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An imaging spectrometer is a spectrometer that collects spectral data over a two-dimensional area, i.e., for each part of the image, information about the intensity for each wavelength is determined. This produces a "cube" of data (three dimensions), two being spatial (the image) and one being the spectral data at the specified point.
There are various types: a common type for astronomy is the integral field spectrograph which incorporates a device called the integral field unit (IFU) to arrange pieces of the two dimensional image along a line, i.e., one dimension. Ordinary spectrometers use a slit to image the spectrum across a line selected from the telescope's two-dimensional image, so the resulting graph (photo or electronically stored image) displays spectral data across one direction (the spectral direction or dispersion direction), with the direction perpendicular to that (parallel to the slit) imaging the spectral data for various points along a line-segment in the sky (along the spatial direction). By rearranging the focused (two-dimensional) image so as to place portions of it (e.g., the image broken into small squares) so they are in a line, the same spectrograph slit can capture spectrums of the whole image, though at a lower spatial resolution. Lenses, prisms, and/or fiber can be used to accomplish the rearrangement. This can be a time-saver compared to taking a series of spectrum-snapshots with a "normal" slit spectrometer, shifting the slit each time, so as to capture the second spatial dimension.
Another type of imaging spectrometer is an imaging Fourier transform spectrometer.
Slitless spectrographs have been very effective producing spectra of separated objects within a field (e.g., stars) but are much less useful for producing spectra over extended sources and are not generally considered imaging spectrometers. There has been some success applying methods to disentangle overlapping spectra produced by slitless spectrographs for extended sources or crowded fields (analogous to PSF fitting).
In X-ray and gamma-ray astronomy, in which collected photons are sparse, there are challenges in producing images, but the commonly-used sensors measure their photon energy as well, making the instruments effectively imaging spectrographs.
Be warned that the term imaging spectrograph has also been used for a spectrograph solely because it has a mode of operating as an ordinary camera, i.e., offers an option of bypassing the prism or grating, so as to use its camera for (non-spectroscopy) imaging. Such an instrument may be described as an "imaging spectrograph" (e.g., the Hubble Space Telescope's STIS) even though in spectrograph-mode, it views only the typical slit's single dimension.