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!

The color of a bunch of dots, Part 4

If you have been wondering for years about TVI bananagrams, you have come to the right blog post. This blog post is the definitive blog post on TVI bananagrams. But if (for some crazy reason) you have not been wondering all your life about TVI bananagrams, then this could still be a watershed moment in your understanding of dot gain and its fuzzy sister, tone value increase.

Cyana bananagram

Dot squish

Pat Noffke and John Seymour presented a paper at TAGA in 2012 entitled A Universal Model for Halftone Reflectance. [1] In the paper, they developed an equation for what they called Dot Area Increase. This is a measure of how much a dot squishes out when it hits the paper. This is similar to what Murray and Davies called dot gain, but there is one novel difference. In both models, dots get bigger when they hit the paper, but in the Noffke-Seymour model, they also get thinner because of the whole law of conservation of ink thing. Because the dots are thinner, the color of the dot is less rich than the solid.

A halftone dot, before and after the steamroller

If there isn't any dot squish for a particular hypothetical printing, then a 30% dot would cover 30% of the area, and the thickness of the dot would be same same as the thickness of the solid ink. This hard dot has no dot area increase. The equation for reflectance in this case is the Murray-Davies equation.

Murray-Davies equation for hard dots

At the other extreme, the dots squish out completely so there is no longer any semblance of the dot structure (think gravure). Perfectly soft dots. Contone - continuous tone. At this extreme, one can use Beer's law to estimate the reflectance of a halftone. You will no doubt recall the equation from my blog on Beer's law, 

Beer's law equation for continuous tone

I should add here, that the two equations here should be applied on a wavelength by wavelength basis.

I should also 'splain a bit about the "Ain" that is being used as an exponent. Normally in Beer's law, this is where you would put something to do with ink film thickness. So, why did I stick the dot area in that spot? Imagine that you start out with perfect, hard halftone dots that are, I dunno, 30% dot area. Since they are perfect, let's just assume that the ink thickness of the dots is the same as the thickness of the solid.

Now, let's say that these dots get stepped on by an elephant. They are squished out so as to cover the whole area uniformly. How thick is the ink now? If the original dot area is 30%, then the new ink thickness is 30% of that of the solid. So, the exponent of Ain represents the thickness of the fully squished out dots.

Poor defenseless halftone dot, about to become a continuous tone

The dot squish equation

The pure genius of the Noffke-Seymour paper is that they considered what happens in between. In the figure below, the left side shows the starting condition. The halftone dot covers 25% of the area, and has the full thickness of the solid. The right side shows what happens after squishing [3]. The dot now covers 39% of the area, and as a result, is thinner by whatever ratio is necessary to preserve ink volume. I dunno? Maybe the ratio is 25% / 39%? I guess that's about 0.64. [4]

A tale of two halftone dots

Using the halftone dot at the right to illustrate the Noffke-Seymour formula, Beer's law is used to estimate the reflectance of the light blue area covered by ink. A thickness of 0.64 (as compared with the thickness of the solid) is used. Since a wandering photon has a 39% chance of hitting the area covered by ink, this reflectance is multiplied by 39%. This accounts for the light that reflects from the ink. The remaining photons will hit the paper, so (in true Murray-Davies fashion) the reflectance of the paper is multiplied by 61%, and this is added to the first number.

The equation below tells the whole story. "Ain" is the dot area going in. In the example above, this would be 25%.. "Aout" is the final area of coverage after squishing, 39% in the example.

The beautiful Noffke-Seymour equation

Note that at the extreme of no squish, Ain = Aout, the equation simplifies to Murray-Davies. At the other extreme, then Aout = 1, and this equation simplifies to Beer's law. 

In the first big finding, the authors of this paper looked at spectra from a big pile of tone curves, and came to the conclusion that pretty much every printing modality (web offset, stochastic web offset, gravure, newspaper, flexo, and ink jet [5]) all fit conveniently between these two extremes. This is huge. (But that's just my opinion.) Tone value increase for any type of printing can be described in terms of how broadly the dots squish. That's all you need to know.

Finally, the bananagram

I have saved the best for last. The other huge finding is shown below, the invention of the bananagram [6]. The bananagram below is an a*b* plot of all possible tone curves for a given cyan ink [7]. The left edge of the banana is the tone curves generated by assuming that the halftone dots are perfectly hard. The right hand side is a similar curve made with the assumption that the dots are perfectly flattened out.

Cyana bananagram

Now, lemme tell you about the rainbow colored lines. The yellow line, as an example, is all possible a*b* values that a 40% cyan halftone dot could take. Starting with a perfectly hard 40% cyan halftone, as you gradually squish it out, you will see it trace out the curve from one side of the banana to the other.

Wow. The position along that line tells you how hard the dots are. If you know the dot hardness (along with the spectra of the solid and the paper, and the original tone value), you can figger out the color of the halftone.

Foreshadowing the next blog post

I need to eventually tie up at least one loose end. I have been throwing dot gain kind of equations around willy-nilly, or perhaps yuley-niely. We have (so far) the following three equations to explain the color of a halftone: Murray-Davies, Yule-Nielsen [2], and Noffke-Seymour.

Murray-Davies (we all know) is a lump of over-cooked turnips when it comes to accurately predicting or measuring color. Yule-Nielsen, seems to be all the rage. Then these young upstarts come along with yet another formula that is gonna save the world! How can this all be reconciled?

Stay tuned for the thrilling conclusion!

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[1]  I know these guys. One of them seems to be around whenever I stop at the bar for a beer. I guess he must hang out there a lot. Anyway, these two guys do bunches of seriously good stuff. And they're modest, too. Well, at least one of them is.

[2]  If you have been paying careful attention, you will notice that I have reverted back to the more common spelling the the latter gentleman's name. In a previous blog, I cited the original paper from the TAGA Proceedings, along with a scan of the heading for this paper. In the original published paper, the name is spelled "Neilsen", which is contrary to virtually every citation of the name. I ranted and raved about how 11,400 patent citations use the misspelling "Nielsen". As such, I would imagine that these 11,400 patents are potentially invalid. Gary Field did some excellent detective work, and has convinced me that the TAGA paper is a transcription error. The gentleman's name is Waldo J. Nielsen. Assignees of these patents can breathe easier.

[3] I keep talking about squishing, but this might not always be the case. In a web offset press, where there is a lot of pressure between the plate and the blanket and the paper, then squishing is probably a valid term. But in the case of gravure, where the ink has a very low viscosity, maybe it's not so much squishing as it is just spreading out. In newspaper, where the paper does not have a coating, maybe the significant effect has more to do with the ink being wicked into the paper. All of these I have put under the umbrella term "squishing." Whatever you squish under your umbrella is your own business.

[4] How should I know what the ratio is?  Am I called John the Arithmetic Guy??!?!?

[5] No data was harmed in the filming of this experiment.

[6] I expect to see bananagram T-Shirts available on the internet. There will be bananagram support groups for people who have family members sucked into the cult. I expect this will be a topic in the next state of the union address, with plenty of polarized commentary on Fox News and MSNBC.

[7] In three dimensions, this is a surface, sort of like a fly's wing or a sail. In other words, the possible range of colors of a halftone of a given ink can be described by a three dimensional figure that looks like a fly's wing.

The color of a bunch of dots, part 3

In this series of blogs on tone value increase, we have been considering two parallel questions: How to predict the color of a halftone, and what to measure about a halftone in order to do process control. In part 3 of the series - the one you are reading right now - I want to consider the process control issue.

For those who can't take the time to read my entire long blog post today (maybe because they have to tend to their sick chameleon?) I will give a spoiler. The Murray-Davies equation, which we use today to compute TVI, has some issues when we try to extend it past CMYK inks printed on a web offset press with standard (AM) screening measured with a densitometer.

TVI depends on wavelength

The graph below might be a bit of a shocker. Then again, maybe not? I guess it depends. The graph is simple enough. I took the spectra of a solid cyan, a 50% cyan, and a solid. I used the Murray-Davies equation that we all know and love to compute the TVI. Rather than using the red channel of a densitometer, I did the computation separately at each wavelength.

Will the real TVI please stand up?

Maybe it's a shocker that the TVI at 630 nm is a healthy 15.7%, and at 500 nm, it's an anemic 2.6%? I think this very clearly shows that TVI is not a direct measure of the dot size. How could the dots be so much bigger if you look at them through a blue green filter instead of a red filter? It's almost like the tone ramp (0%, 10%, 20%, ... 100%) changes in hue... [1]

This is just an oddity, though, right?  It is like the kitten born with two faces, only this kitten is a halftone patch with two different TVI values. Or rather, a halftone with a whole bunch of different TVI values. Still, so long as you make sure that you have the correct setting on your densitometer, you won't have to worry about all the other faces. And process control will work, right?

But, sometimes you don't have density information, particularly in color management. It seems that a lot of data sets have been collected with only colorimetric data; no spectra and no density data. [2] How can you compute density without the spectral data? Can you compute TVI from the XYZ values?

According to ISO 10128, Appendix A, you can compute magenta and black TVI from the Y value and yellow TVI from the Z. To compute the cyan TVI, you need to do a bit more math, but it's not that bad.  There is a formula. Another brilliant researcher presented a paper at TAGA (2008), where he offered up a more complicated formula that works even more better. Brilliant as this researcher is, he stole this formula from another brilliant guy, Steve Viggiano.

So... this is basically just a confusion, right? There isn't really anything here to upset anyone's sense of inner tranquility, right? Keep reading.

TVI does not uniquely define the color of a halftone

Speaking of standards, the ISO standards for print (the ISO 12647 series) defines colorimetric aim points for the four solids and overprints of those solids. But for halftones?  It specifies TVI aim points. This would lead one to believe that, once you get the solid correct, all you need to do is make sure that you TVI is correct. Then you will have the correct color, right? [3]

And if the TVI is not correct, the standard gives you the impression that it is easily corrected in prepress by applying a plate curve. If your 50% has a TVI of 66%, and it should be a 68%, then all you need do is add 2% in a plate curve, right?

Not so fast, boopie.

The graph below is an a*b* plot (a plot looking at color space from above) of cyan, magenta, and yellow tone ramps, along with various other combinations. The blue lines represent the color of tone scales of conventional screening. The red lines represent stochastic screening. It is apparent that certain single color tone scales (cyan and magenta) have a hue shift of several deltaE between the two types of screening.

Comparison of conventional and stochastic screening [4]

When a plate curve is applied, the color is moved along the appropriate curve in color space. (I call this curve the trajectory.) A plate curve cannot make your halftone jump from one trajectory to another. Maybe everyone knows this already, but it bears saying. You cannot use a plate curve to match color between conventional and stochastic screening.

If other forms of printing were to be included, the hue shift in the midtones would be more dramatic. Gravure and ink let printing are still farther from the graph above of conventional printing. To put this another way, hitting the correct TVI does not guarantee the correct color if there is a difference in printing modality. The solids might be spot on and the TVI might be absolutely correct, but the actual color of the 50% tone will not be correct.

TVI of spot colors

What happens if I want to compute the TVI of a spot color (Pantone or PMS)? I just have to pick the right density filter, right? Short answer: For a minority of inks, let's say a quarter of them, this actually works well. But, TVI really sucks at quantifying some spot colors. For others, it's better - somewhere around "lousy".  These are harsh words, but I have data to back them up.

I looked at a set of tone ramps of 394 spot colors. I computed the TVI for each of the tones using the Murray-Davies formula at the wavelength where the spectrum of the solid had the lowest reflectance. Then I sat back and watched the fireworks. Of the 394 inks, almost all of them (353) had a TVI (at 50%) of greater than what we would call "typical", 18%. A total of 70 of the inks had astronomical TVI values of over 35%. Most press operators I know would call that an extreme case of plugging. But it wasn't  plugging at all. The TVI was horribly large, but the color of the halftone looked just fine. Plenty of contrast in the shadows.

The graph below is the spectra of the tone scale of one of the inks that had a TVI of about 35%. Note that, around 600 nm where the reflectance is the very least, the solid, 90%, 80%, and 70% have nearly identical reflectance values. Anywhere in the red part of the spectrum, this looks like an ink with severe plugging.

The spectra of a tone scale of one blue spot color

But have a look at what happens at 450 nm. In the blue part of the spectrum, there is a very clear separation between the shadow tones. In this region, all appears to be right with the world. Despite what TVI tells us, there is contrast in the shadows. [5]

Conclusions

The use of TVI as a process control device has been proven to work for CMYK inks on a web offset press using conventional screening. But...

a) TVI is dependent on the wavelength chosen. If you are dealing with CMYK inks and conventional screening and web offset printing, then so long as you are careful about selecting the correct filter on the densitometer, this isn't a problem. The only time a problem pops up is when TVI is computed from colorimetric data. 

b) When we switch from conventional web offset printing to something else, particularly when there is a difference in the crispness of the dot, having the same color for the solids and the same value for the TVI does not guarantee that the color of the tones will be the same. 

c) When you look at inks other than CMYK, the TVI value sometimes is a useful parameter for process control, but it often is completely misleading.

In the next installment of this series on TVI, I will explain a slightly different equation than the Murray-Davies equation and the Yule-Neilsen equation. In the final installment, I will look at some practical solutions to the issue of process control for halftones of spot colors.

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[1] This isn't just a cyan thing. The same issue can be seen with magenta ink, and to a lesser degree, with black and yellow.

[2] Congress is acting swiftly to ban the disposal of unwanted spectral data, or at least trying to act swiftly. Right now, it is stuck in the quagmire of partisanship with Democrats tying the bill to efforts to save the rain-forests in southeast Kansas, and Republicans demanding that magenta density be removed since the word "magenta" does not appear in the Bible.

[3] Astute readers will note the proliferation of the word "right" added to the end of  a sentence for emphasis.   Some readers might assume that this is a technique used to draw attention to things we might assume but which are not correct, right?

[4] The image is from a presentation by Dr. Bestmann to the ISO technical committee 130 in September of 2011.

[5] This is a drastically shortened version of an article that appeared in IDEAlliance Bulletin magazine, spring of 2012 issue: Measuring TVI of a spot color. If you are interested in obtaining your own subscription to this magazine, have a look at the IDEAlliance website.