Image Processing Reference

In-Depth Information

Chapter 6

Variational Solution

6.1 Introduction

We have studied pixel-based image fusion as a linear combination of multiple input

images. The weights, or more precisely fusion weights, are the data-dependent terms

as they have been calculated from the set of input images (or input hyperspectral bands

in our case). For example, the bilateral filtering-based fusion technique calculates

the fusion weights (
w
) using a predefined function. The Bayesian fusion technique

is based on the computation of the sensor selectivity factor (

) which indicates the

contribution of each pixel toward the fused image. Both of these fusion techniques

explicitly
calculate the weights as a function of the input hyperspectral data. These

functions are also referred to as the weighting functions, while the weights are more

commonly known as the

β

matte in the graphics literature. The fusion weights act as

intermediate variables of the fusion process that define the relationship between the

fused image and the input hyperspectral bands. The purpose of the weighting func-

tion which generates the fusion weights, is to implicitly specify the model for the

fusion process. An explicit computation of fusion weights is, therefore, not required

so long as the underlying model is well specified. In other words, we do not nec-

essarily have to compute the fusion weights independently, if we can appropriately

model the weighting function as a data-dependent term to weigh the hyperspectral

bands. We now explore this possibility, and show how a fusion technique can be

seen that imposing some constraints through a prior related to the smoothness of the

output fused image, gives it a natural and visually pleasing appearance. In order to

incorporate the smoothness constraint, we adopt an approach based on calculus of

variations. This chapter discusses how we can start with an initial estimate of the

fused image, and iteratively converge to obtain the desired resultant image based

on certain constraints as well as the predefined weighting function, without ever

explicitly computing the weights.

In case some of the readers are not familiar, we discuss in very brief about the

calculus of variations in Sect.
6.2
, which will be used as a tool to address the fusion

α