# Illuminant Estimation Using Adaptive Neuro-Fuzzy Inference System
## 4 Experimental Results and Analysis
### Quantitative Results  
We followed the evaluation metrics used in most literature to compare each method's performance. The typical objective measures are based on the angular error (AE) [9; 18]. AE is the angle in degrees between the illumination's actual 3D-chromaticity \(e_{a}\) and its estimation \(e_{e}\), which is defined as:  
\[\gamma(e_{a},e_{e})=cos^{-1}(\frac{e_{a}\cdot e_{e}}{\|e_{a}\|\|e_{e}\|})\times \frac{180^{\circ}}{\pi}. \tag{27}\]  
Since the AE is not normally distributed, the median value is used to evaluate the statistical performance along with the trimean value: \(Trimean=(Q_{1}+2Q_{2}+Q_{3})/4\). The trimean is the weighted average of the first, second, and third quantiles \(Q_{1}\), \(Q_{2}\), and \(Q_{3}\), respectively. We empirically compare our proposed method against a large number of existing statistics based algorithms on the Gehler-Shi and Cube+ datasets. For each dataset, we give a summary of the performance statistics that are available and always includes the state-of the-art results known to us. For completeness, we also compare our method with the latest learning based methods for illuminant estimation. Tables 2 and 3 show the Mean, Median, Trimean, Best 25%, and Worst 25% of AE values obtained by each method on the Gehler-Shi dataset and the Cube+ dataset, respectively. The results of the proposed method in Tables 2 and 3 are from the two separate experiments, where the Gehler-Shi dataset and the Cube+ dataset were trained and tested separately. From Table 2, we can see that on the Gehler-Shi dataset our method outperforms the eight unitary algorithms, with Mean from 3.31 for LSR to 2.96 for ours, and Trimean from 2.87 for LSR to 2.14 for ours. The rate of performance improvement is near 10%. As shown in Table 2, in all statistics based methods listed, there is only one method, CCATI [40], that surpasses ours. Also, our method provides better results than using some learning based methods, comparative with ExemplarCC [41], CNN-based method [42], and Simple feature regression [35], except for Worst 25% is slightly big. In addition, it can be seen that both Best 25% and Worst 25% of our method are smaller than those of most statistics based methods listed. This indicates our method remains a better regression accuracy against the diversity of image features. Based on ANFIS, the estimation models and fuzzy weights developed by our method are robust and adaptive for a wide range of image features. From Table 3, we also have similar observations for the Cube+ dataset. All five statistics of our method are very near the best values from Color Beaver (Gray world) [48]. This further validates the effectiveness of our method, as the Cube+ dataset contains 1707 images, bigger than the Gehler-Shi dataset of 568 images. Thus, on the whole our method achieves competitive performances for using Gehler-shi and Cube+ datasets. It should be noticed that our method does not provide the best estimation, but does provide the performance close to the best. Although the proposed method can be further improved and more experiments should be conducted on other benchmark datasets, we can obtain an initial conclusion that some learning-based methods can outperform even the most carefully designed and tested combinations of statistical and fuzzy inference systems. Even