Dimension Reduction and Remote Sensing Using Modern Harmonic Analysis

Harmonic analysis has interleaved creatively and productively with remote sensing to address effectively some of the most difficult dimension reduction problems of modern times. These problems are part and parcel of fundamental ideas in machine learning and data mining, dealing with a host of data collection and data fusion technologies. Linear dimension reduction methods are the starting point herein, which themselves lead to the formulation of non-linear dimension reduction algorithms necessary to resolve information preserving dimension reduction associated with the likes of hyperspectral imagery and LIDAR data. Harmonic analysis arises in the form of data dependent non-linear kernel eigenmap methods, and it is fundamental to design and optimize techniques such as Laplacian and Schroedinger eigenmaps. These are exposited. Further, the fundamental roles in remote sensing of the theories of frames, compressed sensing, sparse representations, and diffusion-based image processing are explained. Significant examples and major applications are described.