From observations made about the previous gallery, we venture a generalization:
The row/column positions of non-zero coefficients of the spectral domain, as counted from the nearest corners of the Face Card region, engenders waveforms with analogous frequencies along the top/left and bottom/right boundaries of the corresponding spatial image.
Figure 5: q:v Frequencies Produced by Coefficients of an 18x18 Spectral Domain
We're now have a map with sufficient detail to paint waveforms at particular frequencies and orientations. The map is still not complete; going so far, perhaps, as Mercator's 1569 World Map.
There are two significant points on this map: the origin, coefficient 0:0 and the Nyquist rate coefficient N:N, which might compare to the North and South poles of our world maps. We call one the Zero Pole and the other the Nyquist Pole. We probably shouldn't think of a sphere, however, because a torus has a more compelling topography. Notice that we can join Figure 5 on the left and right, then bottom and top edges and the coefficient labels all match up.
Converging upon the origin, 0:0, are the low frequency coefficients, with the origin itself representing an infinite wavelength. At the other extreme, the Nyquist rate coefficient, N:N, marks the highest frequency possible in images of dimension 2N. Clustered around it are the coefficients encoding high frequencies.
In this particular ordering lay the foundations of spectral editing. Elide the coefficients around the Nyquist coefficient for low-pass filtering, but around the origin for high.
| Painting With Waves: Part Two | Intermezzo: Spectral Editing |