Image Processing Reference

In-Depth Information

such as hyperspectral images. We proceed as follows. Let us first assume that fusing

a large sequence of input images does asymptotically yield an acceptable quality

image. The progression of fusion is expected to be smooth and convergent in nature

due to strong inter-band correlations in hyperspectral data. During the progression,

a few initial image bands impart a higher amount of information towards the resul-

tant. The fusion technique captures most of the information from the initial bands.

Thus, the redundancy in the subsequent bands increases as the fusion progresses.

The hyperspectral bands, especially towards the end of the fusion process become

more redundant, and impart a lesser amount of information toward the resultant. This

characteristic of the hyperspectral image data can be easily verified from the analysis

of correlation coefficient presented later in this section. The objective of the perfor-

mance measures presented in this section is to study the progression of the fusion

process as more and more image bands are fused. We consider the final fused image

(i.e.,
F
K
) as the reference during this study. The objective of these tests is to answer

the question: Does the process converge uniformly as desired or does it show any

oscillatory behavior? Further, how fast is the progression? For example, a very quick

convergence of the sequence implies that the subsequent image bands are unable to

provide any additional information during the process. Therefore, a fusion technique

providing a very quick convergence is not desirable. These measures determine the

suitability of a given technique for the fusion of a large number of bands (or images),

but on their own these tests do not provide information about the accuracy of the

technique.

1.
Bhattacharyya Distance:
The Bhattacharyya distance is a measure of similarity

between two probability density functions (pdf) [84]. This measure computes

the extent of overlap between two density functions in order to determine their

closeness. In case of digital images, the normalized histogram of the image is

considered to be equivalent to the density function of its gray values. The Bhat-

tacharyya distance
BD
k
for each of the
k
incrementally fused images with respect

to the final image can be calculated. Through this measure, one can quantify the

similarity or closeness of the
k
th incrementally fused image
F
k
with respect to

the final fused image
F
in terms of their pdfs. As more hyperspectral bands con-

tribute towards the result of fusion, the corresponding incremental fused image

is expected to be increasingly similar to the final fused image. As the similarity

between the images grows, the overlap between their density functions (or his-

tograms) also increases. For the two discrete probability density functions

P
k
(ζ )

and

P
ref
(ζ )

over the same domain

ζ
∈
Z

, the Bhattacharyya distance
BD
k
is

defined by Eq. (
9.5
).

⎛

⎝

ζ
∈
Z

⎞

⎠
,

BD
k
≡

BD

(P
k
,P
ref
)
=−

ln

P
k
(ζ )P
ref
(ζ )

(9.5)

where

P
k
is the same for

the incrementally fused image
F
k
. The plot of
BD
k
as a function of
k
provides

P
ref
is the histogram of the final fused image
F
, and