Biomedical Engineering Reference
In-Depth Information
the correct label for any location
x
in
R
through the mapping
x
→
A
(
T
(
x
))
.
(11.1)
The transformation
T
is parameterized by a
p
-dimensional parameter vector
p
∈ R
p
. The process of finding the vector
p
that describes the “correct” trans-
formation is known as image registration. One of the images,
R
, remains fixed
during registration, while the other,
F
, is transformed in space. The fixed image
R
is commonly referred to as the “reference image”, the transformed image
F
is called the “floating image”.
The terminology used in the remainder of this chapter is as follows. We
refer to the already segmented image as the
atlas image
and the image to be
segmented as the
raw image
. The coordinates of the raw image are mapped by
registration onto those of the atlas image and thereby provide a segmentation
of the former. In the context of non-rigid registration, the atlas image is to be
deformed while the raw image remains fixed. The correspondence between the
common terms for both images in image registration and in the present context
is such that the atlas image acts as the
floating image
during registration while
the raw image acts as the
reference
(or
target
)
image
.
11.3.1
Entropy-Based Image Similarity
It is not usually known a priori, what the correct mapping between the two
images
R
and
F
is. Instead, the correctness of any given transformation is usually
quantified by a so-called similarity measure. This measure is a scalar function
S
:
R
p
→ R
designed so that higher values of
S
correspond to better matches.
That is, if for two parameter vectors,
p
1
and
p
2
, we have
S
(
p
1
)
>
S
(
p
2
), then
the mapping
T
1
parameterized by
p
1
is assumed to be “more correct” than the
mapping
T
2
described by
p
2
. Again, since the correct mapping is not known,
S
can only be a more or less suitable approximation to the true correctness. The
registration is performed by finding the parameter vector
p
that maximizes
S
.
A similarity measure that has been empirically found to be particularly well-
suited for many registration applications is mutual information (MI) [36, 77, 80].
It is based on the information-theoretic entropy concept and is defined as
S
MI
=
H
R
+
H
F
−
H
RF
,
(11.2)
where
H
R
is the entropy of image
R
,
H
F
is the entropy of image
F
, and
H
RF
is the joint entropy of corresponding voxel pairs between the two images. A
