Supplementary Documentation: The Location of Optimal Object Colors with More Than Two Transitions

Published March 11, 2021 by Scott Allen Burns; last updated May 10, 2021

Change Log
3/11/2021 Original publication the Article on arXiv.
3/24/2021 Scope of original paper reduced and surplus material moved here.
3/25/2021 Added plots of high transition solutions for the convexified and truncated CMFs.
3/26/2021 Added comparison of reflectance curves and completed presentation.
3/27/2021 Added an extra figure for Section 7 of the Article and plots of “Object Color Space” color descriptors.
4/30/2021 Prefixed figure numbers with “SD-” to distinguish them from the original Article figure numbers.
5/4/2021 Moved most content back to the journal paper upon request of both peer reviewers.
5/10/21 Updated status to show accepted for publication. Fixed figure numbers.


This page contains supplementary documentation for the article “The Location of Optimal Object Colors with More Than Two Transitions,” which has been accepted for publication in Color Research and Application (to appear). Any reference to “the Article” on this page refers to that publication. Please contact the author for publication requests.

Please note that the content of this supplementary documentation page should not be considered peer reviewed, as it may be updated at any time. This page was last updated on the date shown at the top of the page.

Section 7. Other Illuminants

In Section 7 of the Article, five different highly chromatic illuminants were examined, having chromaticities matching Munsell colors 5R 5/14, 5Y 8/16, 5G 7/10, 5B 6/10, and 5P 4/12. A plot of the chromaticity diagram was presented, showing the high-transition regions associated with each illuminant, plus a sixth equal energy illuminant. Figure SD-1 below presents the same high-transition data shown in Figure 9 of the Article, but plotted on the object color solid (OCS), as viewed from the outside.

different illuminants

Figure SD-1. The effect of five highly chromatic illuminants on the regions of high-transition optimal colors, as view from the outside of the OCS.

The dashed lines are projections of the OCS onto the XZ and YZ planes. This figure was not included in the Article because the Article was getting too long and figure laden. It is shown here for completeness.

Logvinenko’s Object Color Space Color Descriptors

Section 5 of the Article compares the high-transition, LP-generated optimal color to the two-transition Schrödinger color existing along a common ray emanating from the 50% gray point in the center of the OCS. In order to compute the two-transition color, it was necessary to determine the \overline{\lambda} and \delta color descriptors, as developed by Alexander Logvinenko in his “object color space” paper.^{1} Each point on the surface of the OCS has corresponding, unique \overline{\lambda} and \delta color descriptors, and these descriptors can be computed from the spherical coordinates \theta and \varphi. These two color descriptors are then used by the MATLAB code developed by Masaoka and Berns to generate the two-transition color.^{2} Figures SD-2 and SD-3 present contour plots of the two color descriptors on the polar mappings of the spherical coordinates for the top and bottom halves of the OCS.

object color space coords

Figure SD-2. Object color space color descriptors for the top half of the OCS.

object color space coordinates

Figure SD-3. Object color space color descriptors for the bottom half of the OCS.


[1] Logvinenko AD. An object-color space. J Vision. 2009;9(11):1-23.

[2] Masaoka K, Berns RS. Computation of optimal metamers. Optics Letters. 2013;38(5):754-756.