Tuesday, January 16, 2018

Statistical process control of color, a method that works

I have preached quite a bit about the shortcomings of many methods for statistical process control of color data. I reference all the previous blogs at the end of this blog post, just in case you missed them. My take on the topic can be summarized by an anecdote about President Coolidge. After attending a church service, he was asked about the topic of the sermon. "Sin." The followup question was to ask what the minister had to say about sin. "He was against it."

It's time to unveil my technique.

Looking at magenta

Magenta is an interesting ink. It is well known that as you increase the ink film thickness (or pigment concentration), it will change in hue toward red. I know for a fact that this is well-known, since I blogged about the hue shift of inks before. After all, anyone who is anyone reads my blog. 

(Magenta is not such an odd duck, when it comes to colorants. Cyan ink is another printing ink that has this hook. I know from the physics of color that there are a lot of colorants in other industries that will do this same thing.)

The red ellipse in the diagram shows the area of the magenta ink trajectory where magenta is normally run. There are two aspects of this plot that might be disconcerting.

First, the ink trajectory is curved. One might therefor expect that the normal variation in color might wind up being shaped like a kidney bean, rather than a nice ellipsoidal jelly bean. Or maybe... I shudder to think of it... like a cashew! 

Why is this a concern? The idea of SPC is based on being able to tell the difference between typical  and atypical behavior. Ellipsoids are kinda easy to describe mathematically, so it is kinda easy to tell what is inside the ellipsoid and what is outside. I searched through my copy of the NIST Handbook of Mathematical Functions. The word cashew does not appear.

Second, it is clear that, even if the scatter of points for magenta in L*a*b* can be suitably approximated by an ellipsoid, the ellipsoid is not tilted toward the origin, as was the example of yellow in from the previous blog post on this topic.

Whether or not the first issue is a true matter of concern depends upon the amount of curvature present over the range of typical variation, and also the magnitude of other contributors to the variation. I will provide one example of the variation of magenta ink where the hook is not a problem. We shall see in this analysis that the odd direction of the tilt of magenta doesn't make a bit of a difference.

Magenta variation in newspaper data

I spoke before about a data set of test targets printed by 102 newspaper printers. One hundred printers, each printing 928 patches, with (in particular) two magenta solid patches on each one. The image below shows the variation in a*b* of all the solid magentas. 

You will note that it is kinda elliptical, with a decided tilt that is not toward neutral gray. Kinda what we would expect. Just visually, I can't see any sign of kidney-beaning. This, despite the fact that the variation is quite large, something like 10 ΔE from the two extremes. Then again, the other variations may just be doing a great job of hiding any curvature that is present. At any rate, I am going to make the bold assertion that we are not going to be making any beans and rice with this data set.

Below is 3D view of that same data. The height axis of the plot is L*, from 40 to 70. The (mostly) right to left axis is a*, from 30 to 60. The front to back axis is b* from -15 to 15. 

Note that the ellipsoid is not only tilted kinda away from neutral gray, but it is also tilted downward, which makes sense, I guess. As you make the ink richer in color (more pigment) you also make it darker.

A note about the image above. You should see an animation, with the set of points rotating around. If the animation isn't working, try a different browser, or try downloading the image and displaying it with some other app.  I found that Microsoft Office Manager and Paint did not display the animation. Microsoft Photos, Internet Explorer, and Windows Media Player do.

Once again, I don't have much of a problem saying that this data resembles an ellipsoid. At any rate, it looks a lot more like an ellipsoid than a box, which is the assumption that is made if we separately analyze ΔL*, Δa*, and  Δb*.

Previous blog posts

Here is the first post in a series of four blog posts about the futility of using ΔE for statistical process control. The first post ends with a link to the second in the series, and so on. 

I wrote a more recent blog post that looked at a common process management tool, the cumulative relative frequency plot (CRF) with ΔE data. Again, I gave some warnings about trying to make much out of the shape of the curve.

Then I wrote a post about the practice of applying SPC on ΔL*, Δa*, and  Δb*. My conclusion is that this is better in some cases, but there are some very reasonable distributions of color data where the method falls apart.