The color of a bunch of dots, part 2

In the color of a bunch of dots, part 1, I focused on one simple equation for the prediction of the reflectance of a halftone, the Murray-Davies equation. This equation is reasonable and readily understood, but it does not do such a good job at predicting what happens when ink meets paper. The Murray-Davies  equation does such a poor job of predicting reflectance that the prediction error has become one of the most common standard process control parameters for printing.

Dot gain in the headlines

That comment is important enough to repeat. Please read this slowly, carefully articulating every word: The Murray-Davies  equation does such a poor job of predicting reflectance that the prediction error has become one of the most common standard process control parameters for printing. It is called "dot gain" by old pressmen, and "tone value increase" or "TVI" by the intellectual elite. Feel free to decide which group you belong to and use the appropriate phrase.

Of course, we knew one source of error in the Murray-Davies approximation. Halftone dots are bigger in real life than they were in the image file. Ink squishes out between the plate and the blanket and then again when it transfers from blanket to paper, so the dots on the paper are bigger [1]. To be fair to those using the Murray-Davies equation to compute TVI, the degree to which the dots spread is a valid control parameter indicative of how dots squish out.

But, to be fair to people who make brash, negative comments about the Murray-Davies equation (and then have the gall to repeat them in italics), how much the dots squish is only part of the prediction error.

Optical dot gain

The Murray-Davies equation was published in 1936. It was known at least by 1943 that it did not work well for halftone dots on paper. I quote from Yule: "Experimental results do not agree exactly with the theoretical relationships except for screen negatives and positives with sharp dots."

Enter John A.C. Yule and W. J. Neilsen. They presented a paper at the 1951 TAGA conference [2] entitled "The Penetration of Light into Paper and Its Effect on Halftone Reproduction", where they described another reason for the discrepancy. [3]

Yul Brynner, Leslie Nielsen, and Dodd Gayne
(from The King and I in the Cockpit) [4]

The Murray-Davies formula makes the assumption that light either hits a halftone dot or paper. Furthermore - and this is the critical part - that the dots of ink don't effect the color of the paper. Yule and Neilsen point out that this is just not the case. Anyone who says otherwise is itching for a fight.

The diagram below shows the Yule-Neilsen effect. When we look at a halftone, some of the light follows path #1. It passes through the ink once, reflects from inside the paper, and then exits for us to behold its marvelous hue. The ink acts like a filter, so two passes make it a richer color.

Some of the light, however, passes through the ink once and then scatters within the paper. This hapless light then exits from between halftone dots. Since it has only passed through the filter (the ink) once, it will not take on quite as rich a hue.

Dramatization of the  Yule-Neilsen effect

The next diagram illustrates what the result looks like. The paper between halftone dots takes on a richer hue as a result. The magnitude of the effect depends on a few factors. First, obviously upon the amount that the paper scatters light. This is related to the opacity of the paper. A paper that is translucent will tend to scatter light further, enhancing the Yule-Neilsen effect. 

Second, and perhaps not so obvious, is the screen ruling. If the dots are closer together, the light doesn't have to scatter as far to infuse the whole area between halftone dots.

Equally dramatic demonstration of the effect of the Yule-Neilsen effect

These gentlemen, Mr. Yule and Mr. Neilsen, were pretty sharp guys. They knew some math. The graph below shows the equation that they came up with to model this effect. From this equation, it was now possible to predict the reflectance (or density as in the graph) of a halftone from the dot area on the paper and the densities of the paper and solid.

The celebrated Yule-Neilsen equation as it originally appeared

The equation above is written in terms of density and not reflectance. This makes it a bit hard to relate to the Murray-Davies equation. Here is the equation written in a way that makes the correspondence obvious.

Yule-Neilsen equation

Comparing this back to the Murray-Davies equation, we see that the only difference is that in the Yule-Neilsen equation, all the reflectance values are raised to the power of 1/n. The Murray-Davies equation is a special case of the Yule-Neilsen equation with n = 1.

Murray-Davies equation

The appropriate value for "n" depends on the translucency of the paper. One researcher (Pearson) said that it should be between 1.4 and 1.8. Another set of researchers (Qian et al.) had it at 1.3 for their substrate.

Now it gets complicated

This thing that has become known as "tone value increase" thus is comprised of two parts: 

1) The dots on the paper are richer in color than expected because the dots squish out to cover more paper. This is known as physical dot gain.

2) The light spreads between the dots to make the paper take on a tint of the color. In doing so, they also make the color of a halftone richer than expected. This is known as optical dot gain.

That was the easy part. Now for the complicated part. 

Optical dot gain is not as easy to measure as physical dot gain. You need to take a picture of the dots, and assign each pixel to either dot or paper. This isn't all that hard, but it can't be done with a standard spectrophotometer. A second instrument is used, called a planimeter. For "hard" dots - dots that have crisp, well defined edges, the assignment of each pixel isn't that hard. You simply set a threshold gray value somewhere around half way between "paper" and "dot". But for softer dots, the measurement you get depends a lot on how the threshold is chosen.

So, basically, no one in a production environment ever measures physical dot gain. The two types of dot gain get rolled into one. The combination of the two, TVI has become the process control parameter of choice. Any difference between the tone value in the file and the tone value on the paper is undifferentiated.

But researchers generally use the Yule-Neilsen equation to model the relationship between CMYK tone values and reflectance. I have a short list of such papers below [5]. The Holy Grail for these folks is to find a formula that will allow them to compute the whole shooting match. CMYK tone values, along with some press parameters, go into the magic black box. CIELAB values come out.

The Yule-Neilsen equation (and the n value that go with it) are kind of a one-way street. It can be used in prepress to predict what a halftone will look like, but you can't use it in the press room to verify that the print is correct. 

So, for the time being we are stuck with the Murray-Davies equation. Maybe that's not so bad? I will address this issue in the next post in this series. Maybe the current way of measuring halftones is not the panacea that we think it is.

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[1]  I am taking a very web-offset-centric view of this. The same principles apply to other types of printing.

[2]  TAGA (Technical Association of the Graphic Arts) has been running an annual conference since the ten commandments were inscribed via a lithographic process without the help of Yule Brynner. The 2014 call for papers is out. Submissions dues by July 19th, 2013.

[3]  I missed this conference and hence the landmark paper, but I don't quite recall why.

[4] Seriously for the moment. Yule is the print scientist. Yul is the actor who my wife has a crush on. Someday I will shave my head to compete. Neilsen is another print scientist.The actor's last name is spelled "Nielsen". My wife does not have a crush on him, but regardless, I will someday grow my hair white.

The insidious misspelling of Neilsen's name is so pervasive that Google's patent search returns 11,400 results searching on "Yule Nielsen factor" and only 88 on the correct spelling. This just ticks me off.

Scan from the 1951 TAGA Journal, showing correct spelling

[5] Here is a list (incomplete) of papers where the Yule-Neilsen formula is featured as a way to predict the color of a halftone.

Pearson, M. (1980). N-value for general conditions. In TAGA Proceedings, (pp. 415–425).

Viggiano, J. A. S. (1985). The color of halftone tints. In TAGA Proceedings, (pp. 647–663).

Pope, W. (1989). A practical approach to N-value. In TAGA Proceedings, (pp. 142–151).

Rolleston, R., & Balasubramanian, R. (1993). Accuracy of various types of Neugebauer model. In IS&T and SID’s Color Imaging Conference: Transforms and Transportability of Color, (pp. 32–37).

Arney, J. S., Arney, C. D., & Engeldrum, P. G. (1996). Modeling the yule-nielsen halftone effect. Journal of Imaging Science and Technology, 40(3), 233–238.

Hersch, R. D., & Crt, F. (2005). Improving the Yule-Nielsen modified spectral Neugebauer model by dot surface coverages depending on the ink superposition conditions. In IS&T Electronic Imaging Symposium, Conf. Imaging X: Processing, Hardcopy and Applications, SPIE, vol. 5667, (pp. 434–445).

Gooran, S., Namedanian, M., & Hedman, H. (2009). A new approach to calculate colour values of halftone prints. In IARAGAI.

Rossier, R., & Hersch, R. D. (2010). Ink-dependent n-factors for the Yule-Nielsen modified spectral Neugebauer model. In CGIV – Fifth European Conference on Colour in Graphics, Imaging, and MCS/10 Vision 12th International Symposium on Multispectral Colour Science.

Qian, Yiming, Nawar Mahfooth, and Mathew Kyan, (2013) Improving the Yule-Nielsen modified spectral Neugebauer model using Genetic Algorithms, 45th Annual Conference of the International Circle