Constructing adaptive, nonuniform Gabor frames from partitions of unity

Jeff P. Grossman, Gary F. Margrave, and Michael P. Lamoureux


Adaptive molecular decompositions are useful for analyzing nonstationary signals or images. They can be built up from maximally redundant, uniform partitions of unity, the latter being generated by translations of a single atom to each coordinate sample. These decompositions are adaptive in the sense that they respect a given measure of stationarity in an optimal way. The result is a collection of molecules that generate a one-parameter family of analysis and synthesis frames for the Gabor transform. Each such pair of frames ensures an overall amplitude and energy preserving transformation to the Gabor domain and back. Moreover, for a given atom and stationarity criterion, the redundancy of these frames is minimized, and so the computation time for the Gabor transform is optimized.

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