Investigations on the Intrinsic Properties of the Color Domain II*

Riemannian color difference metric for spatial sinusoidal color variations

Several studies report on the sensitivity of human vision to static spatial sinusoidal achromatic and chromatic contrast variations. However, a Riemannian color difference metric, which includes the spatial and colorimetric properties of sinusoidal gratings, is lacking. Such a metric is important for various applications. Here we report on the development of a new Riemannian metric, for the prediction of detection ellipsoids in color space, for spatial sinusoidal gratings as a function of the grating's size, spatial frequency, luminance and chromaticity. The metric is based on measurements and models of achromatic and isoluminous chromatic contrast sensitivity functions available in literature, and the Riemannian metric for split fields which we reported earlier. We find adequate agreement with various data sets of experimental achromatic and isoluminous chromatic contrast sensitivity functions and with experimentally determined threshold ellipses of isoluminous chromatic Gabor gratings.

Color gamut volume and the maximum number of mutually discernible colors based on a Riemannian metric

For the calculation of the color gamut volume and the maximum number of mutually discernible colors, an algorithm based on a Riemannian metric and the densest packing of spheres is proposed. With this algorithm, the color gamut volume was calculated for the conditions of experiments reported in literature. Good agreement was found with the experimental findings of the color gamut volume as a function of the peak luminance. Using the new algorithm, the color gamut volume and the maximum number of mutually discernible colors was calculated for various sets of primary colors corresponding to display standards and various dynamic ranges. Comparisons were made with state-of-the-art methods which are based on the Euclidean metric in approximately uniform color spaces and a simple cubic lattice. It was found that the state-of-the-art methods underestimate the maximum number of mutually discernible colors. However, the relative differences decrease as the primary colors are more saturated. Based on the new algorithm the maximum number of mutually discernible colors was calculated for a range of peak retinal illuminance levels and various sets of primary colors. We found that, for a given set of primary colors, the maximum number of mutually discernible colors is proportional to the logarithm of the ratio of the peak retinal illuminance level and a fitting parameter.

Line element for the perceptual color space

It is generally accepted that the perceptual color space is not Euclidean. A new line element for a 3-dimensional Riemannian color space was developed. This line element is based on the Friele line elements and psychophysical color discrimination models, and comprises both the first and second stage of color vision. The line element is expressed in a contrast space based on the MacLeod-Boynton chromaticities. New equations for the contrast thresholds along the cardinal axes and new metric tensor elements were determined. Visual adaptation effects were incorporated into the model. Color discrimination threshold ellipsoids were calculated with the new line element. Adequate agreement with experimental threshold ellipsoids reported in literature was demonstrated. From a comparison with other color difference metrics a better overall predictability of threshold ellipsoids was found with the new line element.

The non-Riemannian nature of perceptual color space

For over 100 y, the scientific community has adhered to a paradigm, introduced by Riemann and furthered by Helmholtz and Schrodinger, where perceptual color space is a three-dimensional Riemannian space. This implies that the distance between two colors is the length of the shortest path that connects them. We show that a Riemannian metric overestimates the perception of large color differences because large color differences are perceived as less than the sum of small differences. This effect, called diminishing returns, cannot exist in a Riemannian geometry. Consequently, we need to adapt how we model color differences, as the current standard, ΔE, recognized by the International Commission for Weights and Measures, does not account for diminishing returns in color difference perception.

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.

Categorical colour geometry

Ordinary language users group colours into categories that they refer to by a name e.g. pale green. Data on the colour categories of English speakers was collected using online crowd sourcing - 1,000 subjects produced 20,000 unconstrained names for 600 colour stimuli. From this data, using the framework of Information Geometry, a Riemannian metric was computed throughout the RGB cube. This is the first colour metric to have been computed from colour categorization data. In this categorical metric the distance between two close colours is determined by the difference in the distribution of names that the subject population applied to them. This contrasts with previous colour metrics which have been driven by stimulus discriminability, or acceptability of a colour match. The categorical metric is analysed and shown to be clearly different from discriminability-based metrics. Natural units of categorical length, area and volume are derived. These allow a count to be made of the number of categorically-distinct regions of categorically-similar colours that fit within colour space. Our analysis estimates that 27 such regions fit within the RGB cube, which agrees well with a previous estimate of 30 colours that can be identified by name by untrained subjects.

Novel Perceptually Uniform Chromatic Space

Chromatically perceptive observers are endowed with a sense of similarity between colors. For example, two shades of green that are only slightly discriminable are perceived as similar, whereas other pairs of colors, for example, blue and yellow, typically elicit markedly different sensations. The notion of similarity need not be shared by different observers. Dichromat and trichromat subjects perceive colors differently, and two dichromats (or two trichromats, for that matter) may judge chromatic differences inconsistently. Moreover, there is ample evidence that different animal species sense colors diversely. To capture the subjective metric of color perception, here we construct a notion of distance in color space based on the physiology of the retina, and is thereby individually tailored for different observers. By applying the Fisher metric to an analytical model of color representation, we construct a notion of distance that reproduces behavioral experiments of classical discrimination tasks. We then derive a coordinate transformation that defines a new chromatic space in which the Euclidean distance between any two colors is equal to the perceptual distance, as seen by one individual subject, endowed with an arbitrary number of color-sensitive photoreceptors, each with arbitrary absorption probability curves and appearing in arbitrary proportions.

Hyperbolic geometry for colour metrics

It is well established from both colour difference and colour order perpectives that the colour space cannot be Euclidean. In spite of this, most colour spaces still in use today are Euclidean, and the best Euclidean colour metrics are performing comparably to state-of-the-art non-Euclidean metrics. In this paper, it is shown that a transformation from Euclidean to hyperbolic geometry (i.e., constant negative curvature) for the chromatic plane can significantly improve the performance of Euclidean colour metrics to the point where they are statistically significantly better than state-of-the-art non-Euclidean metrics on standard data sets. The resulting hyperbolic geometry nicely models both qualitatively and quantitatively the hue super-importance phenomenon observed in colour order systems.