# Illuminant Estimation Using Adaptive Neuro-Fuzzy Inference System
## 4 Experimental Results and Analysis
### Discussion and Further Analysis
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Cluster Numbers**} & \multicolumn{1}{c}{**Mean**} & \multicolumn{1}{c}{**Median**} & \multicolumn{1}{c}{**Trimean**} & \multicolumn{1}{c}{**Best 25\%**} & \multicolumn{1}{c}{**Worst 25\%**} \\ \hline
2 & 2 & **1.94** & **1.29** & **1.43** & **0.36** & **4.63** \\
2 & 3 & 2.08 & 1.36 & 1.50 & 0.36 & 5.07 \\
2 & 4 & 2.16 & 1.35 & 1.51 & 0.36 & 5.40 \\
2 & 6 & 2.50 & 1.43 & 1.63 & 0.38 & 6.51 \\
3 & 2 & 2.19 & 1.31 & 1.47 & 0.36 & 5.60 \\
3 & 4 & 2.39 & 1.48 & 1.65 & 0.37 & 6.05 \\
3 & 6 & 2.79 & 1.60 & 1.79 & 0.43 & 7.30 \\
4 & 2 & 2.26 & 1.34 & 1.52 & 0.36 & 5.78 \\
4 & 4 & 2.65 & 1.50 & 1.69 & 0.39 & 6.99 \\
4 & 6 & 3.08 & 1.69 & 1.96 & 0.42 & 8.16 \\
6 & 2 & 2.47 & 1.41 & 1.60 & 0.38 & 6.41 \\
6 & 4 & 3.04 & 1.70 & 1.97 & 0.44 & 8.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical metrics on Gehler-Shi and Cube+ dataset with different cluster numbers (the lower, the better). Figure 8: Boxplot and plot of minimum AEs’ distribution: **(a)** the distribution of the minimum AE across the 8 methods, and **(b)** the minimum AE distribution for each method versus minimum AE obtainable with chosen combination of methods. The value near the box of each unitary algorithm indicates the number of the corresponding algorithm with minimum AE across the 8 methods. It can be seen from the left subfigure that for most images there is at least one unitary algorithm to estimate its illuminant color resulting in AE no more than 3. This right subfigure shows the minimum AEs of the proposed method with the chosen combination outperforms most of the eight unitary algorithms, where No. of images 1-100 are the first 100 images in the Gehler-Shi dataset, and No. of images 101–200 the first 100 images in the Cube+ dataset. Due to limited length, we just show the results for 200 images. **Using sparse weight matrix**. In the proposed method, we use CDF weight vector \(\eta\) and IIF weight matrix \(\omega\) to measure what degree an image belongs to each cluster. We can define an integrated weight matrix \(W=[w_{ij}],i=1,2,\cdots,k_{1},j=1,2,\cdots,k_{2}\), where \(w_{ij}=\eta_{i}\cdot\omega_{ij}\) is the final weight for the \(ij\)-th ANFIS predictor. Since commonly there are many elements in \(\eta\) and \(\omega\) much less than 1 and very near to zero, i.e., the weights or possibilities for an image belonging to some clusters are very low, we can set all these elements' values to zero and then recalculate the values of other elements. For simplicity, we set a threshold \(\varepsilon\) for \(w_{ij}\). If \(w_{ij}<\varepsilon\), set \(w_{ij}=0\). \(W\) will become a sparse matrix \(\tilde{W}\). To assure the element summary of each row of \(\tilde{W}\) to be 1, recalculate \(w_{ij}\geq\varepsilon\), \(i=1,2,\cdots,k_{1}\), \(j=1,2,\cdots,k_{2}\), as below:  
\[\bar{w}_{ij}=\frac{w_{ij}}{\sum w_{ij^{*}}}, \tag{28}\]  
where \(\sum w_{ij^{*}}\) represents the summary of all elements greater than \(\varepsilon\) in the \(i\)-th row of \(W\). We performed some experiments using the sparse weight matrix by setting \(\varepsilon\) between \([0.01,0.25]\). We find the results change towards slightly improvement along with the increasing value of \(\varepsilon\), and the best performance corresponds to \(\varepsilon=0.25\), as Table 6 shows. By using the sparse weight matrix, we just need to weight a small number of outputs from ANFIS models, and the computation effort will reduce much more. **Comparison with one-step clustering**. We also conduct experiments to validate the effectiveness of our proposed two-step clustering strategy against the one-step clustering method. In so-called one-step clustering, we keep \(k_{1}\) or \(k_{2}\) to be 1, and just change the value of the other one. Table 7 lists some statistical metrics for the results under different \(k_{1}\) and \(k_{2}\) values.