Generalized windowed transforms for seismic processing and imaging
Charles C. Mosher · 2012
Windowed transforms have been used for many years to provide time/frequency or space/wavenumber decompositions for constructing localized wave operators. Construction of an invertible windowed transform that allows for manipulation of localized plane waves has proven to be a difficult task. As a possible solution, we describe a Generalized Windowed Transform (GWT) framework that collects ideas and algorithms from a variety of sources (i.e. windowed Fourier transforms, filter banks, Gaussian beams, beamlets, wavelet transforms, curvelets, etc.) for constructing localized plane wave decompositions with high sparsity. The GWT framework exploits familiar concepts from signal processing in the Fourier domain along with computational efficiencies of the Fast Fourier transform to construct invertible local plane wave decompositions with low redundancy and reasonable computational efficiency. The windowing framework is based on filter bank theory for wavelet transforms in the frequency domain, with extensions that replace sub-band aliasing in window overlap zones with blending, and a computational structure based on the Fast Fourier transform. The classical normalization and aliasing constraints of the wavelet transform are satisfied by the GWT with redundancy factors less than 2. Multidimensional transforms are constructed in a fashion analogous to Fourier transforms, using repeated application of the 1D GWT along each axis of a higher dimensional object. Shift and derivative operators with reasonable computational complexity are constructed using localization constraints and the FFT butterfly algorithm. Examples are provided that show the application of the GWT to time-frequency analysis, image dip filtering, and image compression. The sparsity of the GWT for a given signal to noise ratio exceeds that of curvelet transform for band-limited seismic data.