Constructing Euclidean Color Spaces based on Color Difference Formulas

This paper shows a computational technique of how to construct a nearly isometric transformation from a color space with a non-Euclidean color difference formula into an Euclidean space. The resulting transformation is a combination of a one-dimensional color lookup table (CLUT) to transform lightness values and a two-dimensional CLUT to transform chroma and hue coordinates. As an example the CIEDE2000 formula and a new optimized color difference formula for CIECAM02 was used and a transformation into an Euclidean space was calculated. The mean isometric disagreement was far below 3%. Color tolerance ellipsoides were plotted for both investigated color difference formulas and transformed into the new Euclidean space to illustrate the performence of the method. A movie of how the CIEDE2000 system is embedded into an Euclidean space is made available at the website: http://munsell.cis.rit.edu/∼pmupci/ Introduction A perceptually uniform color space, in which Euclidean distances highly agree with perceptual color differences, is desired in many fields of imaging science, including color image compression, device gamut mapping and color engineering. In recent years much effort led to the standardization of new color spaces like CIELAB or color appearance spaces like CIECAM02 [1]. Unfortunately, the perceptual uniformity of these spaces is not sufficient for various applications so that new color difference formulas were developed and standardized, such as CMC [2], CIE94 [3] and CIEDE2000 [4, 5]. Visual experiments show that the CIECAM02 space is also not perceptually uniform and new color difference formulas can be applied to enhance the correlation to the visual data. These color difference formulae are only reasonable defined for relatively small color differences, 0-5 ∆E∗ ab [6]. To calculate larger distances the geodesics have to be calculated [7][8] that is complicated since the EulerLagrange differential equation must be solved. If an isometric transformation into an Euclidean space can be found, than the calculation becomes much simpler, because the geodesics in Euclidean spaces are straight lines and even large distances can be calculated by calculating the Euclidean metric. Here the effect of diminishing returns in color difference perception [9] (i.e. perceived large and medium color differences are smaller than the concatenation of threshold differences along the geodesic between the corresponding colors) is not taken into account. Unfortunately, an isometric (length preserving) transformation to an Euclidean space according to such a color difference formula is not possible. The reason is the geometrical property of the space: A necessary condition for the existence of such a isometric transformation is that the Gaussian curvature is zero throughout the space (Theorem Egregium) (see e.g. Wyszecki, Stiles [8], page 658). It can be proven that this condition is not satisfied for each of the formulas mentioned above. Since the Gaussian Curvature of the spaces for each of these color difference formulas is small it is possible to construct a transformation into an Euclidean space that is ”nearly” isometric, i.e. that the disagreement between distances calculated by the formulas and Euclidean distances in the new color space is not zero but small enough for most applications. Since the Gaussian curvature is a local property of a space, the disagreement between Euclidean distances in the new space and color differences in the underlying space is in general larger in regions where the Gaussian curvature is large. It should also be mentioned that the statistical variability of the visual data that is the basis of the color difference formulas is quite high and it is possible to construct transformations into Euclidean spaces for that the isometric disagreement is far below the noise level of the visual data. In this paper we show how to derive an Euclidean color space in high agreement with a color difference formula. Previous approaches used analytical methods to derive closed formulas: for the CMC formula [10], for the CIE94 formula [11, 12] and for the first quadrant of the CIEDE2000 formula [13]. For the whole domain of the complex CIEDE2000 formula an analytical transformation into an Euclidean space could not be found. We present in this article a computational method that leads to a simple color look-up table (CLUT), which transforms a non-Euclidean color space into an Euclidean space with minimal isometric disagreement [14]. As examples we use the CIEDE2000 formula on CIELAB and a new color difference formula defined using CIECAM02. New Color Difference Formula for CIECAM02 A color-tolerance dataset was formed that consisted of the RIT-DuPont 156 color-difference pairs [15] and the Qiao, et al. 44 hue-difference pairs [16]. CIECAM02 JCh coordinates were calculated for D65, 1500 lx, background reflectance Yb = 100, and an average surround. These 200 difference pairs all had the identical visual difference, equivalent to a ∆E∗ ab of unity for a near neutral centered at L∗ of 50. Nonlinear optimization was performed to derive a new formula where the coefficient of variation was minimized. The equation is listed below. For this dataset, this simple equation had equivalent performance to CIEDE2000. (These data 15th Color Imaging Conference Final Program and Proceedings 77 were also used to derive CIEDE2000.) ∆E02−OPT = [( ∆J kJSJ )2 + ( ∆C kCSC )2 + ( ∆H kH SH )2]1/2 (1) SJ = 0.5+(J̄/100)2 (2) SC = 1+0.02C̄ SH = 1+0.01C̄ kJ = kC = kH = 1 Computational Euclideanization Lightness values can be considered separately from chroma and hue values, because lightness differences are independent of chroma and hue for each of the color difference formulas mentioned in the introduction. For this reason the lightness values in the Euclidean space can be simply calculated by integrating the color difference formula along the lightness axis. Calculating the Lightness of the new Euclidean Space For the CIEDE2000 color difference formula the lightness L∗00 in the new euclidean color space is defined by the following integral : L00(L ∗) = ∫ L∗