Image Processing Reference

In-Depth Information

been derived from certain measure related to the local contrast. A similar approach

has been employed for fusion of multi-exposure images to form a high dynamic

range (HDR)-like image in [145]. The basic formulation of the fusion problem using

a variational framework is provided in Eq. (
6.6
).

⎧

⎨

F

2

K

F

(

x

,

y

)
=

argmin

F

(

x

,

y

)
−

w
k
(

x

,

y

)

I
k
(

x

,

y

)

⎩

x

y

k

=

1

+
λ
var
∂

d
x
d
y

2

2

F

(

x

,

y

)

+
∂

F

(

x

,

y

)

,

(6.6)

∂

x

∂

y

where

λ
var
is the regularization parameter, and
F
is the fused image. This parameter

is set by the user to obtain a desired balance between the weightage assigned to the

matte (first term in Eq. (
6.6
)), and the weightage assigned to the smoothness of the

fused image. The second term in the above Equation is known as the regularization

term. In the present case of hyperspectral image fusion, we want the input bands

to merge smoothly without producing any visible artifacts in the fused image. As

most natural images have a moderate to high amount of spatial correlation, any

sharp and arbitrarily discontinuous region generated via fusion may turn out to be

a visible artifact. In order to avoid such artifacts, we would like the fused image

to have a certain degree of smoothness. One of the ways of accomplishing this

objective is to penalize the departure of the fused image from smoothness which

can be characterized by high values of its spatial gradient. We incorporate this in

the form of a regularizing term, and seek for the minimization of the corresponding

functional defined in Eq. (
6.3
). It may, however, be noted that several other types of

regularization or penalty functions have been proposed in the literature. The choice

of the regularization function is determined by nature of the data, and is of utmost

importance for obtaining the desired output.

We have the basic formulation of our fusion problemas given byEq. (
6.6
). Now, we

shall focus on designing an appropriate function for the calculation of fusion weights.

Pixels with higher local contrast bring out a high amount of visual information.

We quantify the local contrast of a given pixel by evaluating the variance in its

spatial neighborhood. Thus, fusion weights
w
k
(

,

)

for the
k
-th hyperspectral band

should be directly proportional to the local variance

x

y

2

σ

k
(

,

)

. However, in case

of homogeneous regions in hyperspectral bands, the variance becomes zero, and

such regions may not get an appropriate representation in the final fused image.

To alleviate this problem, we incorporate a positive valued additive constant
C
to

the weighting function. The constant
C
serves the similar purposes as the constant

from the contrast, we also want to balance the saturation level in the resultant fused

image. The fused image should have very little under- or over-saturated regions. It

is also desired that the fused image be quite close to the mean observation along

the band for radiometric consideration. Hence, we consider the weight
w
k
(

x

y

x

,

y

)

to

2
, with the proportionality constant being

be proportional to

(

F

(

x

,

y

)
−

I
k
(

x

,

y

))