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

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.

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.

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.

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.

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

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.

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".

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,

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!

*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.

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!

Hello,I am a second-year master graduate student in printing engineering in Taiwan. My name is Jerry Chen. I have read your beautiful formula of dot gain, Noffke-Seymour equation. I really respect your creativity. Can you please let me know how your formula is derived? Thank you very much.

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