On the subject of decomposition heirarchies (or Daubechies' "frames" formed using a discrete sublattice), consider the attached image files in sequence. Figure 1 below illustrates the traditional array of horizontal (01), vertical (10) and diagonal (11) components generated as a 2D image is broken down into wavelets (Axx) along with a rescaled (reduced resolution, or "brightfield") image in the upper left corner, which of course may be as shown itself decomposed (Bxx, etc.) at each stage with less direct space but more reciprocal space resolution. One might characterize this as a series of 2x2 decompositions.
Figure 2 simply rearranges that 2x2 heirarchy according to the "locations" in frequency space about which each frame is centered, with the "DC" peak in the field located in the center. Note that the first stage (A) components correspond to higher frequencies, and also higher bandwidths (more resolution) than do the higher stage (e.g. B, C) components -- a feature that distinguishes wavelets from short time Fourier decompositions, for example, and makes them better for wide dynamic range phenomena and "situations where better time-resolution at high frequencies than at low frequencies is desirable". Note also, however, that the angular resolution of frequency (subtended at the DC peak) at any given decomposition stage is very poor.
, Fig 3: 
Figure 3 above does the same "recentering", except this time for a 4x4 decomposition heirarchy. Here, the angular resolution is improved, and the "wraparound" edge effects (most noticable at the corners) affect fewer elements. Because frequency as well as spatial resolution are crucial in our work, if we have nxn images where n>16, then the optimum tiling for us is not 2x2 or 4x4, but something like Sqrt[n]xSqrt[n]. The fourth & fifth images sketch the 8x8 and 16x16 decompositions in this series. Remarkably, in a way which may be largely independent of the basis functions involved, when this decomposition is taken to it's nxn limit, one gets simply the Fourier transform itself.
, Fig 5: 
None of the foregoing discusses the functions used for the decompositions, but only their locations and domains in reciprocal (and scale) space. The simplest conceivable functions to "center and scale" to these domains are sharp edged window functions. This is especially true when constructing electron optical filters: You make for example a 10 micron hole in a thin piece of metal, and electrons are either transmitted with unit amplitude and no phase shift where the hole is, or they are not transmitted at all. Partial transparency, or phase modulation, would require controlled aperture thicknesses at the atomic scale, and hence have not historically been a practical choice. The "darkfield perspective" on position/frequency localization is of course not limited to such "all or none" aperture functions (see below*), but they are a natural place to start because of their convenience and history of use in microscopy.
The natural path between direct image and Fourier transform, that this way of looking at hierarchies provides, is more than just theoretical. Figure 6 below shows what happens when one forms an image of the wave field in an electron microscope just above or below the objective back focal plane of a periodic specimen (in this case sapphire) under plane parallel illumination. (The Fourier transform itself resides in the back focal plane.) As you can see, the decomposition of the previous figures can (it would seem) be found grinning back at us from the midst of the electron wavefield in our scope! Note that here the circular pattern of each inset maps the effective shape of the specimen region being illuminated.
Also note that the symmetry of the "tiling" is not chosen with apertures, but by the specimen itself. This suggests to me that the wavefield in the microscope may be more generally modeled as a kind of image/FT convolution (is Wigner-Ville transformation relevant here?). It masquerades as a Fourier-space tiling here, because this specimen's "point-rich" 3D reciprocal lattice acts (via it's intersection with the incident electron Ewald sphere) as a 2D reciprocal comb (i.e. as a periodic array of delta functions in frequency space).
* The "microscopist's perspective" also provides some clues to the types of more sophisticated Weyl-Heisenberg, Wigner-Ville and/or wavelet-like decompositions (e.g. Meyer-Coifman brushlets) that might prove useful in helping microscopists make the most of their digital images (especially those that show atomic lattice resolution). In this latter case, the fact that the spacing between all atoms in solids is on the order of 2 Angstroms means that precise information on both location and scale size may be desired with a very specific frequency range in mind. The structures we search for (discrete arrays of equally spaced molecules) are also very specific. It turns out in this context, for example, that simple rules define a continuum of tilings between square and hexagonal, that have been optimized (as in Fig. 7) to offer maximum expression of the shape transforms associated with each spot in an arbitrary 2D lattice.
, Fig 8: 
Of course, interest in periodicities and their harmonics is a specialized interest. If on the other hand adaptivity to a wide range of scale sizes takes precedence (especially relevant for gigabyte sized images), logarithmic tilings as in Figs 8-10 (that extend the wavelet strategy of treating frequency as a scale parameter) may be of help.
, Fig 10: 
Note: In Figure 9, the dark circles represent (as do the discs in Figure 6) the shape in direct space of the region being analyzed, while the light blue patches in which they are centered represent the frequency domains from which these "darkfield image patches" draw power. Likewise for the dark hexagons in Figure 10. One of the nice things about sharp window frequency space decompositions (like those commonly used for electron microscope darkfield imaging) is that digital algorithms exist today (some quite efficient) for tiling schemes like those of Figs 7-10.