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The pyramidal (multi-resolution) CCT

Suppose we compute the above CCT (Section 2.6.4) at a few different tile-sizes, and combine these CCTs into a single six-parameter representation. The value of tile-size, NW, may be thought of as a sixth coordinate axis in the chirplet transform parameter space - TF-area. Including this sixth coordinate axis provides us with a hierarchical (multi-resolution) CCT.

To compute the proposed hierarchical CCT, we repeat the computation of the CCT (14) for each of the desired tile sizes and place them in a six-dimensional space, equally spaced along the sixth coordinate axis. Part of the computation involves re-synthesizing a new set of multiple mother chirplets for each value of NW.

Various 2-D slices through the multiresolution CCT may correspond to useful tilings of the TF plane with true parallelograms (true to the extent that the DPSS define a truly rectangular region in the TF plane). For example, the time-scale slice of the multiresolution CCT, taken at a particular resolution, is a wavelet transform based on multiple mother wavelets.

Again, we may use the result of Shenoy and Parksshenoyweyl to generalize the multiresolution CCT by smoothing the TF distribution with a continuously variable parallelogram size. When uncountably many of these parallelogram-smoothed TF planes are ``stacked'', a continuous six-dimensional parameter space results, having parameters time, frequency, scale, chirp, dispersion, and resolution.

Others have done work to further generalize energy concentration to arbitrarily-shaped regions of the TF planetopiwala, rather than just parallelograms. It would therefore be possible to use these results to define more general parameterizable transforms, based on families of multiple analysis primitives acting collectively in the TF plane.