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.  ↩

The development of the projection was motivated as a method to store spherical panoramic images, for potential use as a new format for the Pannellum panorama viewer. Since GPU texture limits are defined in terms of square textures, a square projection is ideal for such a use, and a quincuncial arrangement will in general have lower distortion than something like a cylindrical projection, since the sphere is cut in more places. A quincuncial arrangement is also seamlessly tileable, which can help to avoid sampling issues at the image edges. Additionally, to avoid wasting pixels due to varying levels of spatial detail, an equal-area projection is preferred, eliminating the Peirce quincuncial projection from contention, since it is a conformal projection.² For existing projections, this left the Collignon quincuncial projection and the Gringorten projection. However, the Gringorten projection does not have closed-form solutions, which prevents its use in a GPU shader. This left just the Collignon quincuncial projection, but it has significant angular distortion and cusps, which will increase visual artifacts in the projected panorama view and will likely decrease the effectiveness of lossy image compression, e.g., JPEG compression. Thus, the goal was to develop a new projection that had relatively low angular distortion like the Gringorten projection but had closed-form solutions, allowing for a GPU shader implementation.

After going through many, many pages of scratch paper, I developed the new projection described in the paper, which met the design goals. It has closed-form solutions and has comparable—although slightly higher—average angular distortion to that of the Gringorten projection. However, it has lower maximum angular distortion and has only almost-imperceptible cusps. As it does not have a large, arguably-unsightly cusp along the equator like the Gringorten projection, it is probably a better projection even for cartographic purposes, where the lack of closed-form solutions does not present an issue. A comparison of the new projection with the Collignon quincuncial projection and the Gringorten projection is shown below.

The paper’s supplementary information contains all the code necessary to reproduce the paper’s figures, including a Python implementation of the projection and a D3.js implementation. Before including the projection in Pannellum, a detailed comparison of the new projection to existing cube-based equal-area and near-equal-area projections needs to be completed.

1. Gringorten, Irving I. “A square equal-area map of the world.” Journal of Applied Meteorology and Climatology 11, no. 5 (1972): 763–767.  ↩

2. A projection cannot be simultaneously conformal (angle preserving) and equal area.  ↩

Updated maps are on the campworkcoeman.org maps page. The data is bundled in the Camp Workcoeman Map App. The web app has been updated, and an update to the Android version, which is fully-offline, is forthcoming.

1. Auto-label placement still has a long way to go… ↩

1. Due to Apple’s developer fees and requirement to use OS X for development, there’s no native iOS version; I might reconsider this if there’s enough demand.  ↩