This thesis develops the theory of discrete directional Gabor frames and several algorithms for the analysis of remotely sensed image data, based on constructions of harmonic analysis. The problems of image registration, image superresolution, and image fusion are separate but interconnected; a general approach using transform methods is the focus of this thesis. The methods of geometric multiresolution analysis are explored, particularly those related to the shearlet transform. Using shearlets, a novel method of image registration is developed that aligns images based on their shearlet features. Additionally, the anisotropic nature of the shearlet transform is deployed to smoothly superrsolve remotely-sensed image with edge features. Wavelet packets, a generalization of wavelets, are utilized for a flexible image fusion algorithm. The interplay between theoretical guarantees for these mathematical constructions, and their effectiveness for image processing is explored throughout.