Biomedical Engineering Reference
In-Depth Information
When an undesirable prior
q
(
u
,v
) is given, the cross-entropy, reversed cross-
entropy, or symmetric divergence are maximized. When images are in registra-
tion, we would expect that the voxel values at the same coordinates in the two
images are related. Therefore, unfavorable pdfs can be uniform, proportional
to one of the marginal pdfs, or the product of two marginal pdfs. The case
where the unfavorable pdf is the product of two marginal pdfs is worth noting.
In this case, the voxel values are statistically independent since their joint pdf is
the product of their marginal pdfs. Cross-entropy maximization based on these
three priori estimates degenerate to entropy minimization, conditional entropy
minimization, and mutual information maximization, respectively, [18].
Let's assume the marginal pdfs of
u
and
v
are
h
f
(
u
) and
h
g
(
v
), respectively,
and the unfavorable priori pdf
q
(
u
,v
) is equal to the product of two marginal
pdfs. Putting
q
=
h
f
(
u
)
h
g
(
v
) into Eqs. (10.1) (10.2) and (10.3), one arrives at
p
(
u
,v
)
h
f
(
u
)
h
g
(
v
)
dud
v,
p
(
u
,v
) log
(10.4)
η
CE
=
D
h
f
(
u
)
h
g
(
v
) log
h
f
(
u
)
h
g
(
v
)
p
(
u
,v
)
dud
v,
(10.5)
η
RCE
=
D
and
p
(
u
,v
)
h
f
(
u
)
h
g
(
v
)
dud
v.
[
p
(
u
,v
)
−
h
f
(
u
)
h
g
(
v
)] log
(10.6)
η
SD
=
D
One may notice that Eq. (10.4) is the definition of mutual information. The
measures defined in Eqs. (10.4-10.6) are maximzied when used in image
registration.
10.3
Implementation
Figure 10.1 shows the overall flow chart of the registration process. It applies
to all three similarity measures as well as the minimization and maximization
cases. When the similarity measures are minimized, a favorable priori is used
in the similarity calculation. When they are maximized, the priori estimate is
calculated from the marginal pdfs on-the-fly. As can be seen, no matter which
measure is used, the registration algorithm has the same structure and they can
be implemented in the same manner. Since Eqs. (10.1)-(10.6) are in a continuous
form, they must be discretized to be solved numerically.
