Bayesian Estimation

Abstract

In this chapter, we explore a different aspect of fusion process. Let us take a look at the process of image formation. We assume that the fused image represents the true scene. The elements of the hyperspectral sensor see the same scene, however, capture it partially across the set of bands due to the specific spectral responses of the sensor elements. Thus, each of the hyperspectral bands captures some fraction of the true scene. The objective of the fusion process is then to obtain the resultant image as close to the true scene through combining all available input bands representing the scene partially. The first step towards the solution is to formulate the relation between the true scene (to be obtained through fusion) and each of the constituent bands using a certain model. One may consider the statistical model of image formation which relates the true scene and input images using a first order approximation. According to this model, every sensor element captures only a fraction of the underlying true scene. This fraction is regarded as the sensor selectivity factor in this model, which primarily reflects how well the particular pixel has been captured by the corresponding sensor element of the particular band of the hyperspectral image. The problem of fusion is, thus, equivalent to the estimation of parameters of the image formation model. We do not assume any knowledge about the sensor device, and address this problem blindly using only the available hyperspectral data. We formulate the estimation problem in a Bayesian framework, and obtain the solution through the maximum a posteriori estimate of the corresponding formulation.