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I understand the formula, but what does it even mean for a Fourier transform to convert from time domain to frequency domain? And how is it doing so? Even, in the field of image processing, why do they say that it converts from spatial to frequency domain?

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The Fourier transform decomposes a signal into the weights of orthonormal basis functions: sines and cosines of different frequencies. An analogy would be decomposing a vector in the x-y plane into its x and y components, which are the weights applied to the unit vectors along the x axis and y axis (the orthonormal basis vectors of the x-y plane). Sinusoidal functions can be functions of space as well as time. For image processing you're still decomposing the signal in terms of sines and cosines, the frequencies just correspond to changing in space vs changing in time.

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  • $\begingroup$ Great! But how does it do it? Why, for example, the signal is not transformed into the amplitude domain or phase domain instead of frequency domain? $\endgroup$
    – Shuvo Sarker
    Jan 24, 2018 at 8:17
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    $\begingroup$ As to how: the integral of a product of two cosines or two sines of different frequency or of a product of sine and cosine goes to 0 (they are orthogonal in that sense). When the frequencies match, you get a nonzero answer for the integral. It is this property that plucks off the signed amplitude of that component at that frequency. So the whole transform gives you signed amplitude for the sine and cosine components at all frequencies. From that, treating the signed amplitudes at any one frequency as a complex number (the sine amplitude is imaginary) you can get phase and magnitude. $\endgroup$
    – Andy Walls
    Jan 24, 2018 at 11:35
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    $\begingroup$ As for why frequency domain: sinusoidal functions are orthogonal in frequency (and 90 degree phase: sine vs cosine). They are not strictly orthogonal in amplitude or in phase in general. The orthogonality of sine vs cosine is reflected in the imaginary and real parts of the transform. $\endgroup$
    – Andy Walls
    Jan 24, 2018 at 12:33
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    $\begingroup$ @ShuvoSarker: It's called frequency domain because the axes represent frequencies, just like in the spatial domain they represent location in space. Isolate one particular point in the frequency domain, and you isolate one particular frequency. At that point you do have a complex value, which you can separate into amplitude and phase, but that should not name the domain, just like the spatial domain is not called the "intensity domain", even though it represents an intensity at each spatial location. $\endgroup$
    – Cris Luengo
    Feb 21, 2018 at 17:29

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