"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.

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[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.

Why does my cyan have the blues? (addendum)

I was asked a question about my previous blog about why the hue of ink sometimes changes when the ink film is increased.

I had a lovely plot (see below) that showed that showed that Beer's law doesn't do all that bad of a job at predicting the ink trajectory (and the hook) of a magenta ink. The plot shows how close the match is in a*b*. Erik pointed out that I didn't show what is going on with L*. It could be that Beer's law works well in a*b*, but really messes up when it comes to L*.

The magenta hook, real and estimated

So, I had a look at this same data from a few other perspectives. Here is what the data looks like in the L*a* plane.

And here it is on the L*b* plane.

My conclusion is that it doesn't do so bad. Thanks Erik, for keeping me honest. Naturally, if it hadn't worked out I would have suppressed the results.

Why does my cyan have the blues?

When I started in the print industry as an apprentice to Gutenberg, I noticed that the folks in the press room called the inks red, yellow, and blue. This confused me. Everything I had read in color theory books said that cyan, magenta, and yellow were the subtractive primaries. These were the primaries that you use to make a wide range of colors with pigments and filters. Pigments and filters work by subtracting certain wavelengths of light. On the other hand, red, green, and blue were the additive primaries, and these were used to make all the colors when you are mixing light, as in a TV or computer monitor.

Polaroid snapshot of me working at my first job

Why were those silly printers using some of the additive and some of the subtractive primaries? Didn’t they realize that this reduced their gamut? That was the theory, anyway[1].

Just a naming issue?

Anyone who knows me, or who loves me[2] can attest to the fact that I am a firm believer that ignorance is the main explanation for every cultural and scientific phenomenon. In this case, my previous blog about counting colors provides a clue as to the sort of ignorance that might explain why magenta is so curiously called red.

The eleven people who read my previous blog learned that there are only eleven basic one-word color names in our active vocabulary. Neither cyan nor magenta made that list[3]. Clearly the folks on press were calling the inks “red” and “blue” because they have no other words to describe the colors.

Cyan ink is blue, and magenta is red

In my normal incisive way, it took me a few years to realize that the pressmen were not quite as ignorant as I thought they were. I guess I spent too much time running for buckets of halftone dots to actually put my head in a bucket of ink. When I finally did put my head in a bucket of ink (as part of a hazing[4] experiment) I could see that cyan ink is blue, and that magenta ink is red when you look at them in a bucket.

Cyan and magenta inks are blue and red in the can

Cyan and magenta inks are cyan and magenta on paper

So, this confusion is obviously beyond my original explanation. Just like when a fellow accidently calls his wife by the name of a former girlfriend, you can bet there is something deeper going on.

Beer’s law revisited

In yet another very popular[5] blog of mine, I provided a charming explanation of Beer’s law. This blog post is a prerequisite for the following exciting discussion.

Let’s just say that we have a perfect magenta ink. A perfect magenta ink will reflect all the red light and all the blue light that hits it. As for the green light, a light shade of magenta might reflect about 10% of the green. A rich shade of magenta will reflect about 1%.

Now we bring in Beer’s law. Let’s say we start with that light magenta and add another layer of the same ink. Beer’s law would predict that the reflectance would multiply. Since perfect magenta reflects 100% of red and blue light, Beer’s law predicts that the double layer of magenta will reflect 100% of the red and blue light. Beer’s law would further predict that the green light would reflect at only 10% X 10%, which is 1%. A double layer of light magenta becomes a rich magenta.

Key point here: for this perfect magenta ink, the hue is still that of magenta. It still reflects most of the red and blue light, and absorbs most of the green light.

Let’s just say that we now switch over to a magenta that is less pure. Let’s just say that for some inexplicable reason, the publishers of Schlock magazine are unwilling to spend $100,000 per gallon for their ink. The bargain ink they decide to use does not reflect quite as much blue light as we would hope; maybe it only reflects 40% of the blue light when we put a thin film down, and maybe 10% of the green light. Let’s say that the red light is still reflected at 100%.[6]

What happens when we double the amount of ink on the paper?  Beer’s law takes over, and we see that blue light is reflected at 40% X 40% = 16%.  Green light? The reflectance goes from 10% down to 1%. Red light stays at 100%. The table below summarizes the Beer’s law estimation.

Blue Green Red Thin layer 40% 10% 100% Thick layer 16% 1% 100%

From this table, it would seem that the thick layer of magenta is a lot closer to red. The plot below shows the actual spectra of two magenta patches, one at a larger ink film thickness than the other. The plot leads one to the same impression – that a thick layer of magenta is closer to red in hue than a thin layer.

Spectrum of a magenta ink, normal thickness and thick

The tentative conclusion is that magenta turns red when it is thick because it is impure, or more accurately, because there are several different reflectance levels in the spectrum. When Beer’s law kicks in, the areas of the spectrum where the reflectance is “mid-level” (i.e. 40% reflectance) are grossly effected by the ink film thickness.

The plots below are the spectra of cyan and yellow inks. If the previous rule applies, then we would expect that cyan ink will have an appreciable change in hue as it gets thicker. From the plot of cyan ink, we see that the reflectance values between 500 nm and 600 nm are “intermediate”, somewhere between the highest value and the darkest value. This is the green range. As cyan ink gets thicker, we would expect the amount of green light reflected to drop.

Thus, based on Seymour’s rule of ink hue shift, a quick look at the plot below would suggest that thick cyan ink will be blue, just like thick magenta ink will be red. Yellow ink has very little in the way of intermediate values. It basically has either 75% reflectance or 3%. From that, you would guess that yellow ink will not change in hue. Note that a bucket of yellow ink does indeed look yellow.

Plots of cyan and yellow ink

But spectra can be a bit misleading when trying to discern color. I don’t know many people who can look at a spectrum and tell what the color is. So, I offer a little computational experiment to further validate Seymour’s rule of ink hue shift.

First, I will show the results. Then I will explain how I got them. The chart below shows the a*b* values of a set of ten magenta patches with increasing ink film thickness. These values are the ten blue diamonds in the plot. There is clearly a strong hook. The first five are pretty much along a line without much hue shift. The sixth one goes around the bend, and the last four are changing a lot more in hue than they are in chroma.

The magenta hook, real and estimated For the other views of this data, I have published an addendum to this blog post.

The magenta colored line in the plot is a prediction of what I call the “ink trajectory”. This is the set of all L*a*b* values that an ink will go though as you change the ink film thickness. To compute this estimated trajectory, I started with the spectrum of the sixth patch and that of the paper. (You will note that the magenta line goes right through that point.) I loaded these spectra into a spreadsheet, and used Beer’s law to estimate the spectrum over a range of ink film thickness. You will note that the estimated trajectory comes reasonable close to predicting actual measured values, and definitely predicts the hook.

For those who want more detail, I have a little more description below. This is excerpted from a paper I presented at TAGA in 2008.

Excerpt from Building a Bridge from Dense City to Colorimetropolis

This pretty well settles it in my mind. Magenta ink on paper is magenta. Magenta ink in a bucket is red. I have explained this with some simple ciphering with Beer’s law. This led me to define Seymour’s rule of ink hue shift, which allows you to tell (just by looking at a spectrum), whether an ink will have an appreciable hook.

I then showed some really, really impressive results that show that, armed with just the spectrum of your paper and that of your ink on that paper, you can determine the magenta hook. This is clearly a triumph of modern science.

I have come a long way since I was ransacking the printing plant to find those elusive halftone dots!

Caveats

This is where I admit to some of the lies in the previous section.

First off, Beer’s law is only an approximation. It makes the simplistic assumption that a photon will either pass right through the ink, or get absorbed. It does not make allowances for photons that reflect directly from the surface, or for photons that bounce around a bit in the ink and maybe come out of the ink without ever having visited the paper.

Despite those simplifications is does fairly well. For the standard process inks. I do not have data to see whether it works for Pantone inks. If anyone has a cup of data to spare…

One limitation that I glossed over is that it does not do well at predicting the reflectance of a double layer of ink. Us folks in the know like to say that ink is “sub-additive”, which means that Beer’s law does not do well at predicting the reflectance of a double layer of ink. It will, however, give you a spectrum that is attainable, however. Just not at that particular ink film thickness.

Well, that was kind of a lie as well. There are limitations, especially when you get up to the very high densities. You will note that my hook graph fits the data pretty decently, but it would not be nearly so good if I tried to predict the lightest density from the darkest, or the other way around.

There is one more lie, or one more pair of lies actually, but they are subtle. I demonstrated two ways of deciding whether the spectra of magenta showed a hue change. The first way was kind of hand-wavy. “Look at the spectra and see that it looks a lot like red. Ignore the little bump behind the curtain at 450 nm.”

Well, this argument may fly for someone who has not spent thousands of hours looking at spectra. But, if you have devoted a lifetime to deciphering spectra, you would know that sometimes the stuff happening down at the dark end is important. That little bump at 450 nm might just have a big effect on the color.

In this case, it didn’t. Converting to CIELAB demonstrated that the magenta is definitely turning red.

Or did it turn red? This is where the lie gets very subtle. We are trained from childhood to believe that colors with the same CIELAB hue angle are actually the same hue. But I have stubbornly disagreed with this all along. My first grade teacher almost flunked me over this point. I was glad to come upon a paper by Nathan Moroney where he made an off-hand comment that agreed with me.

The issue has to do with the fact that the CIELAB formula performs a nonlinear function on the XYZ values, which are a linear combination of the actual sensors in the eye, but which probably don’t actually exist in the eye or the brain. But that is grist for another blog.

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[1] Yogi Berra said “In theory there is no difference between theory and practice. In practice there is.”

[2] I am still baffled as to why there are so few people who both know me and love me. Why is there no intersection between these two sets?

[3] Both words came into our language relatively late. Magenta became a word shortly are 1859, and cyan became a word in 1879. You wouldn’t expect them to become common words that quickly, would you? After all, look how long it took “internet”, “email”, and “perifarbe”  to become common words.

[4] “Hazing” of course is some sort of print defect for gravure printing. Nothing to do at all with the old guys picking on the newbie.

[5] Popular? So far, seven people have read the Beer’s law blog post. Well, I should clarify. Seven people stumbled upon the blog post. It is perhaps optimistic of me to expect that all seven of them took the time to actually read the blog rather than just look at the really cool pictures.

[6] Standard process magenta ink is not all that perfect, and there is a ”magenta” ink that is a bit closer to perfect: I am exaggerating just a tiny bit about the price of the alternative. I have not checked the price of Pantone Rhodamine just lately, but I think I can hook you up with a guy who can get you a gallon for something less than $80K a gallon. Unless of course, you are looking for ink jet ink.