How Perceptually Uniform Can a Hue Linear Color Space Be?

We propose a numerical method to determine a transformation of a color space into a hue linear color space with a maximum degree of perceptual uniformity. In a first step, a transformation of the initial color space into a nearly perceptually uniform space is computed using multigrid optimization. In a second step, a hue correction is applied to the resulting color space while preserving the perceptual uniformity as far as possible. The two-stage transformation can be stored as a single lookup table for convenient usage in gamut mapping applications. We evaluated our approach on the CIELAB color space using the CIEDE2000 color-difference formula as a measure of perceptual uniformity and the Hung and Berns data as a reference of constant perceived hue. Our experiments show a mean disagreement of 5.0% and a STRESS index of 9.43 between CIEDE2000 color differences and Euclidean distances in the resulting hue linear color space. Comparisons with the hue linear IPT color space illustrate the performance of our method. Introduction Psychophysical experiments show that observers favor hue-preserving gamut mapping algorithms. Maintaining the perceived hue is therefore an important objective in gamut mapping [1]. Hue linear color spaces, in which the lines of constant hue are straight lines, allow simple access to constant hue curves. Another desirable property for gamut mapping is perceptual uniformity of the color space, meaning that Euclidean distances agree with perceived distances. This is important for adjusting the degree of compression or for preserving contrast ratios. A gamut representation in the perceptually non-uniform CIELAB color space may lead to contrast ratio changes if highly chromatic gamut regions and regions close to the gray axis are treated similarly. In addition, CIELAB is not hue linear, which is especially evident in the blue region (see Fig. 1) [2]. If a gamut mapping is performed in CIELAB, a hue correction of this region is strongly recommended [3, 4]. Other color spaces are especially designed to be hue linear, such as the IPT color space [5], but they exhibit a lack of perceptual uniformity. Color order systems, such as the Munsell system, are also designed to be hue linear, but they cover rather low chroma regions. Unfortunately, there are many indicators that a perceptually uniform color space does not exist [6, 7, 8, 9]. To find a space with optimal perceptual uniformity, Urban et al. proposed a method to transform non-Euclidean into Euclidean color spaces with minimal isometric disagreement [10, 11]. The resulting color spaces show a high degree of perceptual uniformity, provided that the underlying color-difference formulas accurately reflect perceived color differences. Constant hue curves [3, 4] plotted in these approximately perceptually uniform color spaces reveal a significant lack of hue linearity (as shown for the LAB2000 space in Fig. 1). As a consequence, these color spaces are not recommended for gamut mapping — unless colors are mapped along curved trajectories, which requires much greater computational effort. As already mentioned, a hue linear color space with a maximum degree of perceptual uniformity would be beneficial for gamut mapping applications. This requires the creation of a new color space that combines the local property of perceptual uniformity with the global property of hue linearity. Instead of fitting the parameters of analytical functions to visual data, a numerical transformation based on lookup tables is used in this paper. To illustrate the basic concept of our method, we create a transformation of the CIELAB color space using the CIEDE2000 [12] color-difference formula as a measure of perceptual uniformity and the Hung and Berns data [3, 4] as a reference of constant perceived hue. Other color spaces such as the CIECAM02 [13] space and other color-difference formulas such as CIE94 [14], CMC [15] or improved versions of these formulas [16, 17] can be used equivalently. The Color Space Transformation Our initial color space is perceptually non-uniform and not hue linear. We assume that a color-difference formula is defined on this space, and that its color-difference estimations accurately reflect perceived color differences. The proposed method is a two-stage transformation of the initial color space. The first transformation maps the color space to a Euclidean space (Euclidean metric) with minimal isometric (lengthpreserving) disagreement with respect to the color-difference formula. The second transformation maps the resulting color space to a hue linear space while keeping the disagreement small. These transformations can be combined into a single color lookup table for usage in gamut mapping algorithms. In this paper, we use CIELAB as our initial color space, because it is well known and used in many industrial standards. The CIEDE2000 color-difference formula is used to estimate perceived color differences in CIELAB. The transformations can be summarized as follows: T00 : CIELAB Stage 1 7−→ LAB2000 T00,HL : LAB2000 Stage 2 7−→ LAB2000HL, (1) where LAB2000 [10] and LAB2000HL are Euclidean color spaces with minimal isometric disagreement with respect to CIEDE2000, and LAB2000HL is hue linear. The transformations can be turned into a single transformation by composition: T = T00,HL ◦T00. 18th Color Imaging Conference Final Program and Proceedings 97 Stage 1: Perceptual Uniformity The transformation of the CIELAB color space into a Euclidean space with respect to the CIEDE2000 color-difference formula has been described by Urban et al. [10]. We will therefore only sketch the method roughly. The color space transformation for CIELAB and CIEDE2000 is available online [18]. Because CIEDE2000 treats lightness differences independently of hue and chroma differences, the a∗b∗-plane is treated separately from L∗. The L∗ coordinate is transformed into the perceptually uniform lightness coordinate L∗ 00 by numerically integrating the CIEDE2000 formula along the lightness axis. The result is a onedimensional lookup table. The a∗b∗-plane is transformed using a two-dimensional lookup table. This table is calculated using multigrid optimization, starting from two regular grids whose vertices cover the a∗b∗-plane. These grids are designed such that each mesh of a grid encloses exactly one vertex of the other grid. The distance between any two neighboring vertices does not exceed five CIELAB units, the threshold below which CIEDE2000 correlates well with perceived differences [19]. For each mesh, the CIEDE2000 differences are calculated between its four vertices and the enclosed vertex of the other grid. The resulting four color differences are stored and remain unchanged during the subsequent multigrid optimization. In every iteration of the optimization, the vertices of a grid are shifted based on the meshes of the other grid. The objective is to decrease the disagreement between the stored CIEDE2000 differences and the corresponding Euclidean distances. In the first iteration, the vertices of the first grid are shifted based on the meshes of the second grid. In the second iteration, the vertices of the second grid are shifted based on the meshes of the first grid. The optimization continues with alternating grids until the change between subsequent iterations is sufficiently small. The two-dimensional lookup table is then created by mapping the vertices of either starting grid to the vertices of the corresponding optimized grid. Intermediate points are computed using bilinear interpolation. Figure 1 shows a starting grid in CIELAB and the corresponding grid in the LAB2000 space resulting from the multigrid optimization (grids in gray). The resulting color space transformation T00 consists of a onedimensional (lightness) and a two-dimensional lookup table: