## 4 Experiments
### Motion Segmentation on Hopkins155  
In this experiment, we implement motion segmentation to find the independent moving rigid objects by their trajectories. In theory, when a rigid object moves in a video, the points of its trajectories will be formulated as a low dimensional subspace. Thus one can find the moving objects by using subspace clustering technique on the tracked trajectories. From this point, we test our clustering methods on the motion benchmark dataset, Hopkins155, which contains three categories of 156 video sequences with each sequence having two or three moving objects, including indoor objects covered by checkerboard texture, outdoor objects like car, and articulated objects like people. The feature points of the objects are extracted and tracked in all the frames. The ground-truth segmentation is also provided in the dataset for comparison purpose. Some sample images of the dataset are provided in Figure 5. In our experiment, the parameters \(\lambda_{1}=800,\lambda_{2}=\) 10,000 for LLE-SSC and \(\lambda_{1}=200,\lambda_{2}\)= 10,000 for LLE-LRR.  
As there only two or three classes in the experiment, we only report the clustering results in maximal, mean, median and the standard deviation values, as shown in Table 5. On the whole, the performances of all clustering methods are quite good as all methods have clustering errors less than 6% in terms of mean. The reason is two fold. The first is that Hopkins155 is a nearly noise-free dataset. The second is that the number of classes is small in this clustering experiment. It is worth noting that our LLE-LRR method has the best result with the mean error of 1.43%.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Number of Classes**} & \multicolumn{6}{c}{**The Clustering Error (\%)**} \\ \cline{2-9}  & **SSC** & **LRR** & **NSLRR** & **SMR** & **LapLRR** & **LLE-SSC** & **LLE-LRR** \\ \hline
2 & 4.26 & 3.80 & 7.55 & 4.49 & **3.16** & 4.74 & 3.02 \\
3 & 11.45 & 7.64 & 8.75 & 9.77 & 7.03 & **7.23** & 8.34 \\
4 & 21.51 & 11.43 & 15.50 & 12.33 & 14.55 & **8.97** & 9.46 \\
5 & 27.35 & 18.375 & 14.74 & 16.78 & 14.50 & **12.32** & 13.17 \\
6 & 32.14 & 20.05 & 18.84 & 17.91 & 19.08 & **16.42** & 16.74 \\
7 & 34.55 & 20.13 & 19.61 & 19.74 & 20.83 & 20.00 & **17.86** \\
8 & 36.45 & 22.24 & 21.58 & 22.91 & 23.07 & 24.17 & **21.33** \\
9 & 38.17 & 23.76 & **23.68** & 26.07 & 26.36 & 26.40 & 23.84 \\
10 & 39.32 & 28.30 & 26.03 & 29.58 & 27.29 & 28.94 & **24.16** \\ mean & 27.24 & 17.72 & 16.88 & 17.76 & 17.30 & 16.47 & **15.49** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Clustering results on USPS dataset. The clustering errors for \(2,3,4,5,6,7,8,9,10\) classes and their mean error are shown here. The best results are in bold font.  
Figure 5: Some sample images from Hopkins 155 dataset. The cross marked points with different colors indicate the positions of the tracks of different motions at the current images.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **SSC** & **LRR** & **NSLRR** & **SMR** & **LapLRR** & **LLE-SSC** & **LLE-LRR** \\ \hline max & 42.34 & 41.18 & 35.06 & 49.67 & 46.43 & 33.89 & **31.12** \\ mean & 3.11 & 4.83 & 3.74 & 5.56 & 5.36 & 1.57 & **1.43** \\ med & **0.00** & 0.52 & **0.00** & 0.19 & 0.24 & 0.20 & 0.21 \\ std & 7.78 & 9.35 & 7.02 & 10.42 & 10.41 & 3.70 & **3.51** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Clustering results on Hopkins 155 dataset. The max value (max), mean value (mean), median value (med) and the standard deviation (std) of the errors are reported on the total 155 motion sequences. The best results are in bold font.