What’s the big deal about Riemannian color space?

It’s time for a serious essay, for a change…

At the last ISCC Annual Meeting (Nov. 2011, San Jose), Mark Fairchild presented a paper called, “Is there really such a thing as color space? Foundation of uni-dimensional appearance spaces.” I quote from the abstract: “Color science is not devoid of examples of so-called color spaces that were actually descriptions of color perception one dimension at a time.” Mark’s examples ranged from the Munsell system (hue, value, chroma) to CIECAM02 (brightness, lightness, colorfulness, saturation, chroma, hue).

I believe Mark has begun the important exercise of deconstructing color science so as to reconstruct it in more fundamental terms without extraneous formalism. In a sense, we all must undertake such an exercise when asked to explain to a non-expert a complicated concept in color science. What is the essential purpose of the concept? What benefits does it confer, and conversely, what would happen if we abandoned the concept?

I was recently asked to explain to a non-expert audience the concept of a three-dimensional Riemannian color space, as used by two current authors. The authors simply wrote that the workings of tensor calculus made the concept “useful.” This hardly answered the question of purpose or benefit. To establish the historical connection and mathematical correctness, I looked at a retrospective work attempting to explain color-space geometry to theoretical physicists who were already comfortable with Riemannian spaces in other contexts:

“In 1666 Newton discovered that colors form a convex pencil in a linear vector space of three dimensions. In 1891 Helmholtz suggested a metric for that space; and by 1942 MacAdam had surveyed the experimental values of the metric. Meanwhile, Riemann, at the opening of his epoch-making address in 1854 on the foundations of differential geometry, had singled out the space of positions of objects and the space of colors as the only continuous manifolds of several dimensions in common experience. Later in that address he pointed out that a metric represents the effect of factors extraneous to the space itself---the result of what he called ‘forces’ acting upon it to bind it together. Riemann's suggestion has been brilliantly fulfilled in general relativity, where gravitation is the force that imparts a significant metric to space-time. In the space of colors, however, little has been done […] This paper outlines a theory of color space in which Riemann's ‘forces’ are essentially those of natural selection.” [1]

Although dazzling, this passage doesn’t get to the benefit or purpose. And certainly it doesn’t give an intuitive sense of Riemannian geometry per se.

Accordingly, I made a try at an explanation. When we imagine color in a three-dimensional space, we are looking for a consistent arithmetic of color differences expressed as distances in the space. It is not enough to be able to compute a distance between any two colors (as can be done for such formulas as CIE DE2000). Intuition from, say, a topographical map in two dimensions, suggests (i) that there should be a path between the two colors along which the distances add; (ii) that path should comprise the shortest distance between the colors; and (iii) between any two points along the path, the shortest distance should be the part of that same path that connects the two points. Spaces with such defined paths are called Riemannian. If the paths are all straight lines in some coordinate system (as in CIELAB) the space is Euclidean.

After basking in the afterglow of effort, I realized that I still hadn’t answered the original question. Why is the intuition of a contour map especially helpful in color space? How is that need met by a three-dimensional space with a Riemannian distance?  

Perhaps we should start more primitively, and define the utility of a three-dimensional color-order system without the additional encumbrance of a metric. Johann Lambert in his 1760 book, Photometria, gave a compelling use case (quoted by Rolf Kuehni): "Caroline wants to have a dress like Selinda's. She memorizes the color number from the pyramid and will be sure to have the same color. Should the color need to be darker or go more in the direction of another color, this will not pose a problem." [2].

So if you order your colors into a three-dimensional pyramid, you can find the color you want easily (without getting lost) by iterated change in the sensible directions. Seeing all the neighbors of a provisionally-chosen color gives guidance for the next iteration.

Now suddenly it is clear to me what Mark Fairchild’s uni-dimensional scales lack. They don’t allow you to see all of a color’s neighbors. But here’s a surprise: Neither does the Riemannian space, especially in the dehydrated metric-tensor form in which we usually see it. Truly, I can’t determine all of a color’s neighbors from that tensor. If I travel on a shortest path (geodesic) through color space, a neighbor I left long ago may be a neighbor I have right now, yet the locally characterized metric won’t explicitly reveal it. As new Riemannian color spaces emerge (e.g., from “Riemannizing” color-difference formulas) we should be mindful of that fact.

I think Johann Lambert and---a fortiori---Rolf Kuehni identified the purpose of three dimensions, Riemannian spaces, and---most especially---pictures that transcend local description. What every shade sorter knows, we theoreticians should re-learn.

[1] Weinberg JW. The geometry of colors. Gen Rel Grav 1976; 7: 135-169. [I omit references cited by the quoted passage on p. 135.]
[2] Kuehni RG. Color Space and its Divisions. 2003; Hoboken, NJ: Wiley, p. 55.

Michael H. Brill
Datacolor

Perils of the TLA

There’s many a slip twixt the intention and the abbreviation.

In the course of my duties as a subcommittee chair of the ASTM International, I recently shepherded a ballot to withdraw a standard on host computer communications because “the standard is not needed: manufacturers use their own SDKs and users can select an SDK from a menu.” One voter complained that he couldn’t find “SDK” in any dictionary but did find two definitions on the Internet: “Software Development Kit” and “Super Donkey Kong.” He presumed correctly that the former was intended, but he made his point: We should spell out our abbreviations at first occurrence, even if we think everyone should know them.

It’s easy to imagine amusing coincidences from such ambiguity, e.g., the ASA rating on a photographic medium used to record the luncheon meeting of the Acoustical Society of America. Or the National Science Foundation being caught at the bank with NSF (non-sufficient funds). Or the CIA spy who hangs out at the Culinary Institute of America. We could go on and it would look like fun.

Sometimes it isn’t fun. Remember the famous legal trademark dispute between the World Wrestling Federation and the World Wildlife Fund? The giant panda won the WWF fight, as WWE know.
Once I was scheduled to fly from Baltimore-Washington Airport to Miami for a large religious meeting. In preparation for that meeting (and to avoid sunburn), I wore a baseball hat that bore the large letters “NSA.” When I mistakenly ended up at the gate of Fort Meade, Maryland (right next to the airport), I got a strange look from the security guard. Fortunately, I still made my plane.

Undeclared abbreviations can cause confusion even in a narrow field. In connection with medical imagery, I have to inquire regularly whether the American College being called ACR is Radiology or Rheumatology. But my close encounter of the worst kind concerned two methods of solving differential equations, both called SDA: Strong Discontinuity Approach and Spectral Domain Approach. The name of the Russian mathematician Boris Galerkin is associated with both, which deepened the confusion.

“Super Donkey Kong” doesn’t sound so funny anymore.

I think we ought to avoid double meanings of abbreviations at least in the same field. To this aim, I offer the public service of pointing out a new spell-out of LCD that emerged at a recent solid-state-lighting committee meeting: the Light Code Designation (LCD) system for LEDs.  The term doesn’t seem to have reached Google yet, so there may be hope for liquid-crystal displays if we can head this one off at the pass.

By the way, in the title of this essay, TLA means “Three-letter abbreviation.” I haven’t even mentioned two-letter abbreviations (numerous PCs, nm as nanometers versus nautical miles) or four-letter abbreviations with multiple meanings (most notoriously the ISCC).

What are some amusing/confusing TLA’s in your field?

Michael H. Brill
Datacolor

[Note: When I began this essay, I incorrectly used “acronym” in place of “abbreviation.” An acronym is a very special abbreviation that spells a pronounceable word. For example, when NBS changed to NIST (rhymes with “mist”), it graduated to an acronym. NRC and NPL are still just abbreviations. MHB]