## Tuesday, July 11, 2017

### Munsell and the deltaE

I want to correct a common misconception. Well, by "common", I mean that I once had this misconception, and was embarrassingly corrected for being so foolhardy. I may even have blogged on this silly notion. People say that 1.0 DE is a just noticeable difference. That's not exactly the intent.

There are three concepts that are very closely related to the DE color difference:

1. The smallest change in color that we are able to perceive.

2. The acceptable difference in color in manufacturing.

3. The even spacing of colors intermediate between two other colors.

In this (soon to be) highly acclaimed series of blog posts, I will look at how each of these concepts intertwine when we talk about the difference between two colors. I will start with the third concept.

These two colors are just under two inches apart

What is a DE?

The term DE is a strange lexicographic combination. It started with the Greek letter, D (pronounced delta). This has been used by math and engineering folks to stand for difference, usually for a small difference. The second letter is the first letter of the German word Empfindung, which means sensation. So, this lexical chimera literally means difference in sensation. It is defined as a unit of measure of the difference between two colors.

The idea of DE as a measure of color difference dates back to the introduction of the CIELAB color space in 1976. But I get ahead of myself.

Munsell and perceptual uniformity

It really started with Albert Munsell and the Munsell Color System, which was developed in the early 1900's. Munsell provided a color atlas, which was a book with a zillion colors in it. By comparing your color swatch against those in the book, you could give a unique and unambiguous identifier to any color. This was a big leap past "I want my walls to be a lighter shade of apricot, but so so beigey". As you will recall from Munsellology 101, Munsell defined the lightness scale to have ten evenly-spaced steps.

I wrote another whole blog post on the idea that our perception of color is non-linear, and the need and difficulty of finding a way to measure how close one colors is to another. These blog posts just go on and on!

Munsell put a great deal of time and energy into making his color space uniform. Each step in color was the same size. Or put another way, for every set of three adjacent colors, the one in between is halfway between the two outer colors.

The image below is a dramatic reenactment of Albert Munsell putting together a uniform set of colors between gray and burnt orange. In the first step, he created one color between gray and burnt orange. Maybe he started with a 50-50 mixture of the gray and burnt orange pigments and then tweaked from there? When he got a color that he felt was halfway between, he probably jotted down the recipe for this midpoint color; how much of each of the pigments was required to make that color. Then, he stepped back and looked at it. He decided the steps were too big.

Next he tried putting two colors in between the gray and the burnt orange. The result is shown in the second row of the image below, with a ramp of four colors. He use the recipes for gray, burnt orange, and for the previous midpoint as a starting point. Maybe he used some math to interpolate the recipes? He certainly had to do some iterating, since not only does color 2 have to look like it's halfway between colors 1 and 3, but color 3 has to look halfway between colors 2 and 4.

Again he stepped back, and said "the steps are still too big".

I have taken this process to five and to six gradations from gray to burnt orange. Munsell won this contest. He got 12.

This work culminated in the publishing in 1913 of the Atlas of the Munsell Color System, which was a big book with a zillion color patches. If a designer on the West Coast and a printer on the coast of Lake Michigan both bought a color atlas, they could accurately communicate color over the phone by referencing the book.

The Munsell Color Atlas is still available today, albeit under a different name

Now for the boring history lesson

I put this section in to convince skeptics that I really and truly did my homework when I say that the size of a DE, be it the 1976 formula, or a newer formula, is based on the work of Albert Munsell.

In 1933, Munsell, Sloan, and Godlove published a formula that approximated a conversion from tristimulus Y values to Munsell neutral value scale. The Munsell in this trio was Alexander Munsell, not to be confused with bis father, Albert Munsell. I might add, the acronym of this formula is not to be confused with a flavor enhancer that has erroneously been blamed for the hypothetical Chinese restaurant syndrome.

The MSG formula and the MSG formula

The 1933 effort gave the waiting world a formula for the lightness axis, but not for the chroma and hue axes of the Munsell Color System. That work was left to Eliot Adams in 1943. He applied the Munsell-Sloan-Godlove formula to all axes.

In that same year (1943), Newhall, Nickerson, and Judd put a great deal of effort into cleaning up the Munsell system, introducing what amounted to a look up table to convert from Munsell units to tristimulus values. The table is called the Munsell Renotation Data. Their paper included an equation to compute the Y tristimulus value from the lightness component of the Munsell system, which is called value. This equation was a fifth order polynomial. Note that the input to the equation is the Munsell value, and the output is the tristimulus Y value.

Dorothy Nickerson later (1950) merged her own work (the fifth order polynomial) with the work of Adams (a color space) to form a color space which became known as the Adams-Nickerson color space, also known as ANLAB. Very popular at the time. MGM produced one of their big theatrical movies about it, but unfortunately, the era of big production musicals was rapidly waning before the movie took off.

Busby Berkeley's rendition of the ANLAB color space
failed because it was in black and white

The use of a fifth order polynomial allowed for a close fitting of the data, but it caused some problems if you tried to go the other way, that is, from tristimulus to Munsell. (Us math guys call this a trap door problem.) The problem was dealt with by an approximation to the polynomial given in a 1958 paper with the catchy title Cube-Root Color Coordinate System from Glasser, McKinney, Reilly, and Schnelle,  This paper proposed a set of formulas where cube roots are taken of the XYZ values. The cube root function is readily invertible. Coincidentally, I was born in that same year. I am also invertible.

Glasser's cube root formula

The cube-root color coordinate system was modified a bit by the CIE (Commission Internationale L'Eclairage in French, or International Commission on Illumination) when they met in London in 1975. The modification (due to Pauli) replaced the cube root with a straight line below about 1% reflectance. This became CIELAB, also known as L*a*b*. To finish connecting the dots between Munsell and color difference, the official definition of DE is based on CIELAB.

CIELAB L* formula

The original Munsell color space had only ten levels of lightness. CIELAB L* goes from 0 to 100. Somewhere along the way, someone decided that it would be more convenient for there to be 100 shades of gray.

Thus, we see a straight line connecting the Munsell equally spaced color space with the CIELAB color space. Perhaps not all that straight; the path goes directly through at least five papers with a total of eleven authors. I have not counted the number of slightly less direct authors and papers that contributed along the way.

But, the preferred color difference formula today isn't the original DEab (the more precise name for DE). As of 2010, the recommended formula is DE00. Let me provide a little known fact: The DE00 color difference in the L* and in the b* directions at the very center of color space is the same in DEab and DE00. Big shocker: DE00 was scaled so as to agree with DEab at the point {50, 0, 0}. Even DE00 traces back to Munsell.

Just in case you missed it, I blogged about the various color difference formulas before (including DE00). And in case you missed a more recent blog post, in a future blog post I will answer a reader's question about how big of a color difference might be allowable for printing.

For values of L* other than L*=50, the conversion to DE00 is not one-to-one. As a result, there somewhat less than 100 levels of lightness. I count 76 shades of gray.

So, long story short, 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. That's my story, and I'm gonna stick to it. At least until my next blog post on the subject.

1. Very nice story, love it. The evolution of color formulas.

2. I received a comment by email from my good buddy, Robin Myers:

I always enjoy your blogs and I was especially interested in reading your latest blog about ∆E and the Munsell system.

However, I must correct you on one thing. Although Albert Munsell published “A Color Notation” in 1905, at that time he did not produce a book with actual color chips. The first publication of his “Atlas of The Munsell Color System”, with painted chips, was not published until 1915. This publication had hundreds of color patches, but not thousands.

I have measured all 477 surviving patches in my copy (#115). Chart H is missing from my copy, some scurilous miscreant removed it, but from the ghost image of the cover page there were 77 patches on this chart, arranged in 11 columns of 7 patches each. That would bring the total patches originally published to 554.

Robin has a data file with measurements of his copy of the Munsell Color System Atlas onine: http://rmimaging.com/spectral_library/library_index.html You will need a copy of SpectraShop to read the file.

3. I received (via LinkedIn) a correction to this blog post from Michael Huda:

"Hi John, just for clarity, Richard Hunter was the first to develop the De calculation and that was in 1945. It was transferred to CIELab ColorSpace when it was ratified in 1976. Thanks for the interesting article."

I made a small correction to Michael's correction:

The DE color difference goes back just a bit further than that, Michael.

The earliest reference to the DE calculation that I can find is a paper by Deanne Judd of 1939: "Specification of color tolerances at the national bureau of standards". In this paper, he uses the Pythagorean distance formula (same formula as in the 1976 DE calculation) on the UCS color space to derive the "NBS unit of color difference".

I have not been able to find a copy of the Judd 1939 paper, but it is described by Richard Hunter in the paper "Photoelectric tristimulus colorimetry with three filters" from the National Bureau of Standards (1942). Hunter provides six different formulas for DE, all of which have some resemblance to the 1976 formula.

Judd also provided a description of the NBS formula in "Physiological Optics at the National Bureau of Standards" (1967, Applied Optics). In this paper, he traces the concept of the difference between two colors back to a 1909 paper by Nutting.

4. I love following this blog. Thank you, Mr Seymour.
-
Roger Breton

1. Thank you, Roger!

Drop me an email, and I will get you on my mailing list:
john@johnthemathguy.com