Imaging Fourier transform spectroscopy (IFTS) is spectrography using a type of imaging spectrograph (called an imaging Fourier transform spectrometer) that incorporates a Michelson interferometer, apodization and fast Fourier transform-based processing to produce spectra for each pixel of an image. It has the advantage of a very fine grained image, i.e., with each pixel of a typical astronomical-instrument CCD collecting the spectrum for that pixel. This is opposed to the typical integral field unit-based imaging spectrography that produces spectra for only a few pixels.
The spectrum is derived from multiple images taken of light passed through the Michelson interferometer, with the length of one of the interferometer's two light-paths deliberately tweaked between each snapshot (a type of apodization), so in each image, interference affects wavelengths in a different manner. Then the spectrum of each pixel is teased out of the set of intensity values collected for that pixel, each value for that pixel reflecting the result of interference on the wavelengths present, the degree of interference also depending upon wavelength. Considering two or more snapshots, not just any spectra could produce the measured values, and with enough snapshots, a reliable approximation of the spectrum can be calculated. Beyond that, the more snapshots, the better. The instrument typically applies the interference to an image that has been passed through a collimator, with the collimated, interfered image focused on a CCD.
My guess is information about the relative distances of the two light-paths is acquired by passing a beam of light with a known spectrum (likely a laser) through the instrument and incorporating that information into the post-processing.
The method can produce a very fine grained image at the cost of requiring multiple snapshots. An issue is that short-term variations in seeing or in the source itself affect the accuracy of the spectra acquired, and I presume make it unusable for some applications. Another tradeoff is that the wider the spectral band under analysis, the more snapshots are required to produce a reliable spectrum, which is a reason to limit the band under analysis, e.g., with filtering in front of the spectrograph.