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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 "spectrum" 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.

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Nah, it's simpler than that.

The spectrum is the result of the Fourier transform. It's also referred to as the frequency domain. The confusion arises from the common usage of the term (insert flavor here) Fourier Transform to refer to both the operation (the transform) and the results (spectrum).

As for the replication question, I will refer you to Significance of modular arithmetic in DFT?

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