Biomedical Engineering Reference
In-Depth Information
augment the data part of the criterion
E
with a term
E
s
, corresponding to the
potential energy of the springs, and minimize the sum of the two:
E
c
=
E
+
E
s
.
The spring term is:
i
=
1
α
i
g
(
x
i
)
−
z
i
S
2
E
s
=
(9.11)
where
S
is the number of springs,
α
i
are weighting factors corresponding to
their stiffnesses, and
x
i
, resp.
z
i
, are the landmark positions in the reference,
resp., test images. The spring factors
α
i
control the influence of the particular
landmark pairs.
As an example, we present the registration of an MRI slice from an atlas
7
with a sample MRI test image
8
. To identify the same structures in the test image,
we register it with the unlabeled version of the atlas. Once the geometric corre-
spondence is established, the structures and their labels from the atlas can be
projected onto the test image. The unsupervised registration correctly registers
some of the structures but misses others; in particular, the skull boundary (see
Fig. 9.9). If we now help the algorithm by identifying several landmarks in both
images (Fig. 9.10), the semi-automatic version can recover a plausible deforma-
tion, even though the landmark information alone (using e.g., thin-plate splines)
would not have been enough [112].
Adding the spring term privileges likely solutions based on our a priori knowl-
edge and makes the problem better-posed. The points need not to be image-
dependent landmarks. For example, anchoring the four corners of the image
prevents the solution from degenerating. In this way, the springs play in part the
role of a regularization factor.
9.4.8
Implementation Issues
The purpose of this section is to describe some specific aspects of our implemen-
tation. These are mostly independent of the main philosophy of the algorithm
but can have a major impact on its performance.
7
The atlas is a labeled and annotated collection of images. Courtesy of Harvard Medical
School, http://www.med.harvard.edu/AANLIB/home.html
8
We use a proton density MR image from the Visible Human project
http://www.meddean.luc.edu/lumen/meded/grossanatomy/cross section/index.html. Prior to
registration, the histogram of the test image was matched to that of the reference.
