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
