Image Processing Reference

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of the underlying true scene [164]. The true scene is obtained using a MAP estimator.

Yang and Blum have proposed modifications to this model in a multi-scale decom-

posed framework [196]. Kumar [94], and Kumar and Dass [95] have also demon-

strated effectiveness of the Bayesian technique for generalized image fusion. For the

problem of fusion, the images to be fused act as observations, and the fused image

is the quantity of interest. These two are related to each other using an appropriate

image formation model which defines the corresponding likelihood function. Now

we shall explain in detail the image formation model.

5.3 Model of Image Formation

In the past decade, Sharma et al. introduced a statistical technique for generalized

image fusion. Their technique consists of defining an image formation model, the

perturbation noise modeled with a Gaussian distribution, and a Bayesian technique to

solve the fusion problem [164]. This model of image formation is given by Eq. (
5.4
).

I
k
(

x

,

y

)
=
β
k
(

x

,

y

)

F

(

x

,

y

)
+
η
k
(

x

,

y

),

(5.4)

where
I
k
(

x

,

y

)

denotes the observation of a true scene pixel
F

(

x

,

y

)

captured by the

sensor
k
.

is known

as the sensor selectivity factor which determines how well the given observation has

captured the true scene. The maximum value that

η
k
(

x

,

y

)

indicates the noise or disturbance component.

β
k
(

x

,

y

)

β
k
(

,

)

can achieve is unity,

which indicates that the particular pixel is exactly same as it should be in the fused

image (in the absence of noise). On the other extreme,

x

y

can have a value of

zero, when the pixel is essentially pure noise without any contribution towards the

final result.

Yang and Blum have proposed a multiscale transform version of this fusion tech-

nique that can also handle non-Gaussian disturbances [196]. In their model the sensor

selectivity factor can assume values of 0 or

β
k
(

x

,

y

)

1 only. The negative value indicates a

particular case of polarity reversal for IR images [196]. This discrete set of values

brings out only two possibilities- either the sensor can
see
the object, or it fails to

see
it. The

±

β

factor needs to be calculated from the available data only. In [196], the

value of

has been calculated by minimizing a function related to the sensor noise.

Fusion using the statistical model of image formation has been enhanced by

allowing the sensor selectivity factor

β

β

to take continuous values in the range [0, 1]

in [94, 95]. However, the value of

has been assumed to be constant over smaller

blocks of an image [95, 164]. These values of

β

have been computed from the

principal eigenvector of these smaller blocks. The values of

β

, therefore, are constant

over image blocks, but can be totally discontinuous across the adjacent blocks due

to the data dependent nature of eigenvectors. Xu et al. have modeled the sensor

selectivity factor using a Markov random field (MRF), however it can take values

from the discrete set

β

only [192, 193]. They have modeled the fused image

also to be an MRF which acts as the prior for the MAP formulation.

{

0

,

1

}