Biomedical Engineering Reference
In-Depth Information
almost all similarity measure-based image registration. Note that, if there are
more than two images to be registered (see [17, 33]), these definitions can be
easily extended to the n-dimensional case [18].
Assume the joint pdf of random variable (
u
,v
)is
p
(
u
,v
). Also assume a prior
estimation of
p
(
u
,v
) is available and denoted as
q
(
u
,v
). The cross-entropy is
thus defined on a compact support
D
=
D
u
×
D
v
, as in Shore and Johnson [19]
p
(
u
,v
) log
p
(
u
,v
)
η
CE
(
p
,
q
)
=
q
(
u
,v
)
dud
v,
(10.1)
D
where
D
u
and
D
v
are supports of
u
and
v
, respectively.
If the roles of
q
(
u
,v
) and
p
(
u
,v
) are switched, one has the reversed cross-
entropy,
q
(
u
,v
) log
q
(
u
,v
)
η
RCE
(
p
,
q
)
=
p
(
u
,v
)
dud
v.
(10.2)
D
To make the definition symmetric with regard to
p
(
u
,v
) and
q
(
u
,v
), one can
combine cross-entropy and reversed cross-entropy,
[
p
(
u
,v
)
−
q
(
u
,v
)] log
p
(
u
,v
)
η
SD
(
p
,
q
)
=
q
(
u
,v
)
dud
v,
(10.3)
D
which is the definition of symmetric divergence.
In the case where cross-entropy optimization is utilized to perform image
registration, if a favorable (also known as desirable or likely) priori pdf is given,
an estimate of the true pdf can be found by minimizing the cross-entropy. On
the other hand, if an unfavorable priori pdf is given, an estimate of the true
pdf can be obtained by maximizing the cross-entropy. The same would be true
when reversed cross-entropy and symmetric divergence are used as similarity
measures for image registration.
A favorable pdf can be computed based on previous registration results [34].
Theoretical analysis can also provide information regarding a favorable priori.
For example, the voxel values in images of the same modality and of the same
patient are linearly related and it has been shown that MR image can be used
to simulate a PET image [35]. Since the true pdf is expected to be close to a
likely priori, the cross-entropy, reversed cross-entropy, and symmetric diver-
gence will be minimized. It is worthy to note that the cross-entropy, reversed
cross-entropy, and symmetric divergence are not the only frameworks to exploit
the priori knowledge in registration. The recently proposed likelihood maximiza-
tion approach can also systematically use this knowledge (see [34, 36]).
