How big is a deltaE?

Every once in a while, someone in the audience calls on me to ask a question that I know the answer to. That just happened to me, and I am soooo happy!

This is the second blog post in a series about the actual meaning of color differences. I blogged previously about the origin of the DE, in particular, the DE₀₀. I came to the preposterous notion that "the size of a DE color difference is based on the Munsell Color System, which is all about uniform spacing of colors. 1.0 DE₀₀ is one of 76 perceptually equal steps between pure black and pure white."

Today, I want to correct that silly idea. Contrary to what certain bloggers have said previously, the color difference DE₀₀  is a unit of measure which is used for industrial color-difference evaluation. This color difference equation is officially defined in the document Improvement to Industrial Colour Difference Evaluation (CIE 142-2001). The summary of this document starts out with  the sentence "Recommended practice for industrial colour-difference evaluation is presented." (The italics are mine, added strictly to heighten the excitement.)

The question that sparked this series of blog posts

Here is the question that I got from my good buddy, Larry Goldberg. As you will see from his question, he is eloquent, piquant, and just a tad irreverent; three qualities I appreciate in a friend. He works for a little company called Beta Industries, where you can get microscopes and print measurement devices and a variety of other devices for the print industry.

Here is his question:

I'm looking for The Idiot's Guide to delta E, converting scientifically rigorous results to Foolproof Rule o'Thumbs.  Or the more linguistically acceptable Rules of Thumb. Or until the new, improved Fool is released.

Something in simple tabular form such as;

delta E 2000  |  Rule O'Thumb

======================

0                         Deadnuts!

1                         Just Noticeable Difference, depending on who's asking and how much they drank for lunch

2                         Winner Winner Chicken Dinner!

3                         Good commercial color match, Kwitcher belly-achin', looks good to me!

4                         It'll be perfect when you add a little more snap to the magenta.

5                         Whaddya expect on this crappy paper?

6                       TWICE as good as commercial color match, no?  No. The buyer needs another dinner and round of drinks.

7 - 10                Roses are red, violets are blue, the grass is green the sky is blue.  Run it, they'll be wrapping the fish in it tomorrow.

>10                     This is a lot better than when they printed black and white...

If you have a dissertation, or a link, or a suggestion, it would be greatly appreciated.

Boy, have I got a technical paper for you!

The Committee for Graphic Arts Technologies Standards (CGATS) has been feverishly working on a standard that ties numerical color difference to lexical color difference. The title of the standard is "Graphic technology — Printing Tolerance and Conformity Assessment". Those of us in the biz affectionately know it as TR 016. The technical report is free, by the way. Just click on the link in the sentence before the sentence before this one.

This technical report is all about making that critical decision about whether a printed product has conformed to tolerances for color. Yes, this is the stuff of which contracts are made. Come to think of it, citing this document in a contract could simplify an agreement between print buyer and printer. I wonder if the folks on the CGATS committee ever thought about that? I'll hafta mention it during the next meeting I attend.

TR 016 defines four levels of acceptance, with explanations for when these levels are appropriate. Each of the levels has an associated tolerance for color difference (in DE₀₀), and states that 95% of the production samples must have a color difference less than the number. In other words, most of the measurements must be within this tolerance.

Here are the levels:

    Level 1 - "the most color critical applications, e.g., proofing" - 2.0 DE₀₀

    Level 2 - "color critical applications, e.g., commercial printing" - 3.0 DE₀₀

    Level 3 - "utility process color printing" - 4.5 DE₀₀

    Level 4 - "pleasing color" - 6.0 DE₀₀

So, there you have it. The size of a DE color difference is all about tolerances in the industry. For print work, 2.0 DE₀₀ is considered pretty darn good, and 6.0 DE₀₀ is merely "pleasing". Within printing, it is expected that the ink will be kept to a higher tolerance than the print using that ink, and the printing of a proof must also be tighter than the printing of the final product. Kinda makes sense. The variation in the color of the stuff coming out of the print shop can't be any better than the variation of what goes in.

Other industries may have tighter or looser tolerances for color. Please add to the comments below if you know about color tolerances in other industries!

Pondering

Consider this: The tolerances for color are based on a scheme for equal steps of color, which is where the DE came from. Somehow it seems a bit odd to put those two together. But one big benefit of this scheme is that it is based on our perception of color. That's a good thing, since our perception is certainly not linear with reflectance. Another big benefit is that our perception of the size of a DE₀₀ is largely independent of the color that you are looking at. That is, we don't need different tolerances for different colors or directions of color change.

On the other hand, I would argue that our acceptance of a difference in color between two samples is not necessarily the same as our perception of the gradations of color, especially when those two samples are not side-by-side. We are much more discriminating when we see two bags of potato chips sitting next to each other on the shelf.

A color that is slightly off will sit on the shelf until expiration date

And when the colors are not side-by-side? I would argue (without much data to support this) that our brain is much more forgiving of color changes that are strictly changes in lightness or chroma than they our of changes in hue. I would also argue (again without a grain of evidence) that we tolerate differences much better if the whole image has that same sort of shift. And once again without anything to support this, I claim that the brain is much more forgiving of colors in busy images with lots of fine detail.

Comparison of a health food drink ad in a glossy magazine and on newsprint
(Images courtesy of JMG Design Services)

There is a committee in the CIE (TC8-16) that is currently working on trying to quantify Consistent Colour Appearance -- what is it that makes our brain accept the two images above as being "kinda the same", versus if those two images were shifted in hue? There are some bright and knowledgeable minds working on this committee. And then there is one dim-witted slacker who just sits around and writes self-important blog posts all day.

But on the third hand, industrial tolerances with DE₀₀ are amenable to measurement with existing technology. What good is a unit of measure if you can't find a ruler?


That's all for part 2 of this trilogy of blog posts. Stay tuned for part 3, where I revisit the phrase just noticeable difference, and admit that the first two parts of this series were nothing but lies!

How many colors are there - the definitive answer

Here is the quick summary: There are 346,005 discernible colors.

I published a rather popular blog post in October 2012 that posed the slightly whimsical question "How many color are in your rainbow?"  I looked at the question in a number of ways and came up with answers anywhere from 3 to 16,777,216. The intent of the post was to collect some interesting facts into one coherent and entertaining post.

But, I sort of sidestepped the implied question: How many discernible colors are there?  I intend in this post to answer the question a bit more scientifically. Well... quite a bit more scientifically. For this blog post, I used a Monte Carlo technique to determine the volume of CIELAB space, and then modified this volume according to DE₀₀ to account for the nonlinearity of CIELAB. And the number 346,005 plopped out.

General idea

I started by generating zillions of random spectra [1]. Now, I'm not gonna say that I generated every spectra possible. I'm sure I missed a few that were hiding down there in the shadows. But I did look at a whole bunch of them. Half a billion, to be exact.

I converted each spectra into L*a*b* values using D50 illumination and the 2° observer. The resulting L*a*b* values were tabulated into boxes in a three-dimensional array, with each box indicating whether the corresponding region in CIELAB space contained a viable L*a*b* value.  

Next, I counted up the number of boxes checked to establish the volume of CIELAB space. According to my experiment, the volume is just short of 2.2 million. This number fits in reasonably well with two papers cited by Gary Field in my addendum blog post:

Research on the number of colors issue usually starts with reference to the Dorothy Nickerson and Sidney Newhall paper of 1943 (JOSA, pp. 419-422). They conclude that there are about 7,500,000 surface colors at "supraliminal" viewing conditions, and 1,875,000 colors when viewing conditions approximate those used for color matching work.

Mike Pointer and Geoff Attridge concluded that there were about 2,280,000 discernible colors in their 1998 CR&A article (pp. 52-54). 

Thus, my number (2.176 million) corroborates the previous results from Nickerson and Newhall (1.875 million), and from Pointer and Attridge (2.280 million).

But we're not done yet. As we know, CIELAB is just not all that uniform. In particular, two saturated yellow colors might be 5 units apart (according to DE_ab) but might still be perceived by a human as just barely different. Thus, this figure is an overestimate of the number of colors that are actually discernible. Since I have all (or nearly all) the physically realizable colors in boxes, I can compute the volume of each box using DE₀₀. Adding the DE₀₀ volumes of each of the boxes will provide an estimate of the true number of colors, corrected for visual linearity.

Based on this correction, the number of discernible colors is 346,005. I won't attempt to name them in this blog post. That will come in a future post.

Now for some details on how the calculation was done...

Generating spectra

All the spectra were "physically realizable reluctance spectra", which is to say, the reflectance values were all between 0 and 100%. I created spectra from 380 nm to 730 nm, in 10 nm increments. All the spectra I generated were somewhat "smooth", in that they were piece-wise linear functions. I show one example below.

The spectra above is comprised of nine segments. I generated 125 million of these nine-segment spectra, along with the same number of spectra with eight segments, the same number with seven segments, and the same number with six. Thus, there were 500 million spectra en toto.

Initially, I used reflectance values that were uniformly distributed between 0 and 100%. This proved a bit slow to converge (slow to fill the area), since a lot of spectra were generated at the light end where our sensitivity to color difference is rather weak. For this final work, I used random numbers distributed according to the cube root distribution. 

Caveat - This Monte Carlo analysis necessarily will produce only a subset of all possible spectra. First, discontinuous spectra were left out. Second, the fact that "only" half a billion spectra were analyzed leaves open the possibility that some are missed. This would tend to cause my estimate to be a bit low.

I also tried generating purely random spectra, with no correlation between wavelengths. Initially this was slow to converge - perhaps that might have worked out in the long run if I would have just had the patience.

Tallying the number of unique colors

A three dimensional array was created, representing L* values from 0 to 100, a* values from -150 to + 150, and b* values from -150 to +150. All three dimensions were quantized in steps of 5, resulting in 21 X 61 X 61 boxes. Thus, there was a single cube, for example, in CIELAB space representing all colors in the range 20 < L* < 25, -40 < a* < -35, and 80 < b* < 85.

Each of the spectra were converted to CIELAB values using the D50 light source and the 2° observer. The CIELAB values were then converted to a position in the three dimensional array, and the location was marked to indicate that there was at least one viable CIELAB value within the box.

If anyone is interested, I can send you a list of the centers of all the boxes, representing all valid CIELAB colors. Send me an email at john@johnthemathguy.com. If anyone is really interested, I can provide a set of very colorful charts like the one below, that summarize all this data. If enough people are really interested, I will post those to my website for all to marvel at.

Viable a*b* values in the range 55 < L* < 60

(each square represents a 5 X 5 box in a*b*)

Caveat - This discretization causes a bit of an error. It will cause an over-estimation of the number of colors. Why? Let's say that a certain box is at the edge of color space, straddling the line between viable CIELAB values and silly-lab values. If zillions of spectra are tested, then this box will eventually get a tally, despite the fact that only half of its volume should have been counted.

Converting to a count of discernible colors

Now for the novel part, the conversion to DE₀₀. In the previous analysis, the tacit assumption was made that each of those 5 X 5 X 5 boxes had a volume of 125. To be completely correct, the volume of each cube is 125 ΔE_ab³, cubic delta E units. I guess maybe that's not an assumption, that is pretty much just geometry. The assumption comes in when this is interpreted as meaning that each box contains 125 discernible colors. Those who have subscribed to the Color Science Times Newspaper for the last 30 years know that this might not be exactly the case.

So, I computed the volume using a color difference formula that is closer to human visual perception, ΔE₀₀. Theoretically, we could just compute the volume of a box by determining the color difference from top to bottom, from side to side, and from other side to other side. These three numbers would be multiplied together to get the volume of discernible colors in that box. This is reasonable, but it falls just outside of the spec for this color difference. Due to nonlinearity, the warranty on ΔE₀₀ expires at 4. Beyond that point, it may not give reasonable results.

Just to make sure this didn't introduce an error, I divided the cube into eight cubes, each with sides of 2.5 ΔE_ab, and added these up. Now we are within the warranty.

346,005. I'm going to use this for all my computer passwords. Just to make sure I remember it.

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[1] I am talking here in the first person, like I actually generated all the spectra myself. I didn't really. I have better things to do. Like drink beer. I had my assistant Dell Studio generate the spectra. He didn't seem to mind, although he did seem to take his time about it.