## Tuesday, August 1, 2017

### How do you define a color?

I got an interesting question from a good buddy of mine, Mitchell Vaughn, Well, I kinda exaggerate when I say good buddy, cuz I just met him. And it was online, so maybe it doesn't count? But, he said he liked my blog, so I think that's the foundation for a lifelong friendship. Yes. I am that vain.

Here is the question:

I hope you don't mind me asking you a question, which I imagine is a loaded question...but here it is: Are L* a* b* coordinates a color's undeniable "definition"? In other words, is there anything else that needs to be in place to define a color...mathematically speaking? I realize there are several measuring guidelines that need to be met like light source and angle, etc. but wanted to get your thoughts on this.

Thank you, Mitchell

I have three answers, the first one simple and theoretical, the second one complicated and theoretical, and the third one practical.

Color is properly defined as a sensation inside our head. So, once we have defined the relative amounts of light that the three cones in the eye will see, the color has been defined. Well, almost. The eye, brain, and the glop in between need a reference point to establish what white is. All color understanding in the brain is compared against this white reference. But since you're talking about L*a*b* values, this has already been mixed into the soup.

Sealab stew is a hearty meal all by itself!

So, the first answer is that, yes, an L*a*b* value defines a color, provided you know what white is.

Necessary qualifications

But when we are talking about L*a*b*, we are almost always talking about the color of objects -- be it the ink on a package, the paint on a wall, or the color of a plastic part. And (OK, this is gonna sound weird) objects don't have colors.

Consider the red ace of hearts. What color is the heart? Red, of course.

I took three pictures of two aces below. The camera and cards were not moved, all I did was change the lighting. Honest to god... there was no Photoshopping in the images below. No special tricks, other than playing with the lighting.

What color is the ace of hearts?

In the image at the left, taken with "normal" lighting, we see "normal" colors. The heart on the ace of hearts is red. For the middle image, I turned off all the lights in the room and illuminated the cards only with a 456 nm blue LED. The color of the red ace of hearts is now pretty much the same as the ace of clubs; it's black.

The right-most image shows what happened when I swapped in a 626 nm red LED instead of the blue LED. Now the color of the red ace of hearts is white. Or maybe it's red?  I dunno how you would explain it. True statement: The color of the red heart is nearly the same as the color of the card stock. Subjective statements: If you call the card stock white, then the heart is white. If you call the card stock red, then the heart is also red.

I will pause while you consider the implications of this. The color of the heart depends on whether your brain has decided that the card stock is white or red.

This is an extreme example, but all objects, to a greater or lesser extent, will change color as the spectral characteristics of the light changes. I might add, two colors may match under one illumination, but not under another. The ace of hearts matches the ace of clubs at the blue light club, but matches the card stock in the red light district. My wife loves to say the word for that: metamerism. She is not all that fond of saying red light district, or any of the other words for that.

To define the color of an object, we need to specify the spectral characteristics of the light that hits the sample.

To make matters worse, the amount and spectral composition of light that reflects from an object depends to a greater or lesser extent on the angle that the light hits, and the angle from which it is viewed.

The images below are of the same blackberry, with the same camera and camera position, but with different lighting. The image on the left has a point source of light, and the image on then left shows the blackberry illuminated by diffuse lighting. The colors of corresponding parts of the two images are not the same.

Which blackberry looks the most succulent?

To define the color of an object, we need to specify the angles of illumination and of viewing. There are an infinite number of combinations, but a small collection of combinations have been standardized so that we can actually communicate about color values. The most common choices are 45/0 geometry (which is equivalent to 0/45) and diffuse geometry.

Am I done yet? No. Our perception of color depends (slightly) on whether it is a small object (projected onto just the inner circle of the retina, called the fovea) or a larger object (which extends to more of the retina). The relative concentrations of cones are different in the fovea than the rest of the retina, so our perception of color changes.

To define the color of an object, we need to specify whether the object is small (the 2 degree observer) or larger (the 10 degree observer). In case you are not confused enough yet, I discuss standard illuminants and observers in a blog post called How many D65s are there in a 2 degree observer?

In summary, the color of an object is a property of the object itsewlf, but also of the spectral composition of the incident light, the angles of incidence and viewing, and the size of the object. Based on that, once you have specified the L*a*b* value and all of these conditions (by saying, for example, 45/0 geometry, D50 illumination, 2 degree observer), you have defined the color sensation, and the color of the object has been defined.

So the second answer is that, for an L*a*b* value to have a precise meaning, you have to specify the instrument geometry, the illuminant, and the observer (2 or 10 degree).

Note that this does not mean the object won't have a different color under different conditions. Sorry for the double negative. Lemme try again. Objects in the mirror may appear closer than they are. Product is measured by weight and not volume some settling may have occurred during shipping. No warranties are express or implied. And, the color of your tie and sport coat may not match under the funky mood lighting when you get back to your apartment.

There is another important definition for anyone in the business of making stuff that has a specified color. Color is defined as that thing that the customer is willing to pay you for, provided you get it correct. It is whatever is defined in the contract. Without a contract detailed enough to have teeth, the correct color is whatever the customer likes.

The astute print buyer will recognize that his Wheaties package might be sitting on a shelf right next to another Wheaties package that was printed in a different press run or even at a different plant. The astute print buyer will recognize that an off-color package (just like an off-color joke) runs the risk of sitting on the shelf until expiration date, at which time it will get thrown out, much to the dismay of everyone who hates to see good Wheaties go bad.

This astute print buyer will also recognize that metamerism could be an issue if different sets of pigments are used to create the ink on the package. In that case, the astute print buyer might see fit to define the color in terms of spectral values, or in terms of color specifications under multiple illuminants.

So, all those previous answers are just academic if you live in the real world and want to get paid for your print job!

The standards folks, I might add, are pushing for a spectral definition of colors. Various tools are being put into place to allow the standardized communication of desired spectra.

1. Does this discussion also apply to fluorescent colors (e.g. DayGlo)?

My initial guess is yes: You can illuminate a fluorescent color with a defined light source (e.g. D50) and measure the resulting reflectance spectrum, which can then be converted to L*a*b*. Unlike a non-fluorescent color, there will be more energy in certain parts of the spectrum than was present in the light source, but I don't think that should affect the calculations. The color will probably just have a brightness and/or saturation beyond what would be expected given the white point. (Maybe L* could end up greater than 100?)

1. Three years ago, I predicted that you would ask that question, so I blogged on it:
http://johnthemathguy.blogspot.com/2014/11/measuring-fluorescent-inks.html

I agree, the practical answer is that if you have a well defined light source (which is defined down into the UV) that you can have a meaningful definition of the color of something that fluoresces. Such a standard light source was defined for the print industry in ISO 13655.
http://johnthemathguy.blogspot.com/search?q=m3