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

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.

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,

*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,

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.

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]

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.

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.

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*

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**

*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.

[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.

[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.