Riemannian Formulation of the CIEDE2000 Color Difference Formula

The CIELAB based CIEDE2000 colour difference formula to measure small to medium colour differences is the latest standard formula of today which incorporates different corrections for the non uniformity of CIELAB space. It also takes account of parametric factors. In this paper, we present a mathematical formulation of the CIEDE2000 by the line element to derive a Riemannian metric tensor in a color space. The coefficients of this metric give Just Noticeable Difference (JND) ellipsoids in three dimensions and ellipses in two dimensions. We also show how this metric can be transformed between various colour spaces by means of the Jacobian matrix. Finally, the CIEDE2000 JND ellipses are plotted into the xy chromaticity diagram and compared to the observed BFD-P colour matching ellipses by a comparing method described in Pant and Farup (CGIV2010). Introduction A color difference formula or a color difference metric, which measures the difference between two colors, is becoming a prime research topic in the modern colorimetry. The target is to find a good color difference formula, which can give a quantitative measure (∆E) of the perceived color difference correctly. It is also the requirement of many color applied fields such as image analysis, color reproduction, color image restoration and so on. MacAdam [2] described that small or medium color differences can be measured by the Just Noticeable Difference (JND) ellipses, which represent the human perception of threshold colour differences. Since then, many color difference formulae have been developed for measuring the color difference accurately. In 1976, the CIE recommended two color difference formulae, the CIELAB and CIELUV formulae, have become popular in colour industries, but, they fail to measure the visual perception of the color differences sufficiently, although, they are said to be uniform colour difference formulae [3, 4]. In 2001, the CIE recommended the CIEDE2000 formula based on the CIELAB to improve the correlation between measured and human observed color differences. In particular, the CIEDE2000 is the improved version of the CIELAB with specific weighting functions known as lightness (SL), chroma (SC) and hue (SH), parametric factors (kL,kC ,kH), and the rotation term RT to correct chroma and hue differences in the blue region. All these modifications are based on visual data obtained from four different experiments known as BFD-P [5], Leeds [6] RIT-DuPont [7] and Witt [8]. In other words, visual results from these four experiments were adjusted to a common scale by computing scaling factors for each data set and adopting the visual scale of BFD-P as a unit [9]. Luo et al. [10] have described in the detail about the CIEDE2000 formula as an excellent out performing formula, when measured against the aggregate data set, but still it has some issues related to its development [11]. Similarly, Sharma et al. [12] have shown mathematical discontinuity in the formula. Further, field test reports and performance studies on the CIEDE2000 have also not shown conclusively that the latest CIE formula performs better than previous existing formulae [13–15]. In such contexts, it would be useful from many aspects to study the CIEDE2000 color difference formula by the Riemannian approach. First, in this approach, we can map or transfer this formula into other color spaces preserving the subjective property of the formula. Second, the formula does not have its specific or corresponding color space, it is only the improved L,a∗,b∗ color space formulated in terms of lightness (L), chroma (C) and hue (H). So, it is interesting to know how well this advanced formula measures small colour differences in other color spaces. Third, Riemannian space is curved and such space is considered suitable for small to medium color difference measurement because many researchers have described that small color difference calculation using the Euclidean distance does not agree sufficiently with the perceptual color difference due to the curvilinear nature of the color space [16–20]. In this paper, the authors present a method to formulate the CIEDE2000 color difference formula in terms of the Riemannian metric and this metric is used to compute the JND ellipses. Here, the authors take the line element to calculate the color differences dE. To calculate line element, the CIEDE2000 color difference equation should be converted into the differential form. This gives us the Riemannian metric in a non-Euclidean color space. Again, to obtain the Riemannian metric in a new color space, we also need to transform color vectors from one color space to another. This is accomplished by the Jacobian transformation. To illustrate our method, the authors transformed the CIEDE2000 formula into the xyY color space. And, the JND ellipses are plotted into the xy chromaticity diagram. The input data to compute the JND ellipses for our method is BFD-P data sets [5]. BFD-P data sets were assessed by about 20 observers using a ratio method, and the chromaticity discrimination ellipses were calculated and plotted in the xy chromaticity diagram for each set [21]. A comparison has also been done between the computed JND ellipses of the CIEDE2000 formula and the original ellipses obtained from the BFD-P data set. The detailed description, for comparing a pair of ellipses, by calculating the ratio of the area of the intersection and the area of the union can be found in [1]. This method gives a single comparison value which takes account of variations in the size, the shape and the orientation simultaneously for a pair of ellipses. Therefore, this value is an indicator which tells us how well two ellipses match each other. Further, using MacAdam data, the authors also plot the JND ellipses of the formula in the xy chromaticity diagram to see simple visual comparison with the original data set. The authors see a good mathematical technique in this method to study the CIEDE2000 color difference formula. Formulation of the CIEDE2000 Metric Tensor and Color Space Transformation In this section, the authors will describe our method to compute metric tensor of the CIEDE2000 formula in the xyY space. Let us begin the process by defining the standard form of the formula [22].