Image Processing Reference

In-Depth Information

7.3 Optimization-Based Solution to Fusion

We want to obtain a single image representation of the hyperspectral data cube by

selectively merging the useful features from the set of source images. When the

final image is formed as a linear combination of the pixels from all of the input

image bands, which is the usual case, no properties of the resultant fused image are

being explicitly considered in most of the existing fusion methodologies. As our

primary objective is the visualization-oriented fusion of the hyperspectral images, it

is highly desirable to use a fusion strategy which generates the resultant image with

a certain characteristics. The fusion technique presented here is based on some of

the aforementioned characteristics of the fused image. Based on these, we discuss

how a multi-objective cost function has been developed in order to transform the

fusion problem into an optimization framework in the next subsection. The solution

provides an optimal set of weights for the purpose of fusion of the image bands as

has been explained subsequently.

7.3.1 Formulation of Objective Function

The basic approach toward the generation of the fused image remains the same as

that of any pixel-based technique where the fused image is a linear combination of

the input spectral bands. The different sets of weights generated by different fusion

techniques are also popularly known as

α

-mattes in computer graphics literature.

We have already discussed the primary expression of fusion using these

α

-mattes in

earlier chapters of the topic.

The fused image
F

is generated from a linear combi-

nation of the observations across spectral bands at every pixel location as represented

by Eq. (
7.1
).

(

x

,

y

)

of dimension

(

X

×

Y

)

K

F

(

x

,

y

)
=

1
α
k
(

x

,

y

)

I
k
(

x

,

y

)
∀
(

x

,

y

),

(7.1)

k

=

α
k
(

,

)

α

where

-matte for the corresponding pixel which acts as the

fusion weight. The fusion weights

x

y

is the value of

α
k
(

x

,

y

)

should satisfy the following properties-

1. At any given pixel, the sum of all the weights should be equal to unity, i.e.,

K

1
α
k
(

x

,

y

)
=

1

∀
(

x

,

y

).

k

=

2. The weights should be non-negative, i.e.,

α
k
(

x

,

y

)
≥

0

∀
(

x

,

y

).