Image Processing Reference

In-Depth Information

We have already discussed the variance of an image as one of the contrast

measures. A high value of variance indicates a well spread-out values of intensi-

ties in the fused image over the given range, yielding a visually sharper image. A

small value of the variance indicates lack of details in the fused image. An excessive

contrast leads to a non-natural appearance of the image, and thus, a moderate amount

of contrast is necessary to obtain a sharp and visually appealing image. We incorpo-

rate an additional term (

ε
2
) in the objective function for maximizing the variance to

produce high contrast resultant fused images.

x
y

d
x
d
y

⎛

⎞

2

F

(

x

,

y

)

1

XY

⎝
F

⎠

ε
2
(α)
=

(

x

,

y

)
−

d
x
d
y

.

(7.3)

XY

x

y

It can be seen that these two objectives (

ε
2
) are complementary in nature.

The first criterion
pulls in
the far distinct pixels towards the mean gray level
m
I
.The

second criteria
pushes out
the pixels away from the mean. The relative weightages of

these two objectives decide how strong the particular objective is. It converges to the

particular solution where both objectives have been balanced. That is, the resultant

fused image possesses both the characteristics of a higher entropy and a higher local

variance. The right combination of well-exposedness and contrast is what makes an

image visually more appealing.

So far we have not considered any spatial correlation among the pixels while fus-

ing them. The adjacent pixels in the hyperspectral data generally belong to the same

or similar underlying objects in the scene which have similar material composition.

Thus, such pixels in the scene exhibit a high degree of correlation with the neighbor-

hood pixels. A simple and commonly used method to acknowledge this intra-band

correlation is the inclusion of a certain kind of smoothness constraint in the fusion

expression [145]. However, the images are discontinuous at edges and boundaries.

Enforcing a smoothness constraint on the resultant fused image often leads to an

excessive smoothing, thereby blurring edges and washing away the weak and minor

features. This often results in smearing at edges producing visible artifacts in the

fused image. Therefore, enforcing a smoothness constraint on the fused image not

only deteriorates the quality of the result, but also contradicts to one of the objectives

of obtaining high contrast images from Eq. (
7.3
). We want to acknowledge the strong

spatial correlation among the input pixels, but we also want to avoid any smoothing

of fusion result, especially at edges and boundaries. To achieve this, we incorporate

an additional penalty term (

ε
1
and

ε
3
) in the cost function which enforces a smoothness of

the fusion weights (i.e., the

-matte) rather than in the fused image
F
. When the data

in the input hyperspectral bands are smooth, one would expect the corresponding

weights also to be smooth i.e.,

α

α
k
should be smooth along the
x
and
y
directions.

However, when contents of hyperspectral bands exhibit some degree of discontinu-

ity, one would like the corresponding features to get an appropriate representation

in the fused image as well. As we are not imposing any kind of smoothness con-

straint over the output, the features in the input data, representing the discontinuity