I'll add some perspective in addition to the other existing excellent answers.
In a sense, single imaging optical telescopes or even single radio telescope dishes with focal plane arrays can be thought of as Fourier transform telescopes.
Fourier optics tells us that the relationship between the electromagnetic field at the aperture or pupil of an imaging system and the field at the image plane is Fourier in nature. A star viewed on-axis produces a flat plane wave at the pupil, and the FT of that is a delta function at zero. If you treat the finite diameter of the pupil, then the FT of the circular area is an Airy disk, which we say is "due to diffraction". If you have more stuff in the aperture (circular obstructions for secondaries, etc) you can take the FT of that and generate a point-spread function-like plot.
If the star is off-axis, you still have the plane wave with a flat amplitude, but the phase is now ramped across the pupil. The FT of that is the same delta function or Airy disk, but offset in position. Your second star is off to one side on the image plane.
The principle works for microscopes, cameras and projectors as well, the last one being vice-versa; the inverse FT of the projected object represents the field at the output of the projector.
Conventional Radiotelescope Arrays and correlators
With limited resources, an array might have tens or hundreds of dishes or receivers, and to get resolution at small angular distances these are spread out quite sparsely over a very large area. The patterns look like Y's or spirals or random jumbles, usually with most of the dishes near the center but a fraction at very large distances.
The computers that take in all this data are called "correlators" because they perform pair-by-pair correlations of the incoming digitized signals. The results of all of these correlations are then fed into an algorithm that generates an output image, but this requires some serious amount of modeling and careful processing to avoid wrong images or spurious artifacts appearing. An illustration of that is that the results of the Event Horizon Telescope were independently generated by different groups of researchers just to make sure that they came up with the same image after all that processing. There were so few telescopes used in the analysis that it would have been easy to end up with a crazy image.
Fourier Transform Telescope
A Fourier transform telescope would absolutely probably be built with a complete 2^M x 2^N evenly-spaced grid of receiving antennas or telescopes. Now instead of performing pair-wise correlations, you can just dump the whole thing into a computer architecture that has been optimized for performing a parallelized Fast Fourier transform on the data. It's going to be a big computer, but it can take advantage of hardware optimized for FFTs.
This analysis solves the same mathematical problem that a focusing lens or mirror solves, reproducing the Fourier transform of the electromagnetic field at the aperture.
You could build a giant dish the same size as this array, and put a giant 2^M x 2^N array of receiving feed horns at the focal plane of this dish and recover an image that way, but performing the mathematical FT on the output of the flat aperture is thought to be more practical at some point in the near future, when the hardware is fast enough.
below x2: Example of a small dish with a focal plane array, cropped from CSIRO ScienceImage 2161 Close up of a radio astronomy telescope with several more in the background. Click for full size.
There are more radio telescope focal plane array images in this answer and a lonely one in this answer.
That the two different processes can both be used to produce wide-field images when the antennas are on a regular array is shown in Figure 1 of your linked paper, Tegmark & Zaldarriaga 2009 The Fast Fourier Transform Telescope, and some differences are nicely summarized in the caption, along with the unhappy face on the correlator.
FIG. 1: When the antennas are arranged in a rectangular grid as in the FFT Telescope, the signal processing pipeline can be dramatically accelerated by eliminating the correlation step (indicated by a sad face): its computational cost scales as N2 a, because it must be performed for all pairs of antennas, whereas all other steps shown scale linearly with Na. The left and right branches recover the same images on average, but with slightly different noise. Alternatively, if desired, the FFT Telescope can produce images that are mathematically identical to those of the right branch (while retaining the speed advantage) by replacing the correlation step marked by the sad face by a spatial FFT, “squaring,” and an inverse spatial FFT.
When we have two telescopes, optical or radio, and we combine their signals together in hardware (using a beam splitter or electrical equivalent), we can call it an interferometer. If you have more antennas and you combine the signals with incremented phase offsets for each signal, you can also call that an interferometer. But by the time you have an ALMA or VLA we really don't call the combination of all signals as interferometry, because it's done in a more complicated way to recover images, rather than a single output like a stellar interferometer.