Is my color process all wonky?

In a previous post, I looked at how the Zc statistic can be used to isolate individual color measurements that are icky-poopy. Today I look at a slightly broader question: How can I tell if the whole production run is wonky?

I think something went wrong in this production run

I deliberately use the word "wonky" because it has a squishy meaning, which is helpful, since I'm not sure what I want it to mean just yet! So, bear with me while I fumble around in the darkness, not knowing quite what I am doing just yet. 

Here is the premise: Color, in a well-controlled production run, should vary according to some specific type of statistical distribution. (Math mumbo-jumbo alert) I will take a guess that the cloud of points in that ellipsoid of L*a*b* values is a three-dimensional Gaussian distribution, with the axes appropriately tilted and elongated. If this is the case, then the distribution of Zc will be chi with three degrees of freedom. (End math mumbo-jumbo alert.)

If you are subscribed to the blog post reader that automatically removes sections of math mumbo-jumbo, then I will recap the last paragraph in a non-mumbo-jumbo way. In stats, we make the cautious assumption of the normal distribution. Since I am inventing this three-dimensional stats thing, I will cautiously assume the three-dimensional equivalent. But, since this is virgin territory, I will start by testing this assumption.

A quick note about CIELAB target values and DE

This blog post is not about CIELAB target values and DE. Today, I'm not talking about assessing conformance, so DE is not part of the discussion. I am talking about whether the process is stable, not whether it's correct.

A look at some real good data

Kodak produced a photographic test target, known as the Q60 target, which was used to calibrate scanners. The test target would be read by a scanner, and the RGB values which were read were compared against L*a*b* values for that batch of targets in order to calibrate the scanners. When the scanner encountered that same type of film, this calibration would be used to convert from RGB values to moderately accurate L*a*b* values. Hundreds of thousands of these test targets were manufactured between 1993 and 2008.

I think the lady peeking out on the right is sweet on me

We know that these test targets were produced under stringent process control. They were expensive, and expensive always means better. More importantly, they were produced under the direction of Dave McDowell. I have worked with him for many years in standards groups, and I know they don't come more persnickety about getting the details right than him!

Dave provided me with data on 76 production runs of Ektachrome, which was averages of  the L*a*b* values from 264 patches, for a total of about 20K data points. So, I had a big pile of data, collected of production runs that were about as well regulated as you can get.

I applied my magic slide rule to each set of the 264 sets of 76 color values. Note that I pooled at the data for individual colors of patches. General rule in stats: You don't wanna be putting stuff in the same bucket that belongs in different buckets. They will have different distributions.

Within each of the 264 buckets, I computed Zc values. Twenty thousand of them. I hope you're appreciative of all the work that I did for this blog post. Well... all the work that Mathematica did.

Now, I could have looked at them all individually, but the goal here is to test my 3D normal assumption. I'm gonna use a trick that I learned from Dave McDowell, which is called the CPDF.

Note on the terminology: CPDF stands for cumulative probability density function). At least that's the name that it was given in the stats class that I flunked out of in college. It is also called CPD (cumulative probability distribution), CDF (cumulative distribution function), and in some circles it's affectionately known as Clyde. In the graphic arts standards committee clique, it has gone by the name of CRF (cumulative relative frequency).

Here is the CPDF of the Ektrachrome data set. I through all the Zc values into one bucket. In this case I can do this. They belong in the same bucket, since they are all dimensionless numbers... normalized to the same criteria. The solid blue line is the actual data. If you look real close, you can see a dotted line. That dotted line is the theoretical distribution for Zc that 3D normal would imply. Not just one particular distribution -- the only one.

20,000 color measurements gave their lives for this plot

Rarely do I see real world data that comes this close to fitting a theoretical model. It is clear that L*a*b* data can be 3D normal.

More real world data

I have been collecting data. Lots of it. I currently have large color data sets form seven sources, encompassing 1,245 same-color data sets, and totalling 325K data points. When I can't sleep at night, I get up and play with my data. 

[Contact me if you have some data that you would like to share. I promise to keep it anonymous. If you have a serious question that you want to interrogate your data with, all the better, Contact me. We can work something out.] 

I now present some data from Company B, which is one of my anonymous sources. I know you're thinking this, but no. This is not where the boogie-woogie bugle boy came from. This complete data set includes 14 different printed patches, sampled from production runs over a full year. Each set has about 3,700 data points. 

I first look at the data from the 50% magenta patch, since it is the most well-behaved. The images below are scatterplots of the L*a*b* data projected onto the a*b* plane, the a*L* plane, and the b*L* plane. The dashed ellipses are the 3.75 Zc ellipses. One might expect one out of 354 data points to be outside of those ellipses.

Three views of the M 50 data from Company B

Just in case you wanted to see a runtime chart, I provide one below. The red line is the 3.75 Zc cutoff. There were 24 data points where Zc > 3.75. This compares to the expectation of 10.5. This is the expectation under the assumption that the distribution is perfectly 3D normal. I am not concerned about this difference; it is my expectation that real life data will normally exceed the normal expectations by a little bit.

Another view of the M 50 data - Zc runtime plot

So far, everything looks decent. No big warning flags. Let's have a look at the CPDF. PArdon my French, but this looks pretty gosh-darn spiffy. The match to the theoretical curve (the dotted line) is not quite as good as the Ektachrome data, but it's still a real good approximation. 

Another great match 

Conclusion so far, the variation in color data really can be 3D normal!

Still more real world data

I show below the CPDF of Zc for another data set from that same source, Company B. This particular data set is a solid cyan patch. The difference between the real data and the theoretical distribution is kinda bad.

A poor showing for the solid cyan patch

So, either there is something funky about this data set, or my assumption is wrong. Maybe 3D normal isn't necessarily normal? Let's zoom in a bit on this data set. First, we look at the runtime chart. (Note that this chart is scaled a bit different than the previous. This one tops out at Zc = 8, whereas the other goes up to 5.5.) 

A runtime chart with some aberrant behavior
that will not go unpunished!

There are clearly some problems with this data. I have highlighted (red ellipse) two short periods where the color was just plain wonky. Some of the other outliers are a bit clustered as well. Below I have an a*b* scatter plot of that data (on the left), and a zoomed-in portion of that plot which shows some undeniable wonk. 

Look at all the pretty dots that aren't in the corral where they belong

  

I'm gonna say that the reason that the variation in this data set does not fit the 3D normal model is because this particular process is not in control. The case gets stronger that color variation is 3D normal when the process is under control.

Are you tired of looking at data yet?

We have looked at data from Company K and Company B. How about two data sets from Company R? These two data sets are also printed colors, but they are not the standard process colors. There are  about 1,000 measurements of a pink spot color, and 600 measurements of a brown spot color. One new thing in this set... these are measurements from an inline system, so they are all from the same print run.

First we look at the CPDF for the pink data. Yes! I won't show the scatterplots in L*a*b*, but trust me. They look good. Another case of "3D normal" and "color process in good control" going hand-in-hand.

Yet another boring plot that corroborates my assumptions

Next we see the CPDF of Zc for the brown data. It's not as good as the pink data, or the Kodak, or the M 50 CPDF plots, but not quite as bad as the C 100. So, we might think that the process for brown is in moderate control?

Brown might not be so much in control?

The runtime chart of Zc looks pretty much like all the others (I could plop the image in here, but it wouldn't tell us much). The scatter plots of L*a*b* values also look reasonable... well, kinda. Let's have a look.

Halley's comet? Or a scatterplot of variation in brown?

.This data doesn't look fully symmetric. It looks like it's a little skewed toward the lower left. And that is why the CPDF plot of brown looks a bit funky. Once again, we see that the CPDF of Zc values for a set of color variation is a decent way to quickly assess whether there is something wrong with the process.

But why is the brown plot skewed? I know the answer, but we're gonna have to wait for the full exposition in another blog post.

For the tine being, let me state the thrilling conclusion of this blog post.

The thrilling conclusion of this blog post

When a color producing process is "in control" (whatever that means), the variation in L*a*b* will be 3D normal. This means that we can look at the CPDF of Zc as a quick way to tell if we have exited the ramp to Wonkyville.

Munsell - the Father of Color Science? (part 3)

This series of blogs was foretold in a prophecy of April of 2013:

Someday I will write a blog post about how this guy Munsell laid the foundation for the ever-popular color space CIELAB, and came to be known as the Father of Color Science. He was also the father of A. E. O. Munsell, who carried on his work. I don't intend to write a blog post about how Albert became the father of A. E. O.

What I did not foretell in that blog post is that ISCC will be sponsoring the Munsell Centennial Symposium,  June 10 - 15, 2018 in Boston. Or that I would be keynoting this event.

After two previous attempts (Munsell as an educator, Munsell and 3D color space), I am finally have made my way to looking at the most significant work of Munsell. The Munsell Color Space was a model for CIELAB.

First cursory pass

Exhibit A. Richard Hunter's book The Measurement of Appearance, on page 136.

Photo taken at the Color Difference family picnic

This is a family tree of proposed models for determining color difference. Note that the Munsell Color system is in the upper right hand corner, and all arrows come from that box. The only little boxes that are still active today are the two boxes labeled CIE 1976. A similar diagram is on page 107 of that same book, which shows a family tree of color scales. (I have an image of that in a previous blog posts about color difference.) Again, this shows a straight lineage from Munsell Color Scpce to CIELAB.

Is this reliable testimony? Richard Hunter was a fairly knowledgeable guy when it comes to color. I mean, he has his own entry in Wikipedia for goodness sake. CIELAB is (perhaps) the most widely used tool in the color industry. Since Hunter traces the lineage of CIELAB back to Munsell, then I feel pretty confident about putting Munsell on the shortlist of highly influential figures in the history of color science, at the very least.

But, that hides a lot of the fun stuff that happened between the creation of the Munsell color space and the ratification of CIELAB as a standard for color measurement.

What Munsell did

    Munsell Color Space

Munsell's color space is based on some simple principles.

1. Hue, Value, and Chroma

There are three attributes to color in Munsell's color system. While these are implicit in many of the previous color systems (enumerated in a previous blog post), Munsell was intent on tying these to our intuitive understanding of color. (After studying on this for 25 years, I have come to realize that they are indeed intuitive). 

2. A physical standard produced with simple tools, simple math, and a defined procedure

Munsell described the procedure by which his color system could be developed from any reasonable set of pigments. The procedure included a way to assign unique identifiers to each color.  As a result, all colors within the gamut of the chosen pigments could be unambiguously named.    

3. Perceptual linearity

One of Munsell's secondary aims was to create a color space where the steps in hue, value, and chroma were all perceptually linear. Did he meet his goal? Stay tuned.


This color system was used to create the Atlas of the Munsell Color System, which was a book containing painted samples with their corresponding designations of hue, value, and chroma. This book was to be used as an unambiguous way to identify colors, and thus, to provide a standrd way to communicate color.

     Munsell photometer and the gray scale

Munsell invented and patented a photometer which was capable of measuring the reflectance of a flat surface. Well, provided it was a neutral gray. The user would look into a box and see two things: the sample to be measured, and a standard white patch. The sample was illuminated with a constant illumination, and the white standard was illuminated with light through an adjustable aperture. To make a measurements, the size of the aperture was adjusted so as to match the intensity of the dimmed white standard and that of the sample. The width of the aperture, scaled from 1 to 10, was the Munsell Value for the gray sample.

A shoebox with some holes and stuff

Munsell used his photometer to mix black and white paints in steps from V = 1 to V = 10.

     Maxwell disks and the rest of the colors

James Clerk Maxwell invented a creature called the Maxwell disk around 1855. I spent the better part of a day building my own set of Maxwell disks from colored construction paper as shown below. The cool part is the slit. You can slide two or more disks together, and rotate them so as to get any proportion of the colors to show. In the inset, I show the device that I adapted to rotate the disks. Again, the better part of a day was spent assembling a bolt, a couple of washers, and a nut. I first tried a cordless drill, and found it didn't spin fast enough to merge the colors. I had to use my old drill that plugs into the wall.

The Maxwell disks were the inspiration for PacMan

The picture below shows the results of day 3 of my dramatic reenactment of Munsell's landmark experiment. I selected red, green, and blue construction paper, and adjusted the size of the segments in order to get a facsimile of gray. When I saw that gray, I realized that this was four days well spent.

Me, geeking out on the creation of gray from Red, green, and blue

If I were to be doing this on a government grant, I would have spent another day or two actually measuring the sizes of the red, green, and blue areas. For the purposes of this blog, I will be content with just saying that red and blue are each one-quarter, and green is one-half. In other words, this green is half as strong as the others. Thus, Munsell would conclude that the chromas of this red and blue were twice that of this green. Munsell would also have measured this gray with his photometer. Another opportunity for me to get a little more grant money.

In this way, Munsell was able to assign values to the colors.

     Perceptual linearity?

Linear in Value?

Since Munsell's original Value was measured as the width of an aperture, the amount of light let through is proportional to the square of the Value. Conversely, Value is proportional to the square root of the light intensity. The plot below compares this scale against today's best guess at perceptual linearity, CIEDE2000.

Munsell's original Value was kinda sorta close to perceptually linear

Note: The DE2000 scale in the plot above is based on Seymour's formula (L00 = 24.7 Log e (20 Y +1), where 0< Y < 1), which was first presented at TAGA 2015, Working Toward A Color Space Built On CIEDE2000. The height of the curve at the end shows that there are 76 shades of gray, based on DE2000. The Munsell Value has been scaled to that.

Is this perceptually linear? That depends on how gracious you want to be. On the one hand, the linearity is not lousy. Given the tools at hand, Munsell did a fairly decent job of making kinda linear.

On the ungracious side, Munsell merely took what he had handy (the size of the opening of his aperture) and used that. Lazy bum! Surely he would have known about the work of Ernst Weber (1834) and Gustav Fechner (1860) which postulated that all our perception is logartihmically based! 

Really pedantic note: There is some confusion about how the gray scale was set up. My description is based on Munsell's description [1905], as well as comments by Tyler and Hardy [1940], Bond and Nickerson [1940], and Gibson and Nickerson [1940], all of which were based on Munsell's words and measured samples. But in a paper from 1912, Munsell described his assignment of Value as being logarithmic, following the Weber-Fechner law.

Linear in hue?

Munsell started this exercise by selecting five paints with vibrant colors: red (Venetian red), Yellow (raw sienna), green (emerald green), blue (cobalt), and purple (madder and cobalt). He then created paints that were opposite hues for each of these. The opposite hues were adjusted so that the balanced out to gray on the Maxwell disks. Thus, he had a set of ten colors with Value of 5 and Chroma of 5.

What's to say that these paints are equally spaced in hue? I am sure that Munsell selected them with that in the back of his mind, but four of the five are just commonly available, single pigment paints.
From the literature that I reviewed in the bibliography below, I could find no evidence that he put much time into psychophysical testing.

I'm gonna say that the hue spacing in the original Munsell color system is only somewhat perceptually linear.

Linear in Chroma?

Munsell's assignment of Chroma values is all based on simple ratios of areas on the Maxwell disks. Thus, in his original system, chroma is linear with reflectance. I did a bit of testing, comparing Munsell's proposition against DE2000. I will smugly state that our perception is not linear with reflectance.

But Munsell begs to differ with me. He performed some tests of this, and summarized his results in 1909:

These experiments show clearly that chroma sensation and chroma intensity (physical saturation) vary not according to the law of Weber and Fechner, but nearly or quite proportionately, and in accordance with the system employed in my color notation.

This paper seems to have been largely ignored by other color researchers. Deane Judd looked at the question of equal steps in chroma in 1932. His bibliography included Munsell's 1909 paper, but he made no mention of it in the text. The same with several of the papers from 1940 listed below.

My brief test suggests this is not true, and the people who were genuinely interested in the question who were aware of Munsell's suggestion ignored it. The graphs from the 1943 paper (Newhall, et al.) are decidedly non-linear in steps of chroma. Barring further evidence, I would say that the original Munsell Color System was not perceptually linear in chroma.

All in all, I'm gonna rate the claim that the original Munsell system was perceptually linear as "Mostly False".

What happened after Albert Munsell

Albert Munsell passed on in 1918, but a lot of work was done on the Munsell Color System by others after his death.

In 1919 and again in 1926, Munsell's son, A. E. O. Munsell submitted samples to the National Bureau of Standards. These were measured spectrophotometrically. The 1919 data was analyzed by Priest et al., and came along with some suggestions for improvement. They suggested that the Value scale be changed. 

This challenge was taken up by Albert's his own son. In 1933, A. E. O. published a paper describing a modification of the function from which Value was computed. This brought value much closer into line with the predictions of CIEDE2000.

The Munsell Color System was largely ignored in the literature until 1940. At that time, seemingly everyone jumped on the bandwagon. A subcommittee of the Optical Society of America was formed, and the December 1940 issue of the Journal of the Optical Society in America published five papers on the Munsell Color System.

Why the sudden effort? Spectrophotometers were expensive and cumbersome, but were becoming available. The 1931 tristimulus curves were available to turn spectral data into human units. Several of the papers noted a desire to create a system which translated physical measurements into something that made intuitive sense.

The Munsell Color System seemed to be best template to shoot for, since it was "[l]ong recognized as the outstanding practical device for color specification by pigmented surface standards."  (Newhall, 1940)

The efforts of the OSA subcommittee culminated in what has become known as the Munsell Renotation Data, introduced in the 1943 paper by Newhall et al. Inconsistencies of the original data were smoothed out, a new Value scaled was introduced, and a huge experiment (3 million observations) was done to nudge the colors into a system that looked perceptually linear.  The final result is a color system that can indeed be said to be perceptually linear.

Oh what a tangled web we weave, from Newhall (1943)

I'm not gonna take up the rest of the story, from the Renotation Data to CIELAB. That's another long and interesting story, I'm sure. But I am running out of gas!

Conclusion

Here is the firmest entirely factual statement that I can make about this paternity suit involving Albert Munsell and the child named Color Science.

Munsell had a passion for teaching color, especially to children. He sought to bring order and remove ambiguity from communication of color. This passion brought him to create the Munsell Color System. This was not the first three-dimensional arrangement of color, nor was it all that close to being perceptually linear. But it had two great features going for it: It was built on the intuitive concepts of hue, chroma, and lightness, and it came with a recipe for building a physical rendition of the color space. As a result, the Munsell Color Space is both a concept for understanding color, and a physical standard to be used in practical communication of color.

The Munsell Color System saw a number of improvements after his death, resulting in the Munsell Renotation Data. This later became the framework for future development of a magic formula to go from measured specrta to three numbers that define a color. The CIELAB formula is the one that stuck.

I realize that my work over the past 25 years has given me a bias toward the importance of measurement of surface colors, and hence a bias toward thinking that CIELAB is important. The next statement is subjective, and based on my admitted biases.

I think that Albert Munsell deserves to be called The Father of Color Science.

Albert Munsell proudly showing off his very attractive John the Math Guy Award

On the other hand...

I would be remiss if I failed to mention a few other individuals, who might reasonably be on the podium with Munsell.

Isaac Newton - He invented the rainbow, right? Well, actually, he did some experiments with light and came up with the theory of the spectrum. Spectrophotmeters are designed to measure this.

Thomas Young - He first proposed the theory that the eye has three different sensors (red, green, and blue) in 1802. Hermann von Helmholtz built on this in 1894.

Ewald Hering - He proposed the color opponent theory in 1878. Light cannot be both red and green; nor can it be both blue and yellow. His three photoreceptors were white versus black, red vs green, and yellow vs blue. This is explicitly built into CIELAB.

It turns out that all of these are correct, but they are looking at different stages in our perception. Newton's spectrum is a real physical thing. The retina does have three Young-Helmholtz sensors. The cones are not exactly RGB, but kinda. And the neural stuff after the cones in the retina creates signals that follow Hering's theory.

So, maybe one of these gents should get the crown? I dunno... maybe I'll make a few more John the Math Guy awards?

Bibliography

Munsell's papers

Munsell, Albert H., A Color Notation, Munsell Color Company, 5th Edition, 1905, Chap V

Munsell, Albert H., On the Relation of the Intensity of Chromatic Stimulus (Physical Saturation) to Chromatic Sensation, Psychological Bulletin, 6(7), 238-239 (1909)

Munsell, Albert H, A Pigment Color System and Notation, Amer. Journal of Psych, Vol 23, no. 2, (April 1912)

Post Munsell, pre-1940

Priest, Irwin, K. S. Gibson, and H. J. McNicholas, An examination of the Munsell color system. I. Spectral and total reflection and the Munsell scale of value, Tech. Papers of the Bureau of Standards, No. 167 (September 1920)

Judd, Deane, Chromatic Sensibility to Stimulus Differences, JOSA 22 (February 1932)

Munsell, A. E. O., L. L. Sloan, and I. H. Godlove, Neutral Sclaes. I. Munsell Neutral Value Scale, JOSA (November, 1933)

Glenn, J. J. and J. T. Killian, Trichromatic analysis of the Munsell Book of Color, MIT Thesis (1935), also in JOSA (December 1940)

The 1940's flurry

Gibson, Kasson S and Dorothy Nickerson, An Analysis of the Munsell Color System Based on Measurements Made in 1919 and 1926, JOSA, December 1940

Newhall, Sidney, Preliminary Report on the O.S.A. Subcommittee on the Spacing of the Munsell Colors, JOSA, December 1940

Tyler, John E. and Arthur C. Hardy, An Analysis of the Original Munsell Color System, JOSA December 1940

Nickerson, Dorothy, History of the Munsell Color System, Company, and Foundation. II. Its Scientific Application, JOSA, December 1940

Bond, Milton E., and Nickerson, Dorothy, Color-Order Systems, Munsell and Ostwald, JOSA, 1942

Newhall, Sidney M., Dorothy Nickerson, and Deane B. Judd, Final Report of the O.S.A. Subcommittee on the Spacing of the Munsell Colors, JOSA July 1943

More recent

Hunter, Richard S., The Measurement of Appearance, John Wiley, 1975, pps. 106 - 119

Is my color process going awry?

This blog is a first in a series of blog posts giving some concrete examples of how the newly-invented technique of ColorSPC and ellipsification can be used to answer real-world questions being asked by real-world people about real-world problems for color manufacturers.

So, picture this scenario. I am running a machine that puts color onto (or into) a product. Maybe it's some kind of printing press; maybe it mixes pigment into plastic; maybe this is about dyeing textiles or maybe it's about painting cars. The same principles apply.

John the Math Guy really Lays color SPC on the line

Today's question: I got this fancy-pants spectrophotometer that spits out color measurements of my product. How can I use it to alert me when the color is starting to wander outside of its normal operating zone?

An important distinction

There are two main reasons to measure parts coming off an assembly line:

     1. Is the product meeting customer tolerances?

     2. Is my machine behaving normally?

Conformance and SPC (statistical process control). These are intertwined. Generally, one implies the other. But consider two scenarios where the two answers are different. 

It could be that the product is meeting tolerances, but the machine is a bit wonky. Not wonky enough to be spitting out red parts instead of green, but there is definitely something different than yesterday. Should we do anything about this? Maybe, maybe not. It's certainly not a reason to run out of the building with our hair on fire. But it could be your machine's way of asking for a little TLC in the form of preventative maintenance.

Or it could be that your machine is operating within its normal range, and is producing product that is outside the customer tolerances. This the case you need to worry about. Futzing with the usual control knobs ain't gonna bring things in line. You need to change something about your process.

Use of DE for SPC

The color difference formulas, such as DE₀₀, were designed specifically to be industrial tolerances for color. While DE₀₀ may well be the second ugliest formula ever developed by a sentient being in this universe, it does a fair job of correlating with our own perception of whether two colors are an acceptable match. 

But is it a good way to assess whether the machine is operating in a stable manner? I mean, you just track DE over time, and if it blips, you know something is going on. Right? Let's try it out on a set of real data.

The plot below is a runtime chart of just over 1,000 measurements of pink spot color that I received from Company B. These are all measurements from a single run. I don't know for sure what the customer tolerance was, but I took a guess at 3.0 DE₀₀, and added that as an orange dashed line.

It sure looks like a lot of measurements were out of tolerance!

Uh-oh. It looks like we got a problem. There are a whole lot of measurements that are well above that tolerance... maybe one out of three are out of tolerance?

But maybe it's not as bad as it looks. The determination lies in how one interprets tolerance. Here is one interpretation from a technical report from the Committee for Graphic Arts Technologies Standards (CGATS TR 016, Graphic technology — Printing Tolerance and Conformity Assessment):

"The printing run should be sampled randomly over the length of the run and a minimum of 20 samples collected. The metric for production variation is the 70th percentile of the distribution of the color difference between production samples and the substrate-corrected process control aims."

TR 016 defines a number of conformance levels. (For a description of what those values mean, check out my blog on How Big is a DE₀₀) It says that 3.0 DE₀₀  is "Level II conformance", so the orange dashed line is a quite reasonable acceptance criteria for a press run. But a runtime chart is not at all useful for identifying those "Danger Will Robinson" moments. I mean, how do you decide if a single measurement is outside of a tolerance that requires 20 measurements? 

If we want to do SPC, then we must set the upper control limit differently.

Use of DE for SPC, take 2

The basic approach from statistical process control -- the whole six sigma shtick -- is to set the upper control limit based on what the data tells us about the process, and not based on customer tolerances. It is traditional to use the average plus three times the standard deviation as the upper limit. For our test data set, this works out to 5.28 DE₀₀.

The process looks in control now!

This new chart looks a lot more like a chart that we can use to identify goobers. In fact, I did just that with the two red arrows. Gosh darn it, everything looks pretty good.

But I think we need a bit closer look at what the upper limit DE means. The following pair of plots give us a perspective of this data in CIELAB. The plot on the left is looking down from the top at the a*b* values. The plot on the right is looking at the data points from the side with chroma on the horizontal axis and L* on the vertical.

The green dots are each of the measurements. The red diamond is the target color, and the ovoids are the upper limit tolerances of 5.28 DE₀₀. (Note: in DE₀₀, the tolerance regions are not truly ellipses, but are properly called ovoids. One should ovoid calling them ellipses, and also ovoid making really bad puns.)

Those are some big eggs!

The next image is  closeup of the C*L* plot, showing (with red arrows) the small set of wonky points that were identified with the DE runtime chart. I would say that these are pretty likely to be outliers. But look at the smattering of points that are well outside the cluster of data points, but are still within the ovoid that serves as the upper limit for DE. These should have stuck out in the runtime chart, if it were doing its job), but are deemed OK.

Wonkyville

Now, listen carefully... If you are using a runtime plot of "DE₀₀ from the target color", you are in effect saying that everything within the ovoids represents normal behavior for your process. So long as measurements are within those ovoids, you will conclude that nothing has changed in your process. That's just silly talk!

Here is my summary of DE runtime charts: JUST SAY NO! Well... unless your are looking at conformance, and your customer tolerance is an absolute, as in, "don't you never go above 4 DE₀₀!"

Use of Zc for a SPC

I know this was a long time ago, but remember the Z statistic from Stats 101? You compute the average and standard deviation of your data, and then normalize your data points to give you a parameter called Z. If a data point had a Z value that was much smaller than -3, or much larger than +3, then it was suspicious. This is mathematically equivalent to what's going on with the upper limit in a runtime chart.

I have extended this idea to three-dimensional data (such as color data). I call the statistic Zc. This is the keystone of ColorSPC.

Now, remember back when I showed the CIELAB plots of the data along with a DE₀₀ ovoid? Didn't you just want to grab a red pencil and draw in some ellipses that represented the data better? That's what I did, only I used my slide rule instead of a pencil. There is a mathematical algorithm that I call ellipsification that adjusts the axes lengths and orientation of a three-dimensional ellipsoid to "fit" the data. Ellipsification is the keystone of ColorSPC.

Ellipsification charts in CIELAB

The concentric ellipses in the drawings above are the points where Zc = 1, 2, 3, and 4. That is to say, all points on the innermost ellipse have Zc of 1. All points between the innermost and the next ellipse have Zc between 1 and 2.

Zc is a much better way to do SPC on color data. Here is a runtime plot of Zc for this production run. The red dashed line is set to 3.75. That number is the 3D equivalent of the Z = 3 upper limit used in traditional SPC.

Finally, a runtime chart we can believe!

As can easily be seen (if you click on the image, and then get out a magnifying glass) this view of the data provides us with a much better indication of data points which are outside of the typical variation of the process. Nine outliers are identified, and many of them stick out like sore thumbs. Kinda what we would expect from the CIELAB plots.

But wait!

In the previous DE analysis, we computed DE from the target value. In a paper by Brian Gamm (The Analysis Of Inline Color Measurements For Package And Labels Printing Using Statistical Process Monitoring Techniques, TAGA 2017), he pointed out this problem with DE runtinme charts, and advocated the use of the DE, but with DE measured from the average L*a*b* value, rather than the target. The graphs below show the result of this analysis on our favorite data set.

DE00 ovoids based on computing color difference from average

Addendum Feb 22, 2018: 

I would like to update the previous paragraph based on conversations with Brian.

First, he wanted to reiterate something that I have said before, and which bears re-reiterating. Looking at a runtime chart of DE is the correct thing to do when you are doing QA -- if your question is "did my product meet the conformance criteria from my customer?" But his paper (and this blog post) show that DE is not the proper tool for finding aberrant data. Both are necessary and useful.

Second, he advocated something a bit different than what I said. Subtle, but important difference. I said "... but with DE measured from the average L*a*b* value". Brian advocated "... but with DE measured from the initial L*a*b* value". Brian is looking at the drift during a production run. The assumption is made that color was dialed in pretty decent at the start, but may be gradually changing over time.

Thanks, Brian!

It is interesting to note that the DE₀₀ ovoid in a*b* (on the left) is similar to the to the ovoid produced by ellipsifcation. Larger, and not quite as eccentric, but similar in orientation. This is a good thing, and will often be the case. This will not be the case for any pigments that have a hook, which is to say, those that change in hue as strength is changed. This includes cyan and magenta printing inks.

However, it can be seen that the orientation of the DE₀₀ ovoid in C*L* (on the right) does not orient with the data in orientation. This is soooo typical of C*L* ovoids!

So, DE₀₀  from the average is a much better metric than DE₀₀ from target color. If you have nothing else to use, this is preferred. If you are reading this shortly after this blog was posted, and you aren't using my computer, then you don't have nothing else to use, since these wonderful algorithms have not migrated beyond my computer as I write this. I hope to change that soon.

Conclusion

For the purpose of conformance testing, there is no question that DE is the choice. DE₀₀ is preferred to ΔE_ab(or even DE_CMC  or DE₉₄  or DIN 99).

For the purpose of SPC -- characterizing your color process to outliers -- the DE from target metric is lousy. The use of DE from average is preferable, but the best metric is Zc, which is based on Color SPC and fitting ellipses to your data.

Munsell - the Father of Color Science? (part 2)

Albert Munsell has been called the Father of Color Science. In the previous blog post, I looked at whether he earned that accolade through his crusade to put the Science into Color Science Education. I concluded that he would probably have to share this with Milton Bradley -- the board game magnate. I dunno, though. Saying that Munsell and Bradley are both of the Fathers of Color Science might get a bit weird for some.

Before I continue, Munsell is held in esteem by real color scientists, not just color science wannabes who write corny blogs on color in hopes of being invited to the real people parties. One of those cool people parties is the Munsell Centennial Color Symposium, June 10-15, 2018, MassArt, Boston, MA.

So far, I have only gotten as far as being invited to give a webinar for cool people, which is based on this series of posts.  If you are reading this before Feb 21, 2018, then there is still time to sign up. As further.

Today, I try another possible explanation for why Munsell might be due the honor. Albert Munsell developed the Munsell Color System. Unlike previous two-dimensional color systems, the Munsell Color Space is three-dimensional! That made it way cooler. All of the avid readers of my blog even know why color is three-dimensional.

World famous Color Science Model admires the Munsell Color System

But was he the first to bring 3D to color?

Munsell invented the idea of a three-dimensional color space?

Here is a quote from the Introduction of Munsell's book A Color Notation System (1919). (The Introduction was written by H. E. Clifford. Evidently Clifford was his publicist. The world famous Color Science Model shown above is my publicist.)

"The attempt to express color relations by using merely two dimensions, or two definite characteristics, can never lead to a successful system. For this reason alone the system proposed by Mr. Munsell, with its three dimensions of hue, value, and chroma, is a decided step in advance over any previous proposition."

Hue, value, and chroma

That kinda sounds like three dimensional color was Munsell's idea?

Here is another piece of evidence suggesting that Munsell may have been the guy that brought 3D color to a cinema near you. US Patent #824,374 for a Color Chart or Scale was issued to Munsell in 1906. His disclosure states: "It may assist in understanding the order of arrangement of my charts to know that the idea was suggested by the form of a spherical solid subdivided through the equator and in parallel planes thereto, ..."

Doncha just love drawings from old patents?

Fig. 2 above shows a page where the hues of the rainbow are arranged around the perimeter, with them all fading to gray at the center. This is but one page of color. Previous pages would have a brighter version of this, and subsequent pages would be darker. Fig. 1 shows a cut-away version of these pages assembled into a book.

So, he got the patent! Case closed. Munsell deserves to be the Father of Color Science.

Or did he patent the color space?

But... hold on a sec. Another part of the disclosure in the patent refers to "the three well-known constants or qualities of color -- namely, hue, value or luminosity, and purity of chroma..." In the patent biz, we would refer to that hyphenated word well-known as a pretty clear admission of prior art!

Clearly Munsell did not invent the idea of using three coordinates to identify unique colors. This is why I keep telling my dogs that you have to read patents very carefully to understand what is being patented. My cute little puppies are always ready to get out the pitchforks and torches after doing a quick read of a patent.

In Munsell's paper A Pigment Color System and Notation (The Journal of Psychology, 1912), he refers to a number of previous color ordering schemes by "Lambert, Runge, Chevreul, Benson, and others".

A slice of Munsell

So, I did a little investigation. Munsell also mentioned Ogden Rood as an experimenter in color. I dug out a book named Modern Chromatics, by Ogden Rood. I should point out that using the word modern in the title of a book may not be such a good idea if you want the book to be around for a while. This book was published a while ago, like thoroughly before Modern Millie, like in 1879.

The diagrams below are from Rood's book. They look kinda like representations of three-dimensional things to me!

Cross section of Rood's color cylinder and color cone

Not only does Rood's book predate the Munsell patent by about 30 years, but on page 215, he pushed the discovery of three dimensional color back by a full century: "This colour-cone is analogous to the color pyramid described by Lambert in 1772." That was soooo rood of him!

(That was probably the worst pun of my life. I apologize to the anyone whose sense of humor was offended.)

How about these other color systems?

I stumbled on a website called colorsystem.com which chronicles more color systems that you can shake a crayon at. Here is their list of the three-dimensional color systems which predate Munsell. Are you ready?

Johann Heinrich Lambert, 1772

Tobias Mayer, 1775 

Philipp Otto Runge's Farbenkugel (1810)

I just love the name of his color space. In addition to being a world famous Color Scientist Model, my wife makes a pretty decent savory kugel.

Michel Eugene Chevreul, 1839

William Benson, 1868

Benson touts this as both an additive color space and a subtractive one. Orient it one way and you get RGB axes. Orient it another, and you get (what I would call) CMY. He called them yellow, sea-green, and pink. I have used this trick in classes for years. I had no idea that it was invented so long ago.

Wilhelm von Bezold, 1874

Wilhelm Wundt, 1874

So, including Rood's, we have eight different suggestions for a three-dimensional color space, all of which came before Munsell. Oh... wait, I almost forget the earliest one.

Robert Grosseteste, 1230

This gentleman deserves a bit of comment. The colorsystem entry on Grosseteste is a bit sparse, if you ask me. First, Grosseteste has to share a webpage with Leon Battista Alberti and Leonardo da Vinci. I would be honored to share a webpage with da Vinci, but colorsystem didn't mention that Grosseteste's color system was likely the first three-dimensional color system ever conceived.

I do not mean to malign the good folks at colorsystem (although that would be pretty much in line with my reaction to anyone who knows more than I do). I love their website. I think the whole cover-up of Grosseteste's three-dimensional color system was part of a bigger conspiracy to deprive him of his rightful place in the History of Science. In the words of David Knowles (in The Evolution of Medieval Thought, p. 281, "[Grosseteste] is now only a name ... because his chief work was done in fields where he could light a torch and hand it on, but could not himself be a burning flame for ever."

Roger Bacon, who was one of the thinkers that led our way into the renaissance, would become one of the burning flames kindled by Grosseteste. Thus, we see that Robert Grosseteste had two degrees of separation from Kevin Bacon, who was in the movie Apollo 13, which kinda had something to with with science.

Here is a quote from an in-depth study by some people who sound gosh-darn scholarly. The quote is pertinent to the debate over the first three-dimensional color space: "De colore [the paper from Grosseteste] dates from the early thirteenth century and contains a convincing argument for a three-dimensional colour space that does not follow the linear arguments that Grosseteste had inherited from previous philosophers..." 

Back to the Munsell Color Space

It would appear that my original premise was far from being correct. Munsell did not create the first three-dimensional color space.

BUT!!!! The astute picture looker will notice something critical. Rood gave us color spaces that were a cylinder and a cone. Bezold also gave us a cone, and Grosseteste gave us a double cone. Lambert's was a pyramid. Mayer's was a triangular prism. Runge, Chevreul, and Wundt all provided spheres. The Benson color space is a cube.

Please do me the favor of scrolling up to the diagram entitled "A slice of Munsell". Please do me the favor of identifying the shape of that slice. This reminds me of the time when my shrink gave me a Rorschach test. Him: "What does this ink blot look like?" Me: "An ink blot." I failed the test.

Most of the drawings in Munsell's A Color Notation System depict his color space as being a sphere, but there are a few drawings like Fig, 20 (above) that show that his color space is irregular. In his own words, "Fig. 20 is a horizontal chart of all the colors which present middle value (5), and describes by an uneven contour the chroma of every hue at this level."

The last pages of this book are color plates that are slices from his Color Atlas. Note the distinct non-standard-shapedness of this.

Why was Munsell's color space groundbreaking?

We finally come to the unique and revolutionary feature: The Munsell Color Space is not a standard geometric shape. As shown below, the high chroma red hues stick out a lot further than the blue ones. It's hard to see this, but the yellow hues with the highest chroma are near the top, whereas the richest purples are nearer the bottom.

The Munsell color solid

Munsell took the non-intuitive road not taken, and that has made all the difference. That will be taken up in the next exciting installment of this series!