# A Mixed Method with Effective Color Reduction
## 4 Results and Discussion
### Results of Other Methods  
Tables 6-A9 show the MSE values obtained using other quantization methods. Four splitting methods have been applied: [PERSON]'s method (WU) [26], Octree (OC) [28], Median-cut (MC) [24] and the Variance-based method (VB) [25]. In addition, five clustering-based methods were also used: Neuquant (NQ) [33], K-means (KM), the Shuffled-frog leaping algorithm (SFLA) [44], the Artificial bee colony algorithm combined with ATCQ (ABC+ATCQ) [42] and the Firefly algorithm combined with ATCQ (ATCQ+FA) [43]. To complete the comparison, the WATCQ method described in [52] was considered, which applies ATCQ to the partition defined by the [PERSON]'s method. The results reported for each method correspond to 20 independent tests for each image and palette size. The results of the iterative methods (KM, ABC+ATCQ, ATCQ+FA and SFLA) correspond to iteration 25 and include the average MSE and the standard deviation. The best solution for each image appears in bold and the worst solution is underlined. To reduce the length of the article, the execution time of each method for each image and palette size is included as Supplementary Materials. However, the article does include a figure that allows the execution time of the methods to be compared (Figure 4). This figure represents the average execution time of each method for each palette size. Two subfigures have been included to show more clearly the time consumed by the fastest methods. The parameters used for the tests are the following: six food sources have been considered for ABC+ATCQ and six individuals (frogs, fireflies) for SFLA and ATCQ+FA. SFLA was executed considering 2 memeplexes, 4 improvement iterations of each memeplex and a sampling step equal to 10. The ATCQ algorithm combined with other algorithms was executed considering the variant that generates a 3-level tree, since it is faster. In addition, the \(\alpha\) parameter of ATCQ was set to the same values previously described to execute ITATCQ. Finally, the sampling factor of NQ was set to 1, since this is the value that generates the best results for this method. It should be noted that the results obtained using the ATCQ method are not included because the tests published in [46] have already shown that the BS+ATCQ method improves the images obtained by ATCQ. The results included in the tables indicate that BS+ITATCQ3 at iteration 5 obtains better MSE than WU, OC, MC, VB and NQ for all the images and palette sizes. In addition, Figure 5 shows the average percentage of error reduction obtained by BS+ITATCQ3 at iterations 5 and 25 compared to those methods. The summary that shows this figure confirms the superiority of BS+ITATCQ3. Figure 3: Average percentage of error reduction of BS+ITATCQ compared to BS, ITATCQ and BS+ATCQ. ITATCQa compares average errors and ITATCQm compares minimum errors (\(q\): palette size). When the results of BS+ITATCQ3 at iteration 25 are compared to those of the other iterative methods, the proposed solution obtains worse error in some cases. The best result of the comparison is obtained when ATCQ+FA is considered, since the proposed method obtains better error in all the cases. On the other hand, KM, SFLA and ABC+ATCQ are improved in 85, 77 and 31 cases out of 96, respectively. The percentage of error reduction obtained by BS+ITATCQ3 compared to these methods ranges in \([1.15\%,51.29\%]\), \([0.01\%,42.56\%]\) and \([0.04\%,20.87\%]\), respectively. On the contrary, when these three methods obtain better results than the proposed solution, the percentage of error reduction ranges in \([0.11\%,5.49\%]\) for KM, \([0.08\%,6.23\%]\) for SFLA and \([0.13\%,9.07\%]\) for ABC+ATCQ. It can be clearly observed that the upper limit of the intervals is always considerably greater for the cases improved by BS+ITATCQ3. When the results of BS+ITATCQ3 at iteration 25 are compared to those of WAT CQ, the first method obtains better results for 89 out of 96 cases.