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Extending computational harmonic analysis tools from the classical setting of regular lattices to the more general setting of graphs and networks is very important and much research has been done recently. The previous Generalized Haar-Walsh Transform (GHWT) is a multiscale transform for signals on graphs, which is a generalization of the classical Haar and Walsh-Hadamard Transforms. We propose the extended Generalized Haar-Walsh Transform (eGHWT). The eGHWT and its associated best-basis selection algorithm for graph signals will significantly improve the performance of the previous GHWT with the similar computational cost, O(N log N ), where N is the number of nodes of an input graph. While the previous GHWT/best-basis algorithm seeks the most suitable orthonormal basis for a given task among more than (1.5)N possible orthonormal bases in RN , the eGHWT/best-basis algorithm can find a better one by searching through more than 0.618(1.84)N possible orthonormal bases in RN. This dissertation describes the details of the eGHWT/basis- basis algorithm and demonstrates its superiority using several examples including genuine graph signals as well as conventional digital images viewed as graph signals. Futhermore, we display how eGHWT can be extended to 2D signals by viewing them as tensors of graphs and the associated applications to efficient image approximation and nonnegative matrix factorizations.

Yiqun's PhD exit seminar.