I do not understand how this could possibly work. From a camera with RGB filters we get essentially three different integrals over the spectrum per pixel, how would we recover the spectrum from that? Even assuming you can account for the spectrum of the light sources and the color filters in the camera - which should be doable with a color chart with known spectra - I do not see how you could go from three data points to a full spectrum without making assumptions about the possible object spectra.

EDIT: Okay, after going to the actual paper I at least get transmission mode - you photograph the color chart through the sample and this will of course imprint the absorption spectrum onto the know spectrum of the color chart and you can then look at the difference to the color chart without the sample in between. But I do not get the logic behind their reflectance mode.