Starting with prior art, the Planck colormap is a diverging colormap, but it has discontinuities in its perceptual derivative, causing visible banding issues. Here is the output for it from Viscm.

And here is how it performs by the CVD metric I previously developed to evaluate the Turbo colormap (which performs poorly by it). Along with the cusps from the issues with the perceptual derivative, there is a clear imbalance in the metric around the center of the colormap for red and green color-vision deficiencies.

As a first attempt to improve upon this, I fit splines to the color values and applied various levels of smoothing. With small amounts of smoothing, this rounded off the perceptual discontinuities—fixing the worst of the issues—but left much room for improvement; applying more aggressive smoothing further improved the perceptual issues but also more significantly changed the colormap in a manner that my colleagues did not like aesthetically.

For my second attempt, I started by trying to replicate the Planck colormap in Viscm. In the process, I solicited feedback from my colleagues, resulting in tweaks such as moving the mid point from a neutral gray to a slightly warmer color temperature. While this resolved the perceptual uniformity issues, parts of the colormap were less than uniform for individuals with color vision deficiencies. I then iteratively made changes and ran the results through the CVD metric, until the results roughly matched between typical color vision and red and green color-vision deficiencies. Here is the output of Viscm for the new colormap.

And here is how it performs by my previously-developed metric.

While a significant improvement upon the status quo, the new colormap is not perfect, and I think other diverging colormaps such as Peter Kovesi’s CET-D07 colormap (cet_bjy in Python’s Colorcet package), which is both linear and diverging, or some of Fabio Crameri’s colormaps are better choices for most applications. However, the new colormap is probably good enough for CMB maps, where exact values do not need to be read off of the maps by eye, while keeping a similar aesthetic to existing publications.

A Python file that can be imported to use the colormap with Matplotlib or Healpy and the JSCM output from Viscm are available.

1. Y. Li et al., “CLASS Data Pipeline and Maps for 40 GHz Observations through 2022”, arXiv:2305.01045.  ↩

Since rainbow colormaps are best suited for quickly judging values, their most important property is that colors in non-adjacent sections of the colormap are not confused.⁴ To evaluate this quantitatively, I devised the following metric. For each color in the colormap, the perceptual distance in CAM02-UCS⁵ is calculated for every additional color in the colormap. The weighted average of the perceptual distances is then taken, with the squares of the color location distances in the colormap used as weights. For color vision deficiencies, the method of Machado et al. (2009) is used to adjust the colors before the perceptual distance is calculated, as I did for randomly generating color sets and as was done in the development of Cividis;⁶ a severity of 100 was used, indicating deuteranopia, protanopia, and tritanopia. Thus, similar colors in distant locations in the colormap are penalized.

We will start with rainbow colormaps for our evaluation of colormaps by this metric, first considering Jet, the new Turbo colormap, and Matplotlib’s existing Rainbow colormap, which also attempts to address some of Jet’s shortcomings. In the plot legends, the abbreviations “Norm,” “Deut,” “Prot,” and “Trit” are used for normal color vision, deuteranopia, protanopia, and tritanopia, respectively. Higher perceptual distance, ΔE, is better, as are smoother and more consistent discernibility lines.

The discernibility lines for Turbo and Rainbow are much smoother than those for Jet, since both mitigate Jet’s significant banding issues. Although Jet’s banding issues are generally considered problematic, I, as a colorblind individual, find the banding to sometimes be a redeeming quality, since it makes it easier for me to match part of an image to the colorbar or other parts of the image. For normal color vision, Turbo’s discernibility line is smooth and fairly flat, a significant improvement over Jet, and a minor improvement over Rainbow, although Turbo arguably looks better. However, the discernibility lines for various color vision deficiencies are not nearly as uniform, for either Turbo or Rainbow. This means that for colorblind individuals some parts of the colormaps are considerably more difficult to discern than others, making data plotted with them liable to misinterpretation. Thus, while Turbo and Rainbow improve upon some of Jet’s shortcomings, neither is colorblind-friendly.

Next, we will consider cyclic rainbow colormaps. The classic, and severely flawed, version is the HSV colormap, and the improved version is Sinebow; in regards to non-cyclic rainbow colormaps, these are analogous to Jet and Turbo, respectively. Twilight, a perceptually uniform cyclic colormap, is also considered.

In evaluating the metric for these colormaps, their cyclic nature was taken into consideration in the colormap location distance calculation. Sinebow’s discernibility lines are much smoother than HSV’s, but neither does well for color vision deficiencies. Twilight is much more consistent and colorblind-friendly, although at the expense of average discernibility.

Now, we will consider two perceptually uniform linear colormaps, Viridis, the Matplotlib default, and Cividis a derivative designed with color vision deficiencies in mind.

The “V” shape of the metric for these colormaps is expected, since for a linear colormap, the center is closest to the greatest number of other colors. Note that the discernibility of Cividis, which was optimized with color vision deficiencies in mind, is the most consistent between normal color vision and various color vision deficiencies, although Viridis is also okay in this regard, and both are considerably better than any of the rainbow colormaps previously presented.

Finally, diverging colormaps will be evaluated. Here, we consider Matplotlib’s Coolwarm colormap and Peter Kovesi’s Blue-Gray-Yellow colormap.

These show a “V” shape, similar to linear colormaps, although this is less pronounced in Coolwarm. The Blue-Gray-Yellow colormap is linearly increasing in lightness and perceptually uniform, so its discernibility profile is much closer to that of perceptually uniform linear colormaps.

In summary, while Turbo does ameliorate many of the issues with Jet, neither Turbo nor any of the other rainbow colormaps evaluated here are colorblind-friendly, at least per the metric evaluated. It is likely that it is not possible to construct a rainbow colormap with such a property, unlike for linear, diverging, and cyclic colormaps. The Jupyter notebook used to evaluate the colormaps and produce the plots is available.

Edit (2020-07-15): Replaced last plot (and updated Jupyter notebook), since the non-diverging BGY colormap had been accidentally used originally instead of the BJY colormap. All plots were also updated for improved readability.

1. It probably is possible to create a colorblind-friendly rainbow colormap for a particular type of color vision deficiency. However, creating such a colormap that simultaneously works for multiple types of color vision deficiencies as well as for normal color vision is what is likely impossible.  ↩

2. G. W. Meyer and D. P. Greenberg, “Color-defective vision and computer graphics displays,” in IEEE Computer Graphics and Applications, vol. 8, no. 5, pp. 28-40, Sept. 1988. doi:10.1109/38.7759  ↩

3. G. M. Machado, M. M. Oliveira, and L. A. F. Fernandes, “A Physiologically-based Model for Simulation of Color Vision Deficiency,” in IEEE Transactions on Visualization and Computer Graphics, vol. 15, no. 6, pp. 1291-1298, Nov.-Dec. 2009. doi:10.1109/TVCG.2009.113  ↩

4. When differences between adjacent colors are important, a perceptually uniform colormap should be used.  ↩

5. Luo M.R., Li C. (2013) CIECAM02 and Its Recent Developments. In: Fernandez-Maloigne C. (eds) Advanced Color Image Processing and Analysis. Springer, New York, NY. doi:10.1007/978-1-4419-6190-7_2  ↩

6. J. R. Nuñez, C. R. Anderton, and R. S. Renslow. “Optimizing colormaps with consideration for color vision deficiency to enable accurate interpretation of scientific data,” in PLoS ONE vol. 13, no. 7, pp. e0199239, Aug. 2018. doi:10.1371/journal.pone.0199239  ↩