Interesting topic, but one thing should be made clear for people without an engineering/applied mathematics background.
The FFT is not the same as the fourier transform. Most of the time the distinction is irrelevant, but in this case it truly does matter. The transformed signal measured with the lens is the result of a sampled continuous space fourier transform (CSFT). The FFT computed is the discrete space fourier transform. The two are not always equivalent.
I bring this up because the author uses the terms "fourier transform" and "FFT" interchangeably.
You are correct; the FFT or DFT is not the same thing as the continuous-time Fourier transform (CFT).
This technical report[1] examining the relation between the DFT and CFT shows in eq. 20 that the DFT is just the CFT evaluated at w=k2pi/(N*T). As N becomes large (number of pixels is large), this approaches the CFT.
I left this detail out; I wanted to put this in terms the reader knew.
If you just left out one "F" from "FFT" it would be more precise and less confusing. Is there any reason to bring the Fast Fourier Transform algorithm up at all?
The FFT is not the same as the fourier transform. Most of the time the distinction is irrelevant, but in this case it truly does matter. The transformed signal measured with the lens is the result of a sampled continuous space fourier transform (CSFT). The FFT computed is the discrete space fourier transform. The two are not always equivalent.
I bring this up because the author uses the terms "fourier transform" and "FFT" interchangeably.