Gamut mapping is an important process by which colors specified within one device’s color gamut are mapped to fit within or fill out the color gamut of a second device. Typically, gamut mapping involves adjusting the chroma and lightness of colors while holding hue constant because changes in hue are generally more objectionable [17]. Thus, it is important for the color space where gamut mapping occurs to accurately predict perceived hue. The degree to which a color space maps stimuli of the same perceived hue to the same hue angle represents its hue linearity. In a hue-linear color space, it is simple to adjust the chroma of a color without changing its perceived hue by adjusting the color along the line that passes through the achromatic origin (an iso-hue line). In a color space with poor hue linearity, changing chroma or lightness also changes the perceived hue although this crosstalk can be reduced with the use of look-up tables [7]. Thus, performing gamut mapping in a color space with poor hue linearity could lead to hue differences between input and output devices. Thus, hue linearity is an important feature of color spaces used for gamut mapping.

Although CIELAB is a commonly used uniform color space, significant hue non-linearity has been documented in CIELAB for purple-blue (PB) colors [1, 17]. This non-linearity—and the demand for a color space that better predicts perceived hue—has led to the development of color spaces with improved hue linearity such as IPT in 1998 [2] and Jzazbz in 2017 [18]. These color spaces consist of a matrix transform from CIE XYZ coordinates to LMS pseudo-cone space, a non-linear compression in cone space, and a matrix transform from LMS to an opponent coordinate system, where one dimension (I in IPT) corresponds to lightness and the other two dimensions correspond approximately to red versus green (P) and yellow versus blue (T), similar to CIELAB’s coordinate system. The hue of stimuli in IPT is quantified by its polar angle in the PT plane relative to the +P axis.

In general, these color spaces are derived by mathematically fitting transformations to experimental visual data. For example, IPT improved on the hue linearity of CIELAB and CIECAM97s by fitting the transformation from CIE XYZ to IPT to constant hue datasets from Hung and Berns [9] and Ebner and Fairchild [3] in addition to measurements of the Munsell color order system [10]. IPT has proven to be successful in subsequent experiments [19], and it has formed the basis for Dolby’s ICTCP color space [20].

Experimental results from Mizokami et al. [15] opened up a new path to the development of a hue-linear color space. The paper connected spectral properties to perceived hue. Specifically, the researchers found that Gaussian-shaped light spectra of varying bandwidth but the same peak wavelength were perceived by observers to have the same hue [15]. This result would suggest non-linear compensation in the neural coding of color to account for the fact that single-peak-wavelength Gaussian spectra would have different cone excitation ratios at different bandwidths [15]. For neural coding to be connected to Gaussian spectra, though, it would be required that Gaussian spectra serve as an effective representation of the stimuli that we encounter in natural scenes. Further work by this research group explored whether Gaussian spectra “could accurately approximate natural spectra with a small number of parameters” [14]. They found that Gaussian spectra performed similarly to linear models with the same number of parameters [14]. In conjunction, these results establish the plausibility of using features of Gaussian spectra to optimize a color space for hue linearity as opposed to fitting the transformations to visual data.

Work by Mirzaei and Funt [12, 13] related to categorizing object colors added more evidence to the efficacy of using the peak wavelength of Gaussian spectra as a hue predictor. Their approach involves finding a wraparound Gaussian spectrum that is metameric to a stimulus under a specified illumination and using the peak wavelength of the Gaussian spectrum as a direct hue descriptor [13]. (A wraparound Gaussian spectrum is where a function at 780 nm continues at 380 nm up to the wavelength complementary to the peak wavelength. This is one approach that allows the entire chromaticity diagram to be represented by Gaussian functions, including purple and magenta colors, which requires preferential stimulation of S and L cones versus M cones.) The researchers used the peak wavelength hue descriptor to train a genetic algorithm to optimize hue boundaries for Munsell colors [13]. They found that their Gaussian-based system worked better than CIECAM02 for this task. These results agreed with other qualitative measurements [13].

However, there are several issues with using peak wavelength as a direct descriptor of hue. Narrowband Gaussian spectra with peak wavelength at either end of the visible range (e.g., greater than 700 nm) have identical or very similar chromaticities even as the peak wavelength varies. This singularity would lead to ambiguity in the method and would cause visually identical stimuli to be mapped to different hue bins. Additionally, the Mirzaei & Funt method is tailored to being an illuminant-invariant descriptor of object colors and was not developed for use as a color space to perform gamut mapping. As such, the research did not include the standard quantitative assessment of hue linearity that could be directly compared with other color spaces: a measurement of the angular spread of constant hue loci [19].

An alternative approach explored here is to use Gaussian spectra to generate predicted constant hue loci, which can then be used to optimize a color space transform where points on a constant hue locus map to the same hue angle. This approach is similar to the derivation processes for IPT [2] and Jzazbz [18] although where the constant hue loci for those color spaces were the result of visual experiments, these constant hue loci are generated from Gaussian spectra of a single peak wavelength. The color space derived in this article following the above method is referred to as IGPGTG given that the form of the transform was chosen to match IPT’s definition. This allowed a direct comparison of the efficacy of the Gaussian-based hue loci for generating a hue-linear color space with the traditional method of hue loci based on visual data.

The use of a color space to test the Gaussian hue hypothesis posited by previous papers [13–15] addresses the shortcomings of previous attempts described above. Because Gaussian spectra are used as parameters in the optimization process as opposed to being used as direct hue descriptors, problematic singularities for long-wavelength narrowband spectra are avoided. Additionally, the structure of the color space allows for direct use in gamut mapping applications. This structure also requires less computation time than that of the Mirzaei & Funt method, which requires a Gaussian metamer to be calculated for each stimulus [13].

Two experiments were conducted to compare the hue linearity of IGPGTG with that of other established color spaces—CIELAB, CAM16-UCS, and IPT—and the Munsell color order system. The experimental results indicated that IGPGTG matched or exceeded the hue linearity of the comparison color spaces.

Existing visual data related to perceived hue were transformed into IGPGTG as an additional validation of the color space’s hue linearity. In this case, IGPGTG performed much more poorly than the comparison color spaces. The mixed results of the experiment and the visual data analysis indicate that a color space derived from Gaussian spectra is plausible but is unlikely to supplant established color spaces.

Hue linearity is one aspect of broader hue uniformity. Much research has also gone into measuring uniform hue scaling [4], which relates to the uniformity of differences between hue angles. Hue scaling uniformity is a measure of whether the difference in hue between colors with hue angle 0∘ and colors with hue angle 10∘ is the same as the difference in hue between those with hue angles 10∘ and 20∘. Hue-linearity uniformity, as we use it here, is a measure of whether all colors with hue angle 10∘ have the same perceived hue. For the purpose of this article, only hue linearity was evaluated given its special importance in gamut mapping applications as discussed above.

The structure of IPT served as the basis for IGPGTG. IPT coordinates are defined by their transformation from 1931 CIE XYZ coordinates with a D65 white point. XYZ values are first transformed to an LMS cone space using a 3-by-3 matrix transform. Non-linear compression is applied to each dimension in the LMS cone space before another 3-by-3 matrix transforms the LMS coordinates into IPT coordinates, where I represents the brightness/lightness dimension and P and T represent chromatic dimensions of perception (roughly, red versus green and yellow versus blue). For more information on the structure of IPT, see [2]. This structure was deemed appropriate for IGPGTG because of its invertibility, simplicity (while still providing the same number of degrees of freedom for optimization as used to derive other hue-linear color spaces), and the success of IPT as a hue-linear color space.

The first step in optimizing the transform to IGPGTG was to choose Gaussian-shaped spectra to be used for optimization. To fully and evenly sample the gamut of possible stimuli, the peak wavelength and the bandwidth of Gaussian spectra were optimized to fit an evenly spaced grid of coordinates in CIELAB (Figure 1). The simulated spectra were then grouped into sets of equal peak wavelength, which would be expected by our model to have the same perceived hue. Each peak-wavelength group contained at least 32 spectra to ensure full chroma sampling for each group.

Gaussian spectra can be used to generate metamers to any color except colors that correspond to the stimulation of L and S cones without M cone stimulation. (Such colors are typically purple or magenta.) To represent these colors for optimization, spectra consisting of two Gaussian functions were simulated, one with a central wavelength of 380 nm and the other with a central wavelength of 700 nm. The hue of such combined Gaussian spectra can be controlled by the ratio between the bandwidths of the two Gaussian functions. A spectrum’s chroma can be controlled by the overall bandwidth of the function. Since the two Gaussian functions overlap, the greater of the two functions at each wavelength was taken as the radiance of the spectrum at that wavelength. Double Gaussian spectra were simulated to fit an evenly spaced grid of coordinates in CIELAB (Fig. 1). The stimuli were then grouped based on the ratio between the 380 nm and 700 nm functions’ bandwidths with at least 32 spectra in each group. (The bandwidth ratios within each group were set equal.)

For each unique spectrum, the total radiance was adjusted to match nine luminance levels corresponding to Munsell values 1–9 for a D65 white point at 100 cd/m2. CIE XYZ coordinates were calculated for each spectrum using the 1931 Standard Observer with 1-nm sampling.

The transform from XYZ to IGPGTG was then non-linearly optimized in MATLAB to minimize the root-mean-square (RMS) hue angle difference from the mean hue angle for stimuli with the same peak wavelength (or the same bandwidth ratio for double Gaussian spectra) by using IPT as the starting point for optimization. A second optimization objective was to minimize the difference between the IG and I (from IPT) values for each stimulus. Using multiple luminance levels ensured that the optimized formula could handle a variety of luminances. However, this method also involved the assumption that Gaussian spectra of the same peak wavelength but different luminance levels have the same hue. The Bezold–Brücke hue shift effect suggests that this assumption may not hold true for nearly monochromatic spectra [6]. Additionally, to avoid the trivial solution to the optimization problem where all peak wavelengths are mapped to the same hue angle, a third optimization objective was added. The objective was for the transform to evenly space the hue angle of 40 Munsell colors with value 5 and chroma 5. The weight of this objective was minimized as much as possible without causing optimization to revert to the trivial solution. The optimized transform (Eqs. (1)–(5)) had mean and maximum hue angle standard deviations of 2.3∘ and 14.1∘, respectively, for the 31 sets of Gaussian spectra used in optimization (Figure 2). There was a 8.5% root-mean-square difference between IG and I values for the Gaussian optimization spectra. It should be noted that the hue angle spread for these points does not directly represent a failure of hue linearity. Rather, the spread present for certain hues in Fig. 2 merely represents the limits of the optimization process to map Gaussian spectra of equal peak wavelength to the same hue angle. Although it is plausible that including additional parameters in the XYZ to IGPGTG transform would improve the optimization, there was circumstantial evidence that the improvement would have been minimal as the optimization function did not use all of the degrees of freedom provided to it. Additionally, changing the form of the transform would reduce our ability to directly compare this method of fitting to the fitting process used for IPT.

Like IPT, IGPGTG assumes a D65 white point. For stimuli with a different white point, it is recommended to transform the stimuli’s XYZ values using CAT16, the chromatic adaptation transform embedded in CAM16, before converting to IGPGTG [11]. This is done by converting the XYZ values to the LMS cone space and then applying a von Kries type adaptation, where the LMS values of the stimuli are scaled to the LMS values of the test white point. The scaled LMS coordinates are then transformed back into XYZ, where they will have a D65 white point [11].

Once the stimuli have been transformed to a D65 white point, the optimized IGPGTG transform is

Although extant visual data suggested that IGPGTG has worse hue linearity than CAM16, CIELAB, and IPT, our visual experiment indicated that IGPGTG performs equally well as these color spaces on this metric. It should be noted that slightly different versions of the IGPGTG formula were used for the first psychophysical experiment and the comparison with extant visual data (which is the formula presented in Eqs. (1)–(5)). Originally, IGPGTG was optimized using lower-chroma Gaussian spectra to facilitate the optimization process. However, this led to irregularities in how high-chroma points were mapped at the interface between the single and double Gaussian spectra at high chromas. This issue was addressed through the process described in Section 2. Because the main improvements to IGPGTG were made beyond the gamut of colors used in the experiment, we would expect the final IGPGTG formula to present similar performance in the experiments. This expectation was supported by post hoc statistical analysis.

The conflict between experimental results and statistical analysis of extant visual data suggests that deriving a color space based on spectral properties alone is plausible. However, this color space fails to match the hue linearity of existing color spaces. Typically, uniform color spaces are fit to visual data alone. Although this approach has been successful, developing processes by which we can derive color spaces using first principles is attractive because such spaces would not be directly dependent on the accuracy of experimental data and limited to the gamut covered by such data. However, from a performance perspective, this work did not find compelling data to support the use of IGPGTG over the well-established IPT color space. Furthermore, although a direct correspondence between spectral properties of Gaussian stimuli and the perceived hue has not yet been proven, the results of this article add to the growing body of literature exploring potential connections [13–15].

A new color space, IGPGTG, was developed using the premise that Gaussian-shaped light spectra of the same peak wavelength have the same perceived hue regardless of bandwidth. IGPGTG was defined by a transform from CIE XYZ coordinates using the same structure as the well-established hue-linear IPT color space. The transform was non-linearly optimized in MATLAB to minimize the deviation in hue angle for Gaussian spectra of a single peak wavelength. The hue linearity of IGPGTG was then assessed using two methods. First, extant visual data on the perceived hue from the Munsell and NCS color order systems and the Hung and Berns dataset were transformed to IGPGTG. This data analysis indicated that IGPGTG has worse hue linearity than CAM16, CIELAB, and IPT. Second, two visual experiments were performed in which observers were asked to directly assess the hue linearity of these color spaces. In this case, IGPGTG matched or exceeded the performance of CAM16, CIELAB, IPT, and the Munsell system. These ambiguous results show that it is at least plausible to derive a hue-linear color space from first principles without the use of visual data in the derivation process.