Measuring Color Part 3: The Observer Functions

Now that we’ve laid out the problem of taking a color measurement from a continuous spectrum, we can actually execute the conversion. We can start by considering the wavelengths of light that correspond to certain colors. For example, red light has a wavelength of about 650 nm, green light is about 550 nm, and blue light is down around 425 nm. In the actual world, light can exist at any wavelength. Color is more complex than just measuring the intensity of light at one specific wavelength.

The Observer Functions

The human eye perceives red, green, and blue light each over a range of wavelengths. The sensitivity of the eye also varies depending on wavelength. For our project, we will apply some mathematical equations called the “Observer Functions” which approximate the sensitivity of human eyes at each wavelength.

The observer functions show a few interesting things. First, the human eye is far more sensitive to blue light than green or red, but that sensitivity is over a narrower range of wavelengths. Perception of “red” light mostly covers the longer visible wavelengths. However, the red observer function has a secondary peak at about the same wavelength range as blue light. This suggests that the light we think of as “blue” also stimulates human “red” receptors, albeit not as much as the longer wavelengths we associate with red light.

So how do we use these observer functions? Well, we simply need to multiply each of the observer functions by the continuous transmission spectrum that came from our UV/Vis spectrophotometer and compare that result to the original observer function. Thus if a specimen transmitted 100% of light in the “red” range of wavelengths, the observer function would remain unchanged, and we would see that the red receptors in our “eye” would be 100% stimulated.

In the example above, we can see an analysis of a typical red-dyed diesel specimen. The grey curve on the right is the spectrum measured from the specimen, and we can see how the red, green, and blue observer functions are changed by the spectrum. The blue spectrum is practically non-existent. The red spectrum came through at nearly full strength. Finally, the green spectrum was reduced in intensity but has not been suppressed entirely. If we quantify how strongly each range of receptors in our “eye” was stimulated, we could measure the (red, green, blue) color as (0.649, 0.345, 0.004). Mathematically, we accomplish this by summing up the observer functions on the right and comparing them to the observer functions on the left.

So this gives us a color measurement! All that remains is to make some adjustments to make sure our measurement accounts for the conditions of an actual experiment.

Simulating a Standard Experiment

Light Source

While our color measurement does represent the color of the actual specimen, there is an important difference between the UV/Vis spectrophotometer and the standard method: the light source! If we were to follow the standard method, we would use a light source which produces different intensities of light at different wavelengths. The UV/Vis automatically corrects for the light source intensity, so the transmission measurements it provides are as if they were collected using a light source which was equally intense at every wavelength.

For our experiment, correcting for the light source is fairly simple. We just take the spectrum of the standard light source (the “CIE C source” in the plots above) and apply it to our transmission spectrum before we perform our calculations using the observer functions.

Attenuation

There is another difference between our UV/Vis experiment and the standard method: the standard uses a 33 mm thick observation vial and the UV/Vis spectrophotometer uses 10 mm wide cuvettes. So our measurements traveled through less of the specimen than the standard mandates and absorbed less light in total. We can compensate for this using Beer’s Law.

$$A = \epsilon c b$$

Beer’s law states that absorbance, A, scales proportionally with the specimen thickness b. (The other constants are related to the concentration, c, and molar absorptivity, ε, of the specimen, which do not change.) So we know that to simulate the standard measurement, we simply need to scale the absorbance by 3.3 to get an equivalent result! However, our measurement is not in absorbance, it is in transmittance. Fortunately, the conversion between absorbance, A, and transmittance, T, is straightforward.

$$A = log_{10}\left(\frac{1}{T}\right) \\ T = 10^{-A}$$

So dealing with the difference in vial thicknesses is simply a matter of converting to absorbance, applying Beer’s Law to find the equivalent absorbance of a 33 mm vial from the 10 mm cuvette, then converting back to transmittance. Then we will have simulated the standard experiment accounting for both the light source and the thickness of the specimen vial.

Conclusions

I was able to build a color-measurement tool which converted from a continuous spectrum to 3-channel RGB color, which in turn let me match the color to standardized references. Building an automated alternative to the human eye-based standard method required a multi-step conversion.

• Use of the observer functions to convert a continuous spectrum to 3-channel color.
• Simulation of the standard light source called for in the standard method.
• Correction for the smaller-thickness specimen which we used compared to the standard.

To me, the most interesting aspect of this project was understanding how the human eye perceives color. Seeing how the observer functions work showed how in a universe where light can have any wavelength, it is possible to simulate images using only 3 color channels. The usefulness of 3-color images is a result of just which wavelengths of light the human eye is sensitive to.

It is fascinating what this implies. Images displayed on 3-color monitors, printout, and even paintings might look realistic to humans, but any animal with different color sensitivity would likely see them very differently. Representing an image collected by a camera using UV, infrared, X-ray, or other non-visible light could also be accomplished simply by applying a different, specially-designed set of observer functions to convert to 3-channel RGB color. The human eye’s sensitivity is also wavelength-dependent. So a red and blue light might have equal intensities in reality, but a human eye would see the blue light as being brighter.

This all makes me appreciate the human body a little bit more, too. Just “eyeballing” a color measurement seemed like a pretty shaky foundation to build a standardized test on. Using an instrument still gives us a more detailed, objective, and quantitative record of the measurement. However, the eye is actually a rather complex and sophisticated device!