# ABANICCO: A New Color Space for Multi-Label Pixel Classification and Color Analysis
## Appendix A Finding Class Boundaries  
To find the approximate areas where colors were located, we used a color wheel complete with hue modifications, and studied its appearance in the reduced AB color space of CIELAB. The double histogram was simplified, and subsequently skeletonized, to find the color bases. In Figure 11, we can see the results of this method, obtaining the color bases by joining the center with the endpoints and with the achromatic and near-achromatic areas, represented as circles with center \((0,0)\) and radius that of the first and second sets of the branch points. The easiest way to obtain the boundaries between the located color bases was to find each angle bisector. To avoid overlapping with the generated regions, those chromogen coordinate points with angles smaller than \(5^{\circ}\) were merged, assuming they would belong to the same region: this is shown in step 3 of Figure 2. We also tested the primary state-of-the-art classification methods against the bisector option: to this end, we used MATLAB's classification learner. We created classes of 100 points laying over each of the proposed chromogen bases from the center to the limits of the reduced AB color space, with the bases extended to occupy the totality of the AB space: this was done to ensure that the classification methods would learn the regions entirely, instead of leaving the edges undefined. With these points, we trained the following state-of-the-art classifiers, using five cross-validation folds: Decision Trees (Fine, Medium, and Coarse); Discriminant Analysis; Naive Bayes; Support Vector Machines; Nearest Neighbor (Fine, Medium, Coarse, Cosine, Cubic, and Weighted); Kernel Approximations; Ensemble (Boosted Trees, Bagged Trees, Subspace Discriminant, Subspace KNN, and RUSBoosted Trees); and Neural Network (Narrow, Medium, Wide,Bilayered, and Trilayered). For details on the specifics of how each model was trained, refer to the Mathworks page on classification learner: [[https://www.mathworks.com/help/stats/classificationlearner-app.html](https://www.mathworks.com/help/stats/classificationlearner-app.html)]([https://www.mathworks.com/help/stats/classificationlearner-app.html](https://www.mathworks.com/help/stats/classificationlearner-app.html)) (accessed on 10 January 2023). Of all the classifiers, the best results were the Fine and Wide KNN, SVM, and Wide Neural Network classifiers, all of which provided a validation accuracy larger than 0.98, and classified almost all the given points correctly. For comparison, we focused on the best of these: the Weighted KNN (wKNN) and the Wide Neural Network (WNN). The wKNN had a validation accuracy of 99.2% in a training time of 2.3318 s, 10 neighbors, and Euclidean distance metric. The WNN obtained a validation accuracy of 99.6% after training for 1707.9 s with one fully connected layer and ReLu activation. Due to the simplicity of the needed geometry, it was expected that the best models would be the simplest ones, since all we needed to find were radial lines between color bases; however, the resulting regions did not match the expected ones and failed to provide acceptable pixel classification when applied to actual images: specifically, they mixed different colors in the same regions (as seen in comparative Figures 15 and 16). These approaches had good prediction accuracy, but were unsuccessful when creating the expected color regions, especially in the warmer areas of red-orange-pink, and in telling blues and purples apart. This comparative analysis proved that the best regions were those obtained using the geometrical approach of the angle bisectors. Figure 15: Segmentations obtained after using machine learning methods and the bisector approach to find the boundaries between the color bases. This evaluation was carried out in three rectangles depicting the concept of color wheel shadows and tones. Rectangle 1 shows the full-color wheel. Rectangle 3 above was obtained by adding 75% of black to represent shades.