresolution approach leads to a very robust
registration algorithm, there are cases when it is misled by an apparent similarity
of features which do not correspond physically. Therefore, we developed an
extension of the algorithm which can use expert hints. The hints come in the
form of a set of landmarks and are used to gear the algorithm towards the correct
solution. Similar idea appeared also for non-parametric approaches [110, 111].
The landmark information is incorporated in the automatic process using the
concept of
virtual springs
, tying each pair of corresponding points together. We
