The color of a bunch of dots, Part 5

It's about time!  People have been banging on my door, tackling me in parking lots, and calling me at 3:00 AM to demand that I finish my series of blog posts about halftones. Well, maybe the part about being tackled in the parking lot is a bit of a hyperbole. But, there were two comments on the previous blog post in this series.

Picture taken from out of my window

In the first four parts of this series, I described four different mathematical models that allow one to estimate the color of a halftone.

1. Murray-Davies equation

The Murray-Davies equation estimates the spectrum of a halftone from three pieces of information: the spectra of the paper and of the solid, and the percentage of area that is covered by the halftone. The model assumes that the halftone area has either the reflectance of the solid, or the reflectance of the paper, and that the ratio between them is the halftone area on the plate or in the digital file.

2. Murray-Davies equation with dot gain correction

In this model, the straight Murray-Davies equation is augmented with a further assumption - that the halftone dots grew somewhere between the plate (or file) and the substrate. Note that I didn't have to change much in the equation. I just changed Ain (the input dot area in the file) to Aout, which is the effective dot area on the substrate.

3. Yule-Nielsen equation

The Yule-Nielsen equation assumes that there is a leakage of color from the halftone dots to the substrate between the dots. This acts to boost the color beyond what the Murray-Davies formula predicts. There is a parameter, called the n-factor, which is used to adjust the amount of leakage. In the equation below, n is used to characterize the richness of the halftone.

4. Noffke-Seymour

The Noffke-Seymour equation assumes that the volume of ink is equal to the halftone area on the plate or in the digital file. That amount of ink is squished out to some greater or lesser degree. The ink film of the dots is smaller, but the dots cover more area. Beer's law is used to predict the reflectance of the dots. In the equation below, Aout is used to characterize the richness of the halftone.

Assessment of the equations

In part 3 of the series I pointed out that the two Murray-Davies formulas provide a poor prediction of the color of a halftone. (Parents, please cover your children's ears. I am about to spew some apostasy.) It is unfortunate, but Murray-Davies remains the predominant mindset. It has been enshrined even in ISO 12647-2, which provides us with curves for what the TVI should be, despite the fact that the predictions from this are just plain lousy. Yes, I said it. ISO 12647-2 promulgates a dumb idea.

OK, so maybe that's an over-statement? The idea of TVI is good in principle. Using Murray-Davies, you get one number from which to assess how rich the color of a halftone is. Having one number is great for process control. But there are much better alternatives: either equation 3 or 4. Either of these equations have a single parameter to express the richness at which a halftone is printing. The benefit is that they both provide a reasonably accurate estimate of the spectrum of that halftone. Given the spectra or paper and solid, along with one number to describe the richness, you can reconstruct the spectrum of the halftone.

Why does this matter?

Maybe it doesn't matter that we use TVI, which is based on a poor model of reality. Everything is working, right? Well... I would beg to differ.

You really think your TVI formula is working?

First problem: If I compute the TVI of AM and FM tone ramps, TVI would lead me to believe that a simple plate curve can make one look like the other. Or that I could make a magenta tone ramp printed offset look like a magenta tone ramp that was printed gravure or ink jet. Getting the proper TVI is a necessary, but not sufficient, condition to get a color match.

Second problem: If you compute TVI in one region of the spectrum, you won't necessarily get the same TVI from another region. This problem led to the SCHMOO initiative. (SCHMOO stands for Spot Color Halftone Metric Optimization Organization.)

Third problem: How do you compute the spectrum of a halftone? You can't accurately estimate the spectrum (or color) of a halftone using TVI and Murray-Davies.

Which is better?

So, the natural question is, which equation is more better? Yule-Nielsen or Noffke-Seymour?

First, I need to acknowledge my own affiliations. Many of you may be under the impression that my last name is "Math Guy". This is not quite correct. My middle name is "the Math Guy", and my last name is Seymour. Yes... the Seymour of the Noffke-Seymour equation.

Second, I will make a totally impartial statement which can be proved with algebra. At the extremes, the two equations (YN and NS) are identical. At the end of minimum richness, they both simplify to the Murray-Davies equation. At the other end (maximum richness) they both simplify to Beer's law.

Beer is my law

Both equations are based on verifiable assumptions about the underlying physics. Light spreads in the paper, and that causes halftones to be richer. Halftones dots squish out and that causes halftones to be richer. Which one is the predominant effect?

I will make another totally impartial statement. It doesn't really much matter which physical effect is larger. It has been demonstrated through looking at a bunch of data that numerically, the effects are very similar. In between the two extremes, the two equations act very similarly. I suppose some math guy could figger some way to figger just how close the two equations are. But, experience says they are close.

In short, it doesn't much matter which equation (YN or NS) is used. I prefer mine, of course, because. Just "because".

Call to action

What to do about this? To be honest, I don't know. But, the first step to recovery is a trip to the bookstore to buy a shelf full of self-help books. (While you are there, ask about my latest book, How I Recovered from My Addiction to Self-Help Books.)

Part of the problem is that Murray-Davies is such a wonderfully simple equation. You can easily solve it going forward. As it is written above, you can plug in the spectrum of the solid, the substrate and a dot area, and it will give you an estimate of the spectrum of that halftone. But (and here is the cool part) anyone who remembers some of their high school algebra can solve the Murray-Davies equation for the dot area. Plug in the three spectra (solid, substrate, and halftone) and you can solve for the apparent dot area. You can poke this into one cell of a spreadsheet even after a whole evening of experimenting with Beer's law.

This does not hold true for the YN or NS equations. :( These are both "trap door" equations. You can go one way easily, but going the other way requires a bunch more cells, maybe even a whole page. They both require an iterative approach to solving.

If only I knew a math guy who could figger this out!

Getting the most color for your ink, part 1

So, my wife says to me the other day, "Hey Math Guy, you should do a blog about ink mileage - you know, making the most efficient use of pigments." I love it when she calls me Math Guy. It's so much more intimate than John the Math Guy. "Surely you've thunk some thoughts that no one ever thought to thunk before." I don't love it quite as much when she calls me Surely.

But she does have a point. I do have a few thoughts. I have been waiting for an opportunity like this to share them with an eagerly waiting world.

Dot gain is your enemy

I remember years ago hearing a competitive pride in the press room when it came to printing sharp dots. Mushy dots were a sign of a sloppy pressman. Really good pressmen, running on really good presses would produce sharp, crisp dots. Really good pressman would make Felix dots. All pressmen wanted to be Felix. All presses wanted to be Felix presses. All printing inks wanted to be Felix inks. All fountain solutions wanted to be Felix solutions.

Felix halftone dots versus Oscar halftone dots

And so it was the goal to make crisp, Felix dots, and press crews worked at reducing dot gain.When a press crew managed to bring the dot gain down by one point, there would be huge bonuses and wild parties and groupies everywhere. Many of you remember those days. Mick Jagger would show up, and the roll tender would get a call to be on the Carson show. You get it. The whole magilla.

Then those darn pre-press guys started getting involved. Stick-in-the-muds, every one of them. They complained about needing to run a different plate curve for every job. The pre-press folks didn't buy into the whole press room machismo thing. They didn't want to be on late night TV. They just wanted consistency.

And so it came to be that the printing pundits made the rounds, popping in on the Tonight Show and getting little blurbs on the bottom of page 17 of the tabloids. The message went from "Dot gain and Communism are the enemy" to a more subtle one. "Dot gain, just like Kim Kardashian, is inevitable. We can't get rid of either one of them. All we can do is try to control them."

In light of all of this, what I am about to say is heresy. When it comes to ink mileage, dot gain is your friend. Yes, it needs to be controlled and kept consistent, but more is better.

Kim Kardashian proudly displaying high dot gain

Why is there dot gain?

A point to consider: stochastic printing (FM screening) has high dot gain. That means you need to adjust the plates curves to make it print like conventional printing. Get it yet? You need to bring down the 50% when you're printing stochastic. Have you caught my point yet? You need to put less ink on the paper when you are making a halftone. Less ink for the same amount of color.

I think that the fact that stochastic screening requires a different plate curve is common knowledge, but I'm not sure that everyone has connected the halftone dots. Stochastic printing requires less ink. High dot gain means less ink.

But... maybe the "less ink" part is not obvious. Maybe I need to expound on a question that has baffled philosophers of printing science for decades. Why is there dot gain?

Every Phy-Ed major knows why there's air - to fill up volleyballs!

I know of three explanations for why halftone dots come out fuller than one would expect: more ink, more diffusion, and more squish. The physics is probably correct behind all of them, but it is likely that one or two of them are the major factors. 

More ink

The simplest explanation is that the plate simply delivers more ink. The more ink, the more dot gain. As it was explained to me by Herr Gutenberg, it all had to do with ink/water balance. When you put a little more water on the plate, it will crowd out the ink in at the edges of the dot, and there will be less dot gain. If there is a bit less water, the ink will have the upper hand at the edge of the dot, and the dot will grow.

Based on this, the model is this: more dot gain -> more ink -> more color. Simple enough.

This all makes sense, but I have tracked press runs while adjusting water up and down within reasonable limits. I saw a lot of change in dot gain over hours of press time, but little of it was correlated with the amount of water. I don't think that the "ink/water balance at the edge of the dot" theory is the big explanation for dot gain. 

More diffusion

In 1936, Yule and Neilsen came up with the idea that there are two parts to dot gain: physical and optical. They said that the dot on the paper is indeed larger than the dot on the plate, but that there was a second effect. The paper between the dots takes on some of the color of the ink because of light diffusing into the paper.

I won;t go into much detail explaining it here. You can look at my previous blog post for that. I just want to say here that the Yule-Neilsen effect gives you a little extra color for free. That apparent tinting between the dots acts like more ink, more ink that you get for free.

More squish

Noffke and Seymour came up with a little different explanation in 2012 - dot squish. (Some of you may recognize the name Seymour. He has this blog?) A pristine silo of ink is first deposited on the printing plate, and then that nasty old press comes along and presses it flat. Note that there is no change in the volume of each dot, just it's shape. 

Halftone dot transmogrification under imply pressure

But what of the color? Does squishing the dots change the richness of the color? Well, yes. I go into more detail in the blog on the Noffke-Seymour effect, and Pat and I went into excruciatingly painful detail in the TAGA paper. 

(By the way, the call for papers for the 2015 conference is out. The conference is set for gorgeous downtown Albuquerque, March 22 through 25. Email me if you have any questions, or want to discuss an idea. john@johnthemathguy.com)

Does this get you more halftone for your money? Let me motivate the idea a little bit by considering the extreme. Let's take that silo to the extreme. Keep the silo the same volume, but picture it becoming more of a needle - a very tall spire of ink that has a very, very tiny footprint on the paper. Being very tall, the microdensity of that ink is very high. The color at the top of that spire is very rich. But it covers an infinitesimal amount of paper, so the overall reflectance is pretty much the same as the paper. 

That tall narrow spire is the very least efficient use of ink. It is the cleanest, crispest dot possible, but it is absolutely lousy when it comes to ink mileage.

Dot gain is your friend

The first of the three explanation for the cause of dot gain predicts that dot gain is "cost neutral". You put in more ink, and you get just that much more color. The other two explanations predict that dot gain is like getting a little extra pigment in your halftone for free. My own observations are that the first explanation of dot gain is not the major effect.

So, my conclusion is that dot gain is your friend. According to the Yule-Neilsen model, whatever it is on press that causes more spread of light into the paper, like higher line screen or stochastic printing (or perhaps some reformulation of the paper?) can reduce the cost to print a halftone of a given color. 

According to the Noffke-Seymour model, whatever it is on press that causes the halftone dots to spread out more, like decreasing viscosity or increasing pressure, can reduce the cost to print a halftone of a given color. 

Moral of the story -- An efficient halftone dot is a happy halftone dot