Image Processing Reference

In-Depth Information

pixel in the image band has no correlation with the true scene, and hence, it can be

regarded as noise.

The problem of fusion is now equivalent to the estimation of parameters of the

image formation model. Ideally, one can compute these parameters from the pre-

cise information about the sensory system. However, this information is not always

available. Also, the method does not remain generic after incorporation of the device-

specific information. Therefore, let us assume no knowledge about the sensor device,

and address this problem blindly using only the available hyperspectral data. The

parameters of the image formation model should be computed from certain prop-

erties of the data. The model, however, being a first order approximation does not

reflect a perfect relationship between the input and the output images. The modeling

error may lead to an erroneous resultant image. Considering this as random perturba-

tions and to deal with these perturbations in an efficient and systematic manner, we

formulate the estimation problem in a Bayesian framework. The solution is obtained

through the maximum
a posteriori
estimate of the corresponding formulation.

In the next section we explain in brief the Bayesian framework and some of its

applications in vision and image processing. We describe the image formation model

developed by Sharma et al. [164] in Sect.
5.3
. The computation of the parameters of

this model is explained in Sect.
5.4
. The Bayesian solution and its implementation

details are provided in Sects.
5.5
and
5.6
, respectively. Some illustrative results are

provided in Sect.
5.7
, while Sect.
5.8
summarizes the chapter.

5.2 Bayesian Framework

We often come across with problems in various fields of engineering where the data

available to us are either incomplete or imperfect. The data, which might be available

in the formof a set of observations from some system, may contain additional compo-

nents, unwanted for the given application. When one wants to make some inferences

or calculate some parameters from such data, results are likely to be erroneous due to

the unwanted component, known as noise. Alternatively, one may not have an exact

model of the system under investigation, which can also lead to imprecise solutions.

Thus, the uncertainty or the incompleteness of the data or model are primary sources

of errors in the inference.

Bayesian estimation is a probabilistic framework that has been extensively used

for reasoning and decision making under uncertainty. It provides a systematic way of

approaching the solution when the knowledge about the system is incomplete. This

framework takes into account the posterior probability related to a specific event

pertaining to the quantity of interest. A posterior probability refers to the condi-

tional probability assigned after the relevant event has been taken into consideration.

Bayesian estimator provides a solution by maximizing the posterior probability. Due

to incomplete knowledge, one may not be able to obtain the exact solution, but can

obtain a solution that is close to the desired one. The process of Bayesian estimation

can be illustrated as follows. Let

θ

be some underlying variable of interest while we