Table 2. Rigid-body registration results for corresponding point sets with anisotropic noise present in one set of points and isotropic noise present in the other. (Experiment 1B).
| Without Isotropic Initialization | With Isotropic Initialization | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Rot. (deg.) | Alg. | Iter. | Runtime | RE | Inst. | Alg. | Iter. | Runtime | RE | Inst. |
| [0, 15] | Isotropic | 1.0 | 0.0001 | 0.349 | 0 | Isotropic | - | - | - | - |
| Estepar | 14.6 | 0.0100 | 0.332 | 0 | Estepar | 10.4 | 0.0080 | 0.332 | 0 | |
| Balach. | 32.9 | 0.0068 | 0.333 | 0 | Balach. | 14.8 | 0.0039 | 0.332 | 0 | |
| Proposed | 3.7 | 0.0011 | 0.332 | 0 | Proposed | 2.9 | 0.0012 | 0.332 | 0 | |
| [15, 45] | Isotropic | 1.0 | 0.0001 | 0.347 | 0 | Isotropic | 1 | 0.0001 | 0.347 | 0 |
| Estepar | 17.1 | 0.0115 | 0.338 | 0 | Estepar | 10.8 | 0.0076 | 0.338 | 0 | |
| Balach. | 37.5 | 0.0076 | 0.330 | 0 | Balach. | 14.6 | 0.0036 | 0.329 | 0 | |
| Proposed | 4.2 | 0.0012 | 0.330 | 0 | Proposed | 2.9 | 0.0011 | 0.330 | 0 | |
| [45, 90] | Isotropic | 1.0 | 0.0001 | 0.341 | 0 | Isotropic | 1 | 0.0001 | 0.341 | 0 |
| Estepar | 18.7 | 0.0125 | 0.352 | 0 | Estepar | 10.8 | 0.0077 | 0.352 | 0 | |
| Balach. | 40.2 | 0.0081 | 0.325 | 0 | Balach. | 14.4 | 0.0037 | 0.325 | 0 | |
| Proposed | 5.0 | 0.0013 | 0.325 | 0 | Proposed | 2.9 | 0.0011 | 0.325 | 0 | |
| [90, 150] | Isotropic | 1.0 | 0.0001 | 0.345 | 0 | Isotropic | 1 | 0.0001 | 0.345 | 0 |
| Estepar | 22.6 | 0.0149 | 0.366 | 2 | Estepar | 11.8 | 0.0082 | 0.365 | 0 | |
| Balach. | 41.9 | 0.0084 | 0.330 | 0 | Balach. | 14.6 | 0.0037 | 0.330 | 0 | |
| Proposed | 6.1 | 0.0015 | 0.330 | 0 | Proposed | 2.9 | 0.0011 | 0.330 | 0 | |
| [150, 180] | Isotropic | 1.0 | 0.0001 | 0.350 | 0 | Isotropic | 1 | 0.0001 | 0.350 | 0 |
| Estepar | 26.7 | 0.0173 | 0.373 | 60 | Estepar | 16.2 | 0.0109 | 0.385 | 10 | |
| Balach. | 44.5 | 0.0089 | 0.333 | 0 | Balach. | 14.5 | 0.0037 | 0.333 | 0 | |
| Proposed | 8.5 | 0.0018 | 0.333 | 0 | Proposed | 2.9 | 0.0011 | 0.333 | 0 | |
Results report the efficiency (number of iterations and runtime (seconds)), registration error (RE) (mm), and instability (% of trials) of the GTLS method proposed in this paper compared to the closed-form isotropic solution [26] and the prior GTLS methods of Estepar et al. [10] and Balachandran and Fitzpatrick [31]. The tests are binned according to the magnitude of initial misalignment in rotation (degrees), with all bins having a translational misalignment in the range [90, 100] mm; each bin represents average values measured over 1000 randomized trials.