Is building a hyperspectral camera out of projector parts possible?

[void4]

Digital projectors use micromirror devices to project light through a lens onto a surface. By replacing the light source with a camera behind a prism or diffraction grating and actuating these small mirrors one by one, thus inverting the light path, could one record a spectrum for each pixel?

• subtract the recorded light from a calibration frame with no mirrors actuated, and possibly one pixel-specific calibration frame