We could also have shown the 3D FT of f; but, as you may have guessed, it would look exactly like the right plots: by doing a 2D FT followed by a 1D one in the third direction, we have effectively reconstructed the 3D version.

The same technique can be used to reconstruct higher-dimensional FTs. For instance, A 4D transform can be reconstructed as the combination of two 2D ones, a 5D transform as two 2D ones and a 1D one,… More generally, for any positive integer d, one can compute a d-dimensional FT by performing d//2 2D FTs (where // is the Euclidean division) plus a 1D transform if d is odd. In practice, however, the most important cases are d=1 (used, for instance, in signal or time series analysis and in cryptography), d=2 (for classical or deep-learning image processing), and d=3 (for computational fluid dynamics, wave propagation modelling and 3D deep learning CNNs).