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Why RGB and CMYK are so different?

(Note: a shortened version was first published here in Spanish by request of Fundación Gutenberg, written in a hurry as a preview of the then-forecoming 2015 8th Color Management Forum. Here there is a more thoughtful one.)

By now we all have been exposed to classic explanations about color additive model, where three light rays made of primary colors red, green and blue are projected onto a surface, showing in this way composed secondary colors cyan, magenta and yellow. We  can also observe white, made by superposition of all three.

We have also seen the color subtractive model, where mixing of three inks colored cyan, magenta and yellow over a substrate produce colors red, green and blue, and finally black by mixing all three inks.

Modelo aditivo
Additive model classic diagram. Every color in this mix is a light.
Modelo sustractivo
Subtractive model classic diagram. Every color in this mix is an ink.

This way it seems to exist a complete duality between both systems: one additive, the other substractive; former's primary colors are latter's secondary ones; mixing the three primary lights gives white, while mixing the three primary inks yields black...

According to our expertise in graphic arts, we know that last assertion is not quite true. Mixing same quantities of cyan, magenta and yellow inks doesn't give black but a muddy brown. Printers know this fact and (among other things) they try to know which is the right gray balance for their systems, i.e. which (uneven) quantities of colored inks gives a neutral tone. Anyway, if this was all of it, the question may not be worth answering.

A strong difference arise from the fact that while additive model (RGB) requires direct light, subtractive model operates by reflected light, so we can only analyze it if we know not just those inks but that light as well. This makes any CMYK color specification strongly dependent on inks' chemistry and light used in the assessment of the resulting color. For a full, complete duality it would be necessary to have inks whose absorption spectra match the exact opposite to each RGB light. For instance, an ink able to absorb every red light's wavelength (in the right proportion) would appear to us as a perfect cyan; in the same way, having "perfect magenta" and "perfect yellow" also available would make subtractive model not only theory but reality. However, those inks don't exist (may we add "yet": their existence may seem a technologic rather than a theoretic problem) so duality is not quite complete. Proof: we can't get black mixing real, colored inks in even quantities.

Another difference is that RGB model is linear, meaning that if we know each primary colorimetry is possible to calculate the resulting color we'll get by an arbitrary mixing of them (Grassmann laws). On the other hand, it's not possible that kind of "prediction" using inks, because their mix gives colors we cannot accurately predict by knowing just those inks' colorimetry, and we are forced to print those mixings so we can actually measure the resulting color of any combination. This distinction has practical consequences: a RGB ICC profile can be created out of a few measures and can be saved in small size file, while a CMYK system ICC profile requires quite a lot of measures (usually around a thousand) and results in a bigger file.

Last, CMYK color space owns a feature which makes it more complex. While in RGB each set of three numbers represent essentially a unique color, there is a high degree of redundancy in CMYK: a set of four percentages may represent the same color as another, different set of numbers. The addition of black ink (K) to the pure subtractive CMY model, for reasons every printer knows, accounts for this. How much redundancy? Let's take a typical CMYK space such as FOGRA 39. It turns out that, despite a narrow tolerance, just about 6% of all possible numeric combinations correspond to "unique colors". In other words, if we take one CMYK percentage combination at random, in average just one out of 17 will be "unique", i.e. a color only expressible in that way; the other 16 can be obtained by more than one combination. In practice, this redundancy makes possible to choose among several (mathematical infinite) black separation curves when creating CMYK separations from RGB elements, and allows the very existence of ink optimization systems.

These characteristics makes CMYK file creation process a little risky, because it can be made wrong in many ways...


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