# Ordered Subspace Clustering for Complex Non-Rigid Motion by 3D Reconstruction
## 5 Experiments and Analysis
### Expression Clustering on Dynamic 3D Face Expression Sequence  
Facial expressions are very important factors in communications. The BU-4 DFE dataset includes 3D face expression sequences for 101 persons, and each person has six expression sequences. Therefore, the expression clustering aims to cluster the face image sequences to 6 categories regardless of the identities. In this experiment, we selected 6 expression sequence from \(c\) subjects of the 101 persons to test. For the \(i\)-th expression sequence of a subject, we picked continuous 11 frames. Then, 2D features of \(c\) subjects for \(i\)-th expression can be represented as \(\mathbf{X}_{i}\), \(\mathbf{X}_{i}(i=1,2,\ldots,6)\in R^{22\times 76}\). The total face images test dataset is \(X=[\mathbf{X}_{1},...,\mathbf{X}_{6}]\). For one given \(c\) from the set \(c=[2,3,5,8,10,20,30]\), we randomly selected \(c\) subjects to repeat each experiment 30 times and then calculated the average clustering performance as final experimental result. The optimal parameter settings were \(\lambda=0.01,\lambda_{1}=0.02,\lambda_{2}=0.001,\lambda_{3}=0.1\).  
The expression clustering error rates on BU-4 DFE are shown in Table 2. It can be seen that the performance of proposed OSC-NRSfM method outperforms the compared methods, which demonstrates our proposed method can deal with complex sequences data such as facial expressions sequence. We visualized the difference of affinity matrices of \(Z\) when the number of subjects was five in Figure 2. The block-diagonal property of affinity matrices \(Z\) demonstrated the block-diagonal constraint to \(Z\) is functional. The affinity matrices provided by the proposed method are obviously with block-diagonal features and are more numerous within block weights. The clustering results of \(Z\) are shown in Figure 3. Here the same color means the same class it belongs to.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Subject**} & \multicolumn{5}{c}{**The Error Rate(\%)**} \\ \cline{2-7}  & **CNRMS** & **Proposed** & **LRR** & **OSC** & **QOSC** & **SSC** \\ \hline
2 & 14.22 & 6.16 & 34.60 & 0.68 & 0.33 & 47.50 \\
3 & 19.21 & 1.53 & 35.46 & 1.87 & 0.27 & 63.33 \\
5 & 24.94 & 0.96 & 27.37 & 3.92 & 2.13 & 66.67 \\
8 & 28.13 & 0.41 & 28.38 & 5.64 & 4.82 & 78.89 \\
10 & 26.24 & 0.28 & 26.48 & 8.45 & 7.16 & 78.78 \\
20 & 30.14 & 2.93 & 34.20 & 9.18 & 6.65 & 80.11 \\
30 & 34.73 & 5.63 & 36.14 & 10.10 & 7.62 & 82.28 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Face clustering error rate (%) on the BU-4 DFE dataset for 2,3,5,8,10,20,30 subject classes.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Subject**} & \multicolumn{5}{c}{**The Error Rate(\%)**} \\ \cline{2-7}  & **CNRMS** & **Proposed** & **LRR** & **OSC** & **QOSC** & **SSC** \\ \hline
2 & 31.14 & 15.28 & 49.47 & 28.66 & 19.12 & 52.32 \\
3 & 32.00 & 16.35 & 60.51 & 21.30 & 20.27 & 67.44 \\
5 & 45.34 & 20.16 & 66.87 & 26.48 & 21.47 & 77.33 \\
8 & 50.96 & 12.13 & 70.27 & 32.23 & 24.85 & 79.38 \\
10 & 55.86 & 8.76 & 72.30 & 32.41 & 23.08 & 79.44 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Expression clustering error (%) on BU-4 DFE.  
Figure 3: Clustering results of six expressions of five subjects on the BU-4 DFE dataset.  
Figure 2: The visualization results of affinity matrices \(Z\) for expression clustering on the BU-4 DFE dataset.