Image Processing Reference

In-Depth Information

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