Biomedical Engineering Reference
In-Depth Information
10.3.1
Transformation
Although all three registration measures can be applied to more general trans-
formations, we restricted the transformation to rigid-body transformations. For
the rigid-body registration, the registration parameter is a six-dimensional vec-
tor, (
θ
x
,θ
y
,θ
z
,
t
x
,
t
y
,
t
z
), where
θ
x
,θ
y
, and
θ
z
are rotation angles in degrees around
the
x
-,
y
-, and
z
-axis, respectively, and
t
x
,
t
y
, and
t
z
are translation offsets along
the
x
-,
y
-, and
z
-axis, respectively.
There is a 4
×
4 matrix corresponding to any of the three rotations in a homo-
geneous coordinate system [37]. Since a successive application of the rotations
amounts to matrix multiplication which is not commutative, the order of these
rotations is important. It is assumed in this chapter that the rotation angles
are applied around the
x
-,
y
-, and
z
-axis, in that order, and that the rotation is
performed before the translation.
10.3.2
Marginal, Joint, and Priori Probability
Density Functions
The marginal and joint pdfs of gray value pairs were estimated from normalized
histograms. The maximum voxel value of image
f
was found first. The voxel
values in image
f
were then divided into 64 discrete levels. Similarly, the voxel
values in image
g
were divided into 64 discrete values. The size of the joint
histogram is thus 64
×
64. Alternatively, one can change the bin size adaptively
[18]. In the overlapping volume, the histograms of voxel values in images
f
and
g
, and of the voxel pairs were calculated by binning the voxel values and value
pairs. The normalized histograms then give estimations of the marginal as well
as the joint distributions. To compute the cross-entropy, reversed cross-entropy,
and symmetric divergence, these estimated pdfs can be directly substituted into
the formulas defined in Eqs. (10.1)-(10.6).
When a favorable priori estimate is employed in the registration, it can be
computed in the same fashion as that for the joint pdf. The priori estimate shall
be calculated with the same size of histograms as that for the joint histogram.
10.3.3
Interpolation
A grid point in one volume will generally not exactly coincide with another
grid point in the transformed space. Since the voxel values of the reference
