"Density is ink film thickness"

I've heard it said thousands of times. Solid ink density tells you how thick the ink is, and CIELAB tells you the color. Along with this goes the converse: solid ink density does not tell you color, and CIELAB doesn't tell you the amount of ink. I'm writing this post to set the record straight. There is nothing fundamentally different between density and CIELAB, except that there is less information in density.

The start of the scandal

I admit to my own little contribution to this scandal. I have talked about Beer's law. Incessantly. I even wrote a blog about Beer's law. In hindsight, I realize that I should have just stuck with Wien's Law [1]. But there it is, I ordered the Beer.

Beer's law, or Wien's law?

I didn't mention ink in that particular blog, but it wasn't long before I started having a little ink with my Beer. In one post, I used Beer's law to explain why ink sometimes changes in hue when you slather it on. As if that wasn't enough, I pulled out another six-pack to describe how to reach a CIELAB target when all you have control of is ink film thickness or pigment concentration.

Image from the world famous perifarbe blog post

Let's have a look at each of the four myths and see how they stand up.

Myth #1 - Solid ink density is ink film thickness

Now, if there were laws about truth in blogging, I woulda probably shoulda mentioned that Beer's law is a decent approximation, but there are some other things going on that limits it a bit when we are talking about ink.

Beer's law (when applied to ink) assumes that light enters the ink, reflects from the paper underneath the ink, and then goes back through the ink. The more distance it travels in the ink, the more likely it is to be absorbed. Those are the two fates of a photon: it gets caught, or it makes it out of the ink. The thicker the ink, the higher the probability of getting caught. Beer's law puts no limit on this. Given a thick enough ink, the density could be a zillion. (This would correspond, of course, to a reflectance of one in ten to the zillionth. My densitometer doesn't quite go that high.)

There are, however, two other potential fates for a hapless photon. A photon without much hap could bounce off the top of the ink, never having a chance to see the ink at all. This is called specular reflection. Another fate has to do with transparency. Photons could bounce around inside the ink and eventually find their way back out before even seeing the paper.

These two effects guarantee that there will be at least a few wayward photons that wind their way back to the detector. Thus, this puts a limit to the density, so all good proportionality must eventually come to an end. Buy me a beer some day, and I'll tell you everything you want to know about the Tollenaar-Ernst equation [2].

One of my favorite equations, the Tollenaar-Ernst equation

The T-E equation in action

Conclusion? Myth partly busted. 

Myth #2 - CIELAB does not measure ink film thickness

I presented a paper at the 2008 TAGA conference entitled "Building a bridge from Dense City to Colorimetropolis". The paper was dreck, but I am quite proud of the clever title [3].

Photo-realistic drawing of the San Francisco bridge 

In this paper I showed that the color difference between paper and the solid (in deltaE values) correlates reasonably well with (paper relative) density when measuring cyan, magenta, and yellow inks. My conclusion is that CIELAB values contain all the information that density values do.

It is incorrect to say that CIELAB does anything really any different from density. The only issue is that of the software catching up. If spectrophotometers and offline software packages reported the right numbers in a way that could be readily understood by press crews, then the myth would just plain go away.

Myth busted! Here is the correct statement to replace the first two myths: "Both density and CIELAB are indicative of ink film thickness, but neither is completely true, especially when you get to high density." [4]

Myth #3 - CIELAB tells you the color of the ink

Well, duh. CIELAB is color, right?

Some pedantics might argue that CIELAB is not color, but that CIELAB is a good enough approximation to work for many industries. CIELAB tells you a lot about the appearance of an object, but it doesn't take a lot of things into account, like

1. The effect of adjacent colors on our perception of an object
2. The effect of out perception of a white point in our field of view.
3. Goniophotometric effects such as glossiness, opalescence, and metallic luster
4. Eye fatigue
5. Differences  in color vision between people, even among people who are not color-blind

Setting all this sophisticated stuff aside, I'm gonna say that CIELAB is a good measure of what we perceive as color. Myth Confirmed!

Myth #4 - Solid ink density does not tell you the color

All I gotta say is "orange". An orange ink may have exactly the same density as a yellow ink. The blue filter in a densitometer may see exactly the same density on an orange and a yellow ink. But, the orange ink is a different color.

The issue is, solid ink density is only one number, so it can't possibly tell you what the color is. Color is three-dimensional. Well... what if I look at all three density filters, red, green, and blue? A densitometer can report all three of these, right? That gives me three dimensions, so there you go. We have defined the color, right?

I'm gonna say "no". The three filters in a densitometer are different than the three filters in my eye.

Once again, myth confirmed!

My (perhaps unpopular) conclusion

There is nothing magical about density that allows it to put a micrometer on an ink film. Inherently, density and CIELAB are sewn from the same cloth. They are both measures of the reflectance, as measured through specific spectral filters. In one case the filters were selected so as to capture the richest part of the spectrum for specific inks. In the other case, the filters were selected so as to mimic the human eye. Other than that, the only difference is in the math.

If there were just a bit more math applied to CIELAB values to serve as a proxy for ink film thickness, then density would no longer be necessary.

------------------------------

[1] I'm not kidding. Not only is there a law of physics called Beer's law, but there is a law of physics called Wien's law. It says that if you know one black body radiation curve, you know them all. In some sense, they all look alike no matter what the temperature. This is also known as the Wine-goggles effect. After enough wine, all bodies look the same, no matter how hot they really are.

[2] Tollenaar, D. and Ernst, P.A.H., “Optical density and ink layer thickness,” Adv. Print. Sci. Techn., 1962, Bol. 2, pp. 214-233.

[3] I was also proud of the really ornate and detailed drawing, which was my depiction of the San Francisco bridge. The TAGA conference that year was held in San Francisco.

[4] I have an article in the upcoming IDEAlliance bulletin that looks at the traditional Murray-Davies formula for computing dot gain, which is based on the science that went into density measurement. In the article, I show rather conclusively that this paradigm does not work for determining dot gain of spot colors. The density of a 70% for example, is nearly identical to the density of the solid, when it is clear that the 70% and the solid have different CIELAB values.

I don't yet have the data to come to any conclusion about the relationship between density and ink film thickness for spot colors, but I suspect that the same sort of thing applies. If one looks at the density/ink film relationship at the wavelengths with highest density, I suspect that these too will reach a saturation point long before the color stops changing.