Image Processing Reference
In-Depth Information
This definition is nonnegative, but is not a natural extension of the MI between
two variables and requires estimation of a high-dimensional probability mass
function, which is computationally very expensive when the number of involved
images increases. A different approach is suggested by Zhang and Rangarajan [32].
They define a different pseudometric that for two images is:
R (X,Y)
=
H (X|Y)
+
H (Y|X)
(7.16)
This metric is related to MI by the relationship:
R (X,Y)
=
H (X,Y) MI (Y,X)
(7.17)
The main advantage of this metric is the possibility of a straightforward
extension to the multidimensional case. For three images the metric becomes:
R (X,Y,Z)
=
H (X|Y,Z)
+
H ( Y | X , Z )
+
H( Z | X , Y )
(7.18)
And for N images:
N
1
RX X
(
,
,...,
X
)
=
HX X
(
|
,...,
X
,
X
,...,
X
)
(7.19)
12
N
i
1
i
1
i
+
1
N
i
=
To reduce the computational complexity of the metric, an upper bound of R ,
which is also a metric, can be used that does not require the computation of high-
order joint probability. In the case of three images we have:
1
2
RXYZ
(,,)
=
HX YZ
( |,)
+
HX YZ
( |,)
+
HZ XY
( | ,)
(( |
HX
ZZHXZ
)
+
(
|
))
1
2
1
2
+
((|
HYXHYZ
)
+
(|
)
+
((|
HZXHZ
)
+
(
X
) )
1
2
=
(( ,)
RXY
+
RY Z
(, )
+
RX Z
( , )
(7.20)
so that the metric K can be minimized instead of R :
1
2
KXYZ
(,,)
=
((,)
RXY
+
RYZ
(,)
+
RXZ
(, )
(7.21)
Note that the computation of K does not require the computation of multidi-
mensional joint histograms. The definition of upper bound can be extended to
more than three images.
 
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