## 6 Experiments
### Benchmarking of Rotation Search on Synthetic Data  
**Experimental setup:** We tested our solver ICOS for rotation search over synthetic datasets, where our environmental setup was similar to that in [23]. We first randomly generated \(N=\{100,500,1000\}\) vectors \(\mathcal{U}=\{\mathbf{u}_{i}\}_{i=1}^{N}\) with unit-norm, and rotated \(\mathcal{U}\) with a random rotation \(\boldsymbol{R}\in SO(3)\) to obtain a new vector set \(\mathcal{V}=\{\mathbf{v}_{i}\}_{i=1}^{N}\) in another coordinate frame. We then added random zero-mean Gaussian noise with \(\sigma=0.01\) and \(\mu=0\) to all vectors in \(\mathcal{V}\). After that, a portion of the vectors in \(\mathcal{V}\) (from 0% up to 99%) were substituted by randomly generated unit-norm vectors to generate outliers among the correspondences. All the experimental results were obtained over 50 Monte Carlo runs in the same environment.  
To quantitatively represent the errors of rotation, we adopted the geodesic errors such that:  
\[E_{\boldsymbol{R}}=\angle(\hat{\boldsymbol{R}},\boldsymbol{R}_{gt})\cdot \frac{180}{\pi}^{\circ}, \tag{38}\]  
where \"\(\angle\)\" is the geodesic error (21) [51] and the subscript \"\(gt\)\" denotes the ground truth value.  
For benchmarking, we tested our solver ICOS (Algorithm 1) against FGR [22], GORE [21], RANSAC, BnB [38], and RANSIC [49]. For FGR, we merely solved the rotation rather than the whole transformation using the Gauss-Newton method. For RANSAC, we adopted [PERSON]'s minimal method [34] to solve the rotation with the confidence set to 0.995, where we designed two stop conditions: (i) RANSAC with at most 100 iterations, named RANSAC (100), and (ii) RANSAC with at most 1000 iterations, named RANSAC (1000). When \(N=1000\), we excluded BnB as it took tens of minutes per run. QUASAR [23] was also not adopted due to its very long runtime.  
**Results:** Figure 3 displays the benchmarking results on both accuracy and runtime. It is not hard to observe that: (i) both FGR and RANSAC (100) failed at 90% and RANSAC (1000) broke at 98%, while RANSIC and our ICOS were robust against 95% with \(N=100\), 98% with \(N=500\), and 99% with \(N=1000\); (ii) ICOS was (one of) the most accurate  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm 1** & **Algorithm 3** & **Algorithm 4** \\ \hline \(X=2/3\,(N=100)\) & \(X=4\) & \(X=4\) \\ \(X=4\,(N=500)\) & \(A=4.5\sigma\) & \(A=4.5\sigma\) \\ \(X=5\,(N=1000)\) & \(B=5.0\sigma\) & \(B=5.0\sigma\) \\ \(L=2.5\sigma\) & \(C=6.0\sigma\) & \(C=6.0\sigma\) \\ \(G=4\sigma\) & \(D=10.5\sigma\) & \(D=10.5\sigma\) \\ \(F=10.5\sigma\) & \(maxItr_{1}=40000\) & \(maxItr_{3}=X\cdot 400\) \\ \(maxItr_{1}=40000\) & \(maxItr_{2}=400\) & \(maxItr_{4}=400\) \\ \(maxItr_{3}=X\cdot 400\) & \(maxItr_{3}=X\cdot 400\) & \\ \(maxItr_{4}=400\) & \(maxItr_{4}=400\) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter setup of ICOS for the rotation search and point cloud registration problems.  
solvers for all times since it constantly demonstrated the lowest rotation errors; (iii) with \(N=\{500,1000\}\), ICOS was faster than the other solvers excluding RANSIC when the outlier ratio was below 95%, and with \(N=100\), it had a similar speed as GORE when the outlier ratio was no greater than 60%; (iv) though ICOS was slightly slower than RANSIC at 90-98% outliers, it became more time efficient at 99%.