Equalization and Brightness Mapping Modes of Color-to-Gray Projection Operators

In this article, the conversion of color RGB images to grayscale is covered by characterizing the mathematical operators used to project 3 color channels to a single one. Based on the fact that most operators assign each of the $256^3$ colors a single gray level, ranging from 0 to 255, they are clustering algorithms that distribute the color population into 256 clusters of increasing brightness. To visualize the way operators work the sizes of the clusters and the average brightness of each cluster are plotted. The equalization mode (EQ) introduced in this work focuses on cluster sizes, while the brightness mapping (BM) mode describes the CIE L* luminance distribution per cluster. Three classes of EQ modes and two classes of BM modes were found in linear operators, defining a 6-class taxonomy. The theoretical/methodological framework introduced was applied in a case study considering the equal-weights uniform operator, the NTSC standard operator, and an operator chosen as ideal to lighten the faces of black people to improve facial recognition in current biased classifiers. It was found that most current metrics used to assess the quality of color-to-gray conversions better assess one of the two BM mode classes, but the ideal operator chosen by a human team belongs to the other class. Therefore, this cautions against using these general metrics for specific purpose color-to-gray conversions. It should be noted that eventual applications of this framework to non-linear operators can give rise to new classes of EQ and BM modes. The main contribution of this article is to provide a tool to better understand color to gray converters in general, even those based on machine learning, within the current trend of better explainability of models.