The non-Riemannian nature of perceptual color space

Abstract

The scientific community generally agrees on the theory, introduced by Riemann and furthered by Helmholtz and Schrödinger, that perceived color space is not Euclidean but rather, a three-dimensional Riemannian space. We show that the principle of diminishing returns applies to human color perception. This means that large color differences cannot be derived by adding a series of small steps, and therefore, perceptual color space cannot be described by a Riemannian geometry. This finding is inconsistent with the current approaches to modeling perceptual color space. Therefore, the assumed shape of color space requires a paradigm shift. Consequences of this apply to color metrics that are currently used in image and video processing, color mapping, and the paint and textile industries. These metrics are valid only for small differences. Rethinking them outside of a Riemannian setting could provide a path to extending them to large differences. This finding further hints at the existence of a second-order Weber–Fechner law describing perceived differences.

Keywords: Riemann; cognition; color space; diminishing returns; metric.

Fig. 1.

Diminishing returns imply that large color differences appear less than the sum of their parts. This image is a purely figurative illustration of how this phenomenon could potentially occur, even though this specific geometry is not suggested in this paper. If an isoluminant plane through color space (Upper) would, for example, have the shape of a curved two-dimensional submanifold embedded in a 3D space (Lower), then the 3D Euclidean metric would produce the inequality of diminishing returns.

Fig. 2.

Geodesics (shortest paths computed with the shooting method) between the corners of the sRGB cube in CIE LAB with its non-Euclidean metric $\mathrm{\Delta }{E}_{2000}^{*}$ (79) illustrate what has long been known for perceptual color spaces in general, namely that in contrast to a Euclidean setting (e.g., $\mathrm{\Delta }{E}_{1976}^{*}$), the geodesics do not form straight lines. In this paper, we go one step further and show what CIE LAB does not capture: that in contrast to a Riemannian setting, their lengths do not even coincide with the distances between their endpoints.

Fig. 3.

Thurstonian distance perception (58) of a triad modeled with (solid) and without (dashed) additivity with an example scaling function $f(\mathrm{\Delta }\psi )=\sqrt{\mathrm{\Delta }\psi }$. If additivity is assumed and the gray-shaded Gaussians are appropriately centered, the difference between the means can be subtracted to give the perceived difference between the standard and either test (shown as the shaded green and orange triads). The difference between these means would be the perceived difference of differences, which can be plotted on a Gaussian centered at zero, as shown in Lower, to predict how often participants would be incorrect. However, if diminishing returns should exist, the means of the green and orange Gaussians should be scaled as demonstrated by the dashed distributions in the second row. The effect of the scaling function is an increased rate of predicted incorrect responses by more than an order of magnitude.

Fig. 4.

Two triads with the same difference in differences $\mathrm{\Delta }d=12.5$ in $L^{*}$ units. (Upper) $L^{*}=45,50,67.5$, respectively. (Bottom) $L^{*}=20,50,92.5$, respectively.

Fig. 5.

Degree of consensus is the absolute value of selecting a given test minus 50%. The negative slope of the lines indicates that participants’ responses tend toward chance with increasing average difference in the triad, despite the difference of differences remaining constant. A 95% CI around each line is shown in gray.

Fig. 6.

Accuracy of optimal models.

Fig. 7.

Optimal perceptual function g from Eq. 17 mapping $L^{*}$ values to perceived grayness, ψ. The shaded region indicates the middle 95% of all learned models across test sets and bootstrapped training sets.

Fig. 8.

Optimal difference scaling function f from Eq. 17 using the spline with four control points. The shaded region indicates the middle 95% of all learned models across test sets and bootstrapped training sets.

Fig. 9.

Optimal difference scaling functions using the hypothesized functions. The shaded region indicates the middle 95% of all learned models across test sets and bootstrapped training sets.

Fig. 10.

Photograph of Bernhard Riemann used to illustrate the impact of the non-Riemannianness for the computation of the intrinsic mean.

Fig. 11.

Representation of the triads used for simulations and experimental study. Each point represents five distinct experimental triads, each with a different $L^{*}$ value for the standard. The values for the tests are calculated using the x and y values.