Hirn, Matthew JohnWe investigate two aspects of frame theory, one of a theoretical nature, the other very much on the applied side. In the former, we enumerate all harmonic frames of prime order, and develop partial proofs concerning the structure of the symmetry group for this subset of frames. In the latter, we develop frame theory in the context of kernel eigenmap methods, merging the two theories in a practical manner and applying new algorithms to hyperspectral imagery data for the purposes of material classification. These two problems, while seemingly separate, are united by frame theory and serve to illustrate both the beautiful theoretical nature of frames as well as their practicality in dealing with real world problems.en-USDissertationMathematicsdimension reductionframesharmonic frameshyperspectral imagery datakernel eigenmap methods