Earlier spectrophotometers used transmission filters, which meant the wavelength that matched the color sample would be reflected through the filter, and all of the other wavelengths would be absorbed. An example of this is if you were to shine a white light on a red apple. The apple will absorb all of the other colors, but will reflect red back to you.
Nowadays, spectrophotometers in paint matching use interference filters, where only the color of the sample reaches the filters -- all other wavelengths are reflected away. This increases the accuracy of matching the sample because absorption can interfere with readings. The wavelength representing the correct color match passes through the correct filter and is then picked up with fiber optics and piped to a photo diode. The photo diode converts the information to an electronic signal, which is then sent to computer software that formulates the exact amount of pigments needed to make the match. And that's how ya do it, folks.
Though the technology may seem complicated, the process of using a spectrophotometer is not. It's hooked up to a touch screen computer monitor that asks all the questions, making the process user-friendly for the operator. You pick your paint base, and if you want the best match, it's wise to steer clear of the cheap stuff. Mid-grade and high-grade paint work best.
These machines are pretty adept at matching colors from a variety of sources, though the best matches happen with samples that don't have much texture or sheen. For example, glossy magazine clippings can cause a glare, and a busy wallpaper pattern or a high pile carpet present challenges in detecting the true color. Photos aren't great either because the colors aren't precise.