Dimension Reduction and Remote Sensing Using Modern Harmonic Analysis

Dimension reduction (DR) refers to removing data from a given set, in such a way as to have the possibility of extricating desired intelligent messages embedded in that set, by making an appropriate analysis of the remaining smaller set. Remote sensing is a means of collecting data about objects without coming in contact with them. These two ideas have been quantified in many ways; and they are fundamentally related because, in many applications, data sets obtained by remote sensing are often too large for effective understanding and computation without first engaging in dimension reduction. In this regard, ideas and techniques from modern harmonic analysis have played a critical role in accounting for significant advances in the field. The theme of this chapter is to exposit the role of modern harmonic analysis in understanding the relationship between dimension reduction and remote sensing. Typically, the setting for remote sensing is the earth’s surface and atmosphere, and the concept of remote sensing is associated with garnering information dealing with geology, meteorology, oceans and glaciers, natural disasters, climate change, and the classification and detection problems associated with man-made issues. Remote sensing technology is phenomenally varied. For example, airborne photography technology, going back to Tournachon’s aerial photographs in 1858 from a balloon, has given rise to more recent satellite, RADAR, and LIDAR methodologies to collect data. However, this technology comes with certain limitations. One of them is the sheer volume of data that arises in some applications, e.g., in the analysis of hyperspectral imaging data, and which underscores the need for computationally efficient and reliable DR methods. There is a natural evolution of ideas we shall highlight to develop our theme: linear DR methods (Section 2), non-linear DR methods (Section 3), the theory of frames (Section 4), compressed sensing and sparse representation (Section 5), and diffusion-based image processing (Section 6). Each of these topics can be considered in the realm of DR, and each of them is a broad, deep, highly developed theory. In each section, except Section 2, we shall provide current applications in remote sensing associated with each of the topics. The material