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