What do we mean by spectrum? Consider the context of image processing and computer vision, and, in particular, discrete Fourier transform.
For example, in the sentence

In the Discrete Time Fourier Transform the forward Fourier Transform correspond to a discrete function of a sequence $x[k]$, however the inverse transform still remains continuous:
$$x(t) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \hat{x}(\omega) e^{j \omega t} d \omega, \; \forall t \in \mathbb{R}$$
the inverse Fourier Transform is typically limited to integration on $[−\pi, \pi]$ (with $T = 1)$ as frequencies outside of the interval just correspond to replicas of the original spectrum produced by the sampling procedure.

or in the sentence

To make the inverse transform treatable by modern digital computer, we need to discretize the spectrum of the signal as well

The word &quot;spectrum&quot; is used. I have seen it being used in several other places. However, I still do not get its meaning.
What is the spectrum? I think this is related to the concepts of time, frequency, spatial and spectral domains, but how exactly?
I have a little understanding of Fourier transform, but I haven't yet fully grasped the concept of discrete Fourier transform. Furthermore, my knowledge of signal processing, image processing and computer vision is very limited.
The "spectrum" refers to the frequency content of the signal (both phase and amplitude/power). For a time series or 1 dimensional signal, it basically represents the presence or non presence of the different possible gradients in the time domain signal.

For an image the spectrum simply represents how quickly/slowly the pixels change in contrast/color/values in both the spatial dimensions.