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Analogous to the 1-D FT, the 2-D FT will extract the repetitive patterns (frequencies in 1-D signls) present along a certain direction at incremental rates. These patterns are manifested as a gradual smooth transitions in color intensity (just like sine waves). Just like 1-D FT, the whole operation is a mere projection for the 2-D picture on a new plane whose kernel is the 2-D set of all special frequencies. Very similar to saying that the set [14 8] is equal to [2 4](*)(7 2) (a 1-D decomposition analogy). In its essence, the whole projection thing is a form of compression if you like. In the previous example, saying [2 4] is obviously simpler than saying [14 8] provided that you know the kernel on which I performed my operator (7 2), which is by the way a fictitious operator I came up with.
Hope this is beneficial.
PS: FFT is the Fast FT and is meaningless without the original famous Fourier Transform.