Image Processing Reference

In-Depth Information

the average intensity values of the input and the output to remain the same, in

which case the fusion is said to preserve the radiometric mean of the data. This

preservation could be of importance in some remote sensing applications. The

radiometric fidelity facilitates a comparison and visual inspection of two different

datasets which is not possible with the earlier mapping strategy. To achieve this

goal, we would like to map the pixels from the input toward the mean gray value

of the input data (i.e., hyperspectral data cube). This mapping helps maintaining

the average intensity level of the output close to that of the input hyperspectral

data. The fused image will have a low value of the relative bias which has been

defined in [92] as the deviation of the mean of the fused image from the mean

of the input hyperspectral bands. Thus, the fused image has a lesser degree of

radiometric distortion, and is often preferred in remote sensing applications.

As our primary goal is to generate a fused image for visualization purposes, we

follow strategy
S-I
. Thus, the fusion weights should be calculated in such a way

that the gray values of the pixels of the fused image will be close to the central gray

level. From an information theoretic point of view, this problem can be related to the

problem of entropy maximization. We are addressing this problem for the mapping

of the dynamic range of the hyperspectral scene across the wavelength spectrum to

generate a single fused image. We define our objective function to compute a set

of weights

{
α
k
(

x

,

y

)
}

that will maximize the entropy

ε
1
of the fused image
F

(

x

,

y

)

given by Eq. (
7.2
).

log
F

d
x
d
y

(

x

,

y

)

ε
1
(α)
=−

(

,

)

,

F

x

y

(7.2)

0

.

50 e

x

y

where the factor of 0

50 e in the denominator has been introduced to force the trivial

solutiontogoto0.50asexplainedin
S-I
. It should be noted that in the present context,

the entropy of an image has been defined over the normalized pixel values of the

hyperspectral data. This should not be confused with the general andmore commonly

used definition of the image entropy as a measure of average information content in

the picture. The definition of entropy as in Eq. (
7.2
) is commonly used in restoring

astrophysical images under the terminology maximum entropy reconstruction [80].

If one wants to implement the algorithm for strategy (
S-II
), one should replace

the denominator in Eq. (
7.2
) (i.e., 0

.

.

50 e) by
m
I
e, which is the mean of the entire

hyperspectral data
I
.

Let us take a look at the trivial solution for the maximization of Eq. (
7.2
), that is

the solution one obtains after the iterative system converges. In this case, the system

converges to provide a solution that is constant, i.e.,
F

.

Although, a practical fusion system may not reach an exactly constant solution, it

definitely indicates that the output image has a poor contrast. Thus, the maximization

of entropy alone is not enough to obtain a sharply fused image. Therefore, it is

essential to incorporate a complementary objective to the optimization function that

would provide the necessary balance between the entropy maximization and the

contrast of the fused image
F
.

(

x

,

y

)
=

0

.

50

∀
(

x

,

y

)