## 6 Experiments
### Benchmarking of Point Cloud Registration on Real Data  
**Experimental setup:** We evaluated ICOS against other state-of-the-art methods over the \"bunny\" and the \"Armadillo\" point clouds from the Stanford 3D Scanning Repository [52], and the environmental setup was inspired by [24,25]. We first randomly downsampled the point cloud to 1000 points and resized it to be placed in a cube of \([-0.5,0.5]^{3}\) meters, which can be regarded as the initial point set \(\mathcal{P}=\{\mathbf{P}_{i}\}_{i=1}^{1000}\). Then, we generated a random rigid transformation: \((s,\boldsymbol{R},\boldsymbol{t})\), in which scale \(s\in[1,5)\), rotation \(\boldsymbol{R}\in SO(3)\), and translation \(||\boldsymbol{t}||\leq 3\) m, to transform point set \(\mathcal{P}\). We added random Gaussian noise with \(\sigma=0.01\) and \(\mu=0\) to the transformed point set, as the point set \(\mathcal{Q}=\{\mathbf{Q}_{i}\}_{i=1}^{1000}\). To create outliers in clutter, a portion of the points in \(\mathcal{Q}\) were substituted by random points inside a sphere of diameter \(s\sqrt{3}\) m according to the outlier ratio (from 0% up to 99%), as exemplified in Figure 4. All the experimental results were obtained over 50 Monte Carlo runs in the same environment. The estimation errors in all experiments were computed as:  
\[ \begin{split}& E_{s}=\big{|}s-s_{gt}\big{|},\\ & E_{\boldsymbol{t}}=\|\boldsymbol{\hat{t}}-t_{gt}\|,\end{split}  \tag{39}\]  
and \(E_{\boldsymbol{R}}\) is the same as (38), where subscripts \"\(gt\)\" indicate the ground truth. **Known-scale problems:** We adopted FGR, GORE, RANSAC, TEASER, RANSIC, and ICOS for the known-scale point cloud registration problems. As for RANSAC, we used [PERSON]'s minimal (3-point) method [34] to solve the rigid transformation, and the confidence was set to 0.995. There were two stop conditions for RANSAC: (i) RANSAC with at most 1000 iterations, named RANSAC (1000), and (ii) RANSAC with at most 1 min of runtime, named RANSAC (1 min). In terms of TEASER, we adopted the GNC heuristic version  
Figure 3: Benchmarking of rotation search over the synthetic data. (**a**) Results with \(N=100\) correspondences w.r.t. an increasing outlier ratio (0–99%). (**b**) Results with \(N=500\) correspondences w.r.t. an increasing outlier ratio (0–99%). (**c**) Results with \(N=1000\) correspondences w.r.t. an increasing outlier ratio (0–99%). as in [25], which uses GNC-TLS to solve the large robust SDP problem, rather than using convex relaxation, but differently: (i) we did not apply parallelism programming in order to ensure the fairness in the runtime evaluation for all the tested solvers; (ii) TEASER was completely implemented in MATLAB, not in C++, using the fast maximum clique solver [53]. **Unknown-scale problems:** For unknown-scale registration, we only compared ICOS with RANSAC and RANSIC because: (i) FGR and GORE were not designed for scale estimation originally; (ii) the scale estimator Adaptive Voting in TEASER was too slow for handling 1000 correspondences (requiring tens of minutes per run) and was also unable to tolerate over 90% outliers, thus no need to be compared. **Results:** The main benchmarking results are shown in Figure 4 (over the \"bunny\" point cloud) and Figure 5 (over the \"Armadillo\" point cloud). We can observe that: (i) in the known-scale registration problem, GORE, TEASER, RANSIC, and our ICOS were all robust against 99% outliers, but GORE was hardly accurate; (ii) RANSAC (1000) failed at 90%, while FGR and RANSAC (1 min) failed at over 95%, not as robust as ICOS; (iii) in the unknown-scale registration problems, ICOS could still address 99% outliers highly robustly; (iv) ICOS was the fastest solver when the outlier ratio was below 98% (note that though FGR seemed faster at over 96%, it already became no longer robust) and was constantly faster than RANSIC when the outlier ratio was high (e.g., 90-99%). Figure 4: Benchmarking of point cloud registration over the “bunny” point cloud [52].