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| There is another course, rarely taken, of launching undulating waves, which extend everywhere off to the limits of space, with frequency (which could be zero) or wavelength (which could be infinite), amplitude, phase or polarization each as carefully “tuned” as marks are colored, shaped or placed in the spatial course… | …then launching several more waves, each as aptly tuned as the other… |
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| …and even more waves, each always a precise tuning… | …and yet even MORE waves, varying in frequency, direction, amplitude, phase and polarization, reinforcing each other there, but interfering with each other here… |
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| until after awhile (it could be a long while), a picture emerges, something like the one above. | |
This latter case is the spectrally oriented course, and just as surely as we can plot points in space, we can set waves to constructively reinforce or destructively interfere with one another and in each case – so long as we are careful with the tuning of these waves – arrive at a wedge of cheese and some antipasto.
The processes which connect these two diverse courses of image making are, respectively, Discrete Fourier Analysis and Synthesisi. Fourier Analysis takes an image composed in the spatial domain (colored dots arranged in space) and expresses an equivalent arrangement of coefficients arrayed in the spectral domain. Each spectral domain coefficient represents a sine wave signal of a particular frequency, amplitude, phase, propagation direction and polarization, extending out to the limits of space.
| The Spatial And The Spectral | The Spectral Domain |
i There are equivalent Fourier analysis and synthesis processes which operate in the complex real number plane, ℂ2 but images are composed of discrete samples called pixels, so everywhere we say “Fourier Analysis” or Fourier Synthesis” without being specific, we are referring to the Discrete, and not Continuous variant.