What is the relation between time, frequency, spatial and spectral domains?

Question

What is the relation between time, frequency, spatial and spectral domains?

I think that time and spatial domains are the "original" domains (e.g. if the signal is a greyscale image, then the domain would e.g. be all possible grey intensities for all pixels), whereas the frequency and spectral domains are the "Fourier" domains, that is, the domains of the functions after having applied the Fourier transform to the original function or signal. Is this right?

I also think that we use spatial when the input, in the original domain, are images, whereas we use "time" in a more general sense, but maybe we should be using it only when the original domain only represents time (we could say a similar thing regarding the frequency domain). The spectral domain, I think, is a more general term, which refers to the domain after having applied the Fourier transform, so it can e.g. refer to the frequency domain. Is this right?

I am more interested in knowing the meaning of these concepts in the context of image processing and computer vision. I think it would be interesting to have a diagram that explains the transitions between these domains (using e.g. the FT).

Answer 1

A signal (e.g. a time series or an image) is a function $f: A \rightarrow B$, where $A$ and $B$ are respectively the domain and codomain of the signal $f$, which is also often denoted by $x$. A time series signal is often denoted by $x(t)$ to emphasise that the signal $x$ depends on the time $t$.

In general, the expressions "time domain" and "spatial domain" refer to the domain of the original or given signal, which can 1D or 2D (or of higher dimensions). The expression "spatial domain" can be used to more specifically refer to the domain of a given image (which is a subset of the $2D$). On the other hand, "time domain" is a more general term and can be used to refer to the domain of a time series signal or an image, even though an image does not really depend on time (but, by convention, we can still refer to the domain of the image using the expression "time domain").

In general, the expressions "frequency domain" and "spectral domain" refer to the domain of the transformed signal using a Fourier transform. In other words, the domain of the Fourier transform of a given signal, which is another signal, can be referred to as any of these expressions. It can also be called "Fourier domain". So, "frequency domain", "spectral domain" and "Fourier domain" are synonyms. However, the expression "frequency domain" is often used in conjunction with the expression "time domain".