## 6 Experiments
### Application 2: 3D Object Localization  
We tested ICOS over RGB-D scenes datasets [56] for tackling the 3D object localization (pose estimation) problems, as operated in [24; 25]. We first extracted the 3D point cloud of the target object (_cereal box, cap_, and _table_ in this case) from the scene according to the labels and imposed a random rigid transformation (with both a known scale and unknown scale), as well as random noise with \(\sigma=0.01\) and \(\mu=0\) on the target object. We utilized FPFH to build putative correspondences between the scene and the transformed object. We found that the correspondences constructed by FPFH contained over 80% outliers in all the scenes tested; even worse, the outlier ratios even exceeded 90% in most scenes. For each object-scene, we applied ICOS, RANSAC (1000), and RANSAC (1 min) to estimate the transformation (relative pose) of the object w.r.t. the scene and then reprojected the object back to the scene with the transformation estimated by each solver, respectively. The qualitative results are demonstrated in Figures 11 and 12, and the quantitative results are specified in Tables 2 and 3, in which the unit for \(E_{R}\) is degrees (\({}^{\circ}\)) and the unit for \(E_{t}\) is meters, computed as in (39). In each table, the best results are labeled in **bold** font. From the results, ICOS was able to estimate the object poses robustly and accurately even with over 95% outliers among the 548 correspondences (e.g., the last scene in Table 2); however, the two RANSAC solvers failed in nearly half of the cases. Thus, in addition to its superior performance in standard benchmarking, ICOS manifested strong practicality for real problems. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scene No.** & **ICOS** & **RANSAC (1000)** & **RANSAC (1 min)** \\ \hline _Scene-02_, \(s=1\) & \(E_{R}=\mathbf{0.196}\) & \(E_{R}=154.820\) & \(E_{R}=0.727\) \\ \(N=597\), 93.80\% & \(E_{t}=\mathbf{0.004}\) & \(E_{t}=1.120\) & \(E_{t}=0.013\) \\  & \(t=\mathbf{0.071}\) & \(t=1.559\) & \(t=34.308\) \\ \hline _Scene-05_, \(s=1\) & \(E_{R}=\mathbf{0.202}\) & \(E_{R}=61.278\) & \(E_{R}=0.991\) \\ \(N=606\), 90.92\% & \(E_{t}=\mathbf{0.003}\) & \(E_{t}=1.092\) & \(E_{t}=0.024\) \\  & \(t=\mathbf{0.057}\) & \(t=1.578\) & \(t=11.110\) \\ \hline _Scene-07_, \(s=1\) & \(E_{R}=\mathbf{0.188}\) & \(E_{R}=0.449\) & \(E_{R}=1.736\) \\ \(N=249\), 81.53\% & \(E_{t}=\mathbf{0.005}\) & \(E_{t}=0.013\) & \(E_{t}=0.056\) \\  & \(t=\mathbf{0.055}\) & \(t=0.579\) & \(t=0.582\) \\ \hline _Scene-09_, \(s=1\) & \(E_{R}=\mathbf{0.144}\) & \(E_{R}=3.319\) & \(E_{R}=0.546\) \\ \(N=284\), 90.16\% & \(E_{t}=\mathbf{0.003}\) & \(E_{t}=0.048\) & \(E_{t}=0.012\) \\  & \(t=\mathbf{0.066}\) & \(t=1.663\) & \(t=21.465\) \\ \hline _Scene-02_, \(s\in(1,5)\) & \(E_{R}=\mathbf{0.046}\) & \(E_{R}=63.040\) & \(E_{R}=0.257\) \\ \(N=618\), 94.82\% & \(E_{s}=\mathbf{0.001}\) & \(E_{t}=5.389\) & \(E_{t}=0.233\) \\  & \(E_{s}=\mathbf{0.001}\) & \(E_{s}=4.754\) & \(E_{s}=0.004\) \\  & \(t=\mathbf{0.039}\) & \(t=1.621\) & \(t=59.095\) \\ \hline _Scene-05_, \(s\in(1,5)\) & \(E_{R}=\mathbf{0.030}\) & \(E_{R}=152.878\) & \(E_{R}=48.347\) \\ \(N=555\), 95.14\% & \(E_{t}=\mathbf{0.003}\) & \(E_{t}=5.094\) & \(E_{t}=5.093\) \\  & \(E_{s}=\mathbf{0.001}\) & \(E_{s}=3.669\) & \(E_{s}=3.667\) \\  & \(t=\mathbf{0.185}\) & \(t=1.453\) & \(t=60.001\) \\ \hline _Scene-07_, \(s\in(1,5)\) & \(E_{R}=\mathbf{0.057}\) & \(E_{R}=123.266\) & \(E_{R}=122.522\) \\ \(N=185\), 92.43\% & \(E_{s}=\mathbf{0.001}\) & \(E_{t}=8.723\) & \(E_{t}=8.721\) \\  & \(t=\mathbf{0.066}\) & \(t=0.535\) & \(t=6.393\) \\ \hline _Scene-09_, \(s\in(1,5)\) & \(E_{R}=\mathbf{0.067}\) & \(E_{R}=3.633\) & \(E_{R}=143.444\) \\ \(N=548\), 95.26\% & \(E_{t}=\mathbf{0.005}\) & \(E_{t}=0.241\) & \(E_{t}=4.081\) \\  & \(E_{s}=\mathbf{0.002}\) & \(E_{s}=0.004\) & \(E_{s}=3.247\) \\  & \(t=\mathbf{0.142}\) & \(t=1.429\) & \(t=60.001\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantitative object localization results on cereal box.