Four questions on color

Question

First, I apologize for being a mathematician and having no scientific background in physics. The following questions on color came up in a discussion at lunch and I would be very happy to get some answers or corrections of my understanding.

• The first question is, what a color the human eye can see is in the first place. According to this wikipedia article, the human eye has three cone cells $L$, $M$ and $S$ responsible for color vision which can be stimulated each in a one dimensional way, i.e. ''more'' or ''less''. Combining the stimulation of all three, one gets a $3$-dimensional space called the LMS color space. Is a color the human eye can see just a point (in a specific range) in this LMS-space? A negative answer could be that time plays a role (switching between e.g. two points of the LMS-space with a certain frequency defines a color). The model is made in the way that only one point in the space at a time represents a state of the eye at that time and not many.

For the following questions, I suppose that the answer to the above question is positive, e.g. a color the human eye can see is defined by a specific point in the LMS-space. By experiments, one could possibly determine the body in the LMS-space, the human eye can see (this is what I meant by the ''specific'' point). Of course, this may differ from person to person but perhaps one can average over some and get an average body $AB$ in the LMS-space. Points inside are now exactly the colors the human eye can see.

• Now there is this CIE color space which is vizualized by this $2$-dimensional picture looking like a sole of a shoe. As far as I understand, there is a bijection (continuous?) from the LMS-space above to this CIE-space where two axes describe the so called chromaticity and one axis describe the so called luminance. What is the figure shoe sole now exactly? Is it a certain hyperplane in the $3$-dimensional CIE color space to which the luminance-axis in perpendicular? (Perhaps one can say something like ''it is the color at full luminance''. Note, that black isn't in here.)

• The RGB color space has the drawback of having a gamut not filling the whole visible range. Why doesn't one use the frequencies of the $L$, $M$ and $S$ cone cells as the primary colors for technical devices like screens? Couldn't one describe the whole visible range then (i.e. wouldn't the gamut be the whole shoe sole then)?

• My last question concerns this old painters belief, that every color can be obtained by mixing red (R), yellow (Y) and blue (B) or the belief that every color can be obtained by mixing cyan (C), magenta (M) and yellow (Y). Both models, the RYB and the CMY model, are subtractive. When one mixes these colors, doesn't one ignore the luminance? I.e. don't the painters want to say that they can obtain each chromaticity by mixing these colors? (Of yourse, if they want to say this, it is also not really true since the gamut of these three colors determine a certain region in the shoe sole plane from the previous questions which is not the complete sole.)

Answer 1

Yes, a color is a point in LMS space. At least, that's the signal that the eye tells to the brain starting signal which is post-processed by neurons in the eye and brain. For example, the brain does some inferences on what the lighting condition is etc., so that an object looks like it has one color even if half of it is in sunlight and half of it is in shadow.

There is a linear transformation between LMS and the xyY of the CIE color-space. It's some 3x3 matrix. The horseshoe / sharkfin / shoe-sole is a slice of xyY space at constant Y surface in xyY space projected onto the xy plane. Below left is the slice at Y ~ 1, below right is the slide at Y ~ 0.5 Below left is the surface showing the highest possible Y for each xy, below right is the surface showing a somewhat lower Y at each xy.

The slice at Y = 0 would be a completely black horseshoe.

If there was a frequency of light that ONLY stimulated the L cells, and a different frequency that ONLY stimulated M, and a different frequency that ONLY stimulated S, then it would be possible to create a kind of RGB space that fills the entire visual gamut. Unfortunately, that is NOT the case.

If you're making a projector with three color lights, the rules are (1) Each of those three colors has to be composed of a non-negative amount of each frequency, and (2) Each pixel on the screen has to have a non-negative amount of each color. These two requirements are mathematically incompatible with recreating every possible SML color stimulus. (The lack of negative numbers means that your linear algebra intuition does not apply here.)

The "old painters belief" is certainly not true, and I doubt that any painters really believe that. Ask a painter to grab red, yellow, and blue tubes of paint, and then mix them to get black, and then mix them a different way to get white. They will surely acknowledge that it's impossible. Basically, I agree with what you said.

Answer 2

Is a color the human eye can see just a point (in a specific range) in this LMS-space? A negative answer could be that time plays a role (switching between e.g. two points of the LMS-space with a certain frequency defines a color).

I think the answer to this is basically yes, with a couple of caveats. (1) There are imaginary colors, which are possible sets of cone-cell outputs that don't correspond to any possible physical stimulus to the eye under normal conditions. However, it is possible to see imaginary colors by doing things like fatiguing one set of cones. (2) There may be more than one point in this space that produces the same sensation of color (see below).

Note that (L,M,S) is not the set of signals that gets sent to the brain. There is a second layer of processing before that, modeled mathematically by the Hurvitch-Hering-Jameson opponent process model.

As far as I understand, there is a bijection (continuous?) from the LMS-space above to this CIE-space where two axes describe the so called chromaticity and one axis describe the so called luminance.

It's not immediately obvious to me that it's a bijection. For example, a mixture of red and blue is perceptually similar to violet. Whether "perceptually similar" is close enough to be considered the same os a different issue. Psychologists in the lab measure "just noticeable differences," but those may not be the same as jnd's in ordinary conditions.

What is the figure shoe sole now exactly? Is it a certain hyperplane in the 3-dimensional CIE color space to which the luminance-axis in perpendicular?

I think the WP article answers this.

Why doesn't one use the frequencies of the L, M and S cone cells as the primary colors for technical devices like screens?

This is just a matter of the available phosphors.

When one mixes these colors, doesn't one ignore the luminance?

In additive and subtractive models, you start with black or white and then add or subtract. By applying a thin wash of paint to a white canvas, I can get something bright. Thicker paint would darken it up. In practice, I think they often mix in white or black paint, but that's not an objection in principle.

Answer 3

My last question concerns this old painters belief, that every color can be obtained by mixing red (R), yellow (Y) and blue (B) or the belief that every color can be obtained by mixing cyan (C), magenta (M) and yellow (Y). Both models, the RYB and the CMY model, are subtractive. When one mixes these colors, doesn't one ignore the luminance? I.e. don't the painters want to say that they can obtain each chromaticity by mixing these colors? (Of yourse, if they want to say this, it is also not really true since the gamut of these three colors determine a certain region in the shoe sole plane from the previous questions which is not the complete sole.)

Chromaticity only is not interresting for painters, one has to know the level of luminance associated to a given chromaticity to correctly represent a color. For example, a dark green and a "lemon" yellow can have the same (or almost the same) chromaticity, but they are certainly not the same colors.

There is an ambiguitiy in the word "color" itself that cannot be understood by the sole concept of chromaticity : there is nothing "dark yellow", no material is at the same time of an "intense blue" AND light (blue paints are always dark, the more blue the darker...)

Chromaticity is practical when you want to characterize the "color" of light sources, if you want to talk "real colors" (light sources on a screen or material colors), you need a 3D space.

The CIELAB colorspace gives you a more intuitive picture for this.