Image Processing Reference
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an estimate of the convergence rate of the technique in terms of the pdfs of
the images F k .Thevalueof BD k reaches the minimum for the final image F ,
which is obviously zero, and it is maximum at the start. It is expected that the
Bhattacharyya distance should monotonically decrease as more image bands are
considered indicating a gradually increasing similarity between the images F k
and F .
2. Jensen-Shannon Distance: The Bhattacharyya distance requires a reference pdf,
and the similarity measure is evaluated with respect to the same. The Jensen-
Shannon distance is another method of measuring the similarity between two
probability distributions, which eliminates the requirement of the reference dis-
tribution [106]. This entity is derived from the Kullback-Leibler divergence that
measures the average number of extra bits required to represent the given probabil-
ity distribution from the reference probability distribution. The Jensen-Shannon
divergence, JSD is the symmetricized modification of Kullback-Leibler (KL)
divergence. The KL distance of a discrete probability distribution
P Z 1 (ζ )
from
another distribution
P Z 2 (ζ )
defined over the same domain
Z
is given by:
log P Z 2 (ζ )
D KL (
Z 2 ,
Z 1 ) =
P Z 2 (ζ )
P Z 1 (ζ ) .
(9.6)
ζ Z
The Jensen-Shannon distance can be calculated as [106]:
1
2 D KL (
1
2 D KL (
JSD
(
Z 2 ,
Z 1 ) =
Z 2 ,
Z mid ) +
Z 1 ,
Z mid ),
(9.7)
1
where
. Similar to the Bhattacharyya dis-
tance, the JSdistance between each of the k incrementally fused images and the
asymptotic reference image can be calculated by Eq. ( 9.8 ).
P Z mid (ζ ) =
2 (P Z 1 (ζ ) + P Z 2 (ζ ))
1
2 D KL (
1
2 D KL (
JSD k
JSD
(
F
,
F k ) =
F
,
Z mid ) +
F k ,
Z mid ).
(9.8)
The plot of JSD k should again monotonically reduce to zero, which indicates that
the histograms of the incrementally fused images asymptotically approach the
histogram of the final image. The JS distance along with the previous measure
(BD) provide the extent of similarity in terms of the histograms (or pdfs) of the
images.
3. Correlation Coefficient: The correlation coefficient CC has often been used
to measure the degree of similarity between two functions [49, 176, 206]. This
measure has been used to evaluate the closeness of the fused image to the con-
stituent bands (or images). The performance assessment of the pan-sharpening
techniques has also been carried out using the correlation coefficient. We employ
this measure CC to study the relation between the incrementally fused images

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