Table 1.

Registered translations ${\widehat{s}}_x$, ${\widehat{s}}_y$, and ${\widehat{s}}_z$ and rotations ${\widehat{\alpha }}_{\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}}$, ${\widehat{\alpha }}_{\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}}$, and ${\widehat{\alpha }}_{\mathrm{y}\mathrm{a}\mathrm{w}}$ from the investigated sample implementations of the ICP algorithm for the tested phantoms plane, 3plane, bowl, and torso and treatment table shifts s _d in different directions d = {x, y, z}. Translations with an absolute translational registration error |e _d ^trans| > e _d ^trans _max = 1 mm and rotations with an absolute registration error e _r ^rot > e _d ^rot _max = 0.5° are marked boldface.

Phantom	Shift		Registration
			Translation/mm			Rotation/°
	s d/mm	d	${\widehat{s}}_x$	${\widehat{s}}_y$	${\widehat{s}}_z$	${\widehat{\alpha }}_{\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{l}}$	${\widehat{\alpha }}_{\mathrm{p}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}}$	${\widehat{\alpha }}_{\mathrm{y}\mathrm{a}\mathrm{w}}$
Wilm algorithm
Plane	0.5	z	0.4282	0.0087	0.5482	−0.0001	−0.0012	0.0000
	1.0	z	1.4503	0.4656	1.0627	0.0000	−0.0123	0.0000
	10.0	z	0.4760	0.8346	10.0927	0.0002	−0.0022	−0.0005
	20.0	z	0.0827	1.1255	20.0975	−0.0001	0.0213	−0.0007
3plane	0.5	z	−0.0100	0.1440	0.5325	−0.0000	0.0001	−0.0003
	1.0	z	−0.0395	0.1362	1.0882	0.0001	0.0004	0.0001
	10.0	z	−0.1475	0.1217	10.0550	0.0003	0.0001	−0.0003
	20.0	z	−0.0665	−0.0903	20.9796	0.0005	0.0012	−0.0006
	10.0	y	−0.1237	9.9743	0.0157	0.0001	0.0004	0.0001
	−10.0	y	−0.1991	−9.6955	0.1602	0.0002	−0.0001	0.0004
	10.0	x	9.8886	0.3100	0.1109	0.0013	−0.0004	0.0001
Bowl	0.5	z	0.0749	0.0702	0.4491	−0.0006	0.0013	−0.0000
	1.0	z	0.0596	0.1477	0.9348	−0.0010	−0.0018	−0.0011
	10.0	z	−0.0084	0.0054	10.1927	0.0061	0.0121	0.0004
	20.0	z	0.3618	0.0843	20.0056	0.0005	0.0382	−0.0001
	10.0	y	−0.1668	9.9264	0.1325	0.0035	−0.0678	−0.0033
	−10.0	y	0.0885	−9.7892	−0.3034	0.0004	0.0246	−0.0006
	10.0	x	9.9217	−0.0522	0.0799	−0.0014	−0.0400	−0.0024
	−10.0	x	−9.8553	0.1800	−0.0533	−0.0005	0.0854	−0.0005
Torso	0.5	z	−0.0239	−0.0895	0.5171	0.0033	1.5783	−1.5785
	1.0	z	−0.0986	−0.1490	1.0508	0.0022	1.6005	−1.6007
	10.0	z	0.0303	0.1723	9.9136	0.0090	1.1490	−1.1480
	20.0	z	0.0995	0.1012	19.8599	0.0017	0.8377	−0.8362
	10.0	y	0.1819	9.9194	−0.1830	0.0009	0.9571	0.9572
	−10.0	y	−0.1553	−9.5344	0.0316	0.0020	−1.4725	1.4724
	10.0	x	10.1230	−0.0174	−0.0374	0.0047	1.5300	−1.5299
	−10.0	x	−9.8724	−0.0488	−0.0401	0.0034	−1.6261	1.6263
Kroon algorithm
Plane	0.5	z	−0.4031	−0.1523	0.5499	−0.0001	−0.0001	−0.000
	1.0	z	−0.9254	−0.2737	1.0675	0.0002	0.0005	−0.0001
	10.0	z	1.3403	1.2951	10.0872	0.0002	−0.0015	−0.0004
	20.0	z	−3.7160	0.7362	20.0891	−0.0001	0.0007	−0.0008
3plane	0.5	z	0.0023	−0.1280	0.3746	−0.0005	0.0000	0.0000
	1.0	z	−0.0470	−0.2840	0.8252	−0.0001	0.0003	0.0001
	10.0	z	−1.5692	−5.8559	7.3710	−0.0125	−0.0045	−0.0043
	20.0	z	1.0803	−4.9535	18.1806	1.0803	0.0014	−0.0021
	10.0	y	0.0793	6.5978	−1.3214	−0.0066	−0.0004	−0.0003
	−10.0	y	−0.1597	−6.6084	−1.3826	0.0062	0.0004	0.0006
	10.0	x	0.7600	−0.0439	0.0272	0.0002	0.0008	−0.0018
Bowl	0.5	z	0.0818	−0.0440	0.2600	0.0009	0.0004	−0.0004
	1.0	z	0.0572	−0.0864	0.5162	0.0015	0.0002	−0.0004
	10.0	z	−1.4703	−0.2302	8.9576	0.0087	0.0022	−0.0420
	20.0	z	−1.2950	−0.1217	19.0383	0.00206	0.0035	−0.0554
	10.0	y	−0.2905	8.8071	0.0682	0.0369	−0.0581	−0.0027
	−10.0	y	0.2614	−8.6043	0.3671	−0.0368	0.0533	0.0006
	10.0	x	9.4381	−0.4111	0.0864	0.0023	0.0202	−0.0048
	−10.0	x	−9.1155	0.5139	0.1217	0.0004	−0.0187	0.0473
Torso	0.5	z	0.3474	1.4598	0.1776	−0.0003	0.0003	−0.0128
	1.0	z	0.3474	1.5383	0.0214	−0.0012	0.0013	−0.0214
	10.0	z	0.0014	0.2854	10.4856	0.0004	0.0018	−0.0006
	20.0	z	0.2456	0.8967	20.8858	0.0016	0.0010	−0.0012
	10.0	y	0.1391	0.8154	0.5305	0.0032	−0.0002	−0.0007
	−10.0	y	0.0276	−0.7041	−0.3240	−0.0025	0.0002	−0.0003
	10.0	x	10.7852	0.6038	0.0172	0.0002	0.0026	−0.0353
	−10.0	x	−10.6538	−1.5451	0.2848	0.0005	−0.0080	0.0327
Renoald algorithm
Plane	0.5	z	−0.3951	−0.1486	0.5499	−0.0001	−0.0001	−0.0000
	1.0	z	−0.8802	−0.2661	1.0675	0.0000	0.0006	−0.0001
	10.0	z	1.2549	1.1879	10.0882	0.0002	−0.0010	−0.0005
	20.0	z	−2.0547	−0.4494	20.0903	−0.0001	−0.0040	−0.0006
3plane	0.5	z	0.0041	−0.1311	0.3722	−0.0005	0.0000	0.0000
	1.0	z	−0.0469	−0.2864	0.8242	−0.0006	0.0003	0.0001
	10.0	z	−0.0744	−4.5305	8.2095	−0.0099	−0.0003	−0.0039
	20.0	z	0.9315	−6.7487	16.8058	−0.0134	0.0058	0.0033
	10.0	y	0.1794	4.0410	−1.1857	−0.0124	−0.0004	−0.0002
	−10.0	y	−0.2608	−5.4906	1.3925	0.0088	0.0009	0.0005
	10.0	x	0.7315	−0.0448	0.0321	0.0002	0.0009	−0.0018
Bowl	0.5	z	0.0818	−0.0440	0.2600	0.0009	0.0004	−0.0004
	1.0	z	0.0571	−0.0831	0.5143	0.0015	0.0003	−0.0004
	10.0	z	−1.2139	−0.1348	8.8075	0.0069	0.0014	−0.0393
	20.0	z	−1.5751	−0.1130	18.7352	0.0056	0.0045	−0.0556
	10.0	y	−0.4060	7.5070	−0.0636	0.0356	−0.0402	−0.0014
	−10.0	y	0.3518	−8.5498	0.2583	−0.0358	0.0521	0.0009
	10.0	x	8.8252	−0.2963	−0.1059	−0.0000	0.0147	−0.0470
	−10.0	x	−8.9940	0.4230	0.1382	0.0007	−0.0171	0.0477
Torso	0.5	z	0.2920	0.4220	0.7578	0.0005	0.0012	−0.0321
	1.0	z	0.2378	0.4817	1.0779	0.0003	0.0009	−0.0315
	10.0	z	−0.0037	0.2426	10.4506	0.0001	0.0018	−0.0006
	20.0	z	0.3008	−0.1655	18.7455	−0.0011	0.0022	−0.0001
	10.0	y	0.1177	0.7804	0.4367	0.0028	−0.0002	−0.0004
	−10.0	y	0.0045	−0.6522	−0.2843	−0.0022	0.0001	−0.0001
	10.0	x	10.4139	0.4050	0.0860	0.0004	0.0015	−0.0327
	−10.0	x	−10.4768	−0.6117	0.1427	−0.0003	−0.0019	0.0334
Bergström algorithm
Plane	0.5	z	0.0818	−0.0440	0.2600	0.0009	0.0004	−0.0003
	1.0	z	−0.9302	−0.2768	1.0675	0.0000	0.0005	−0.0001
	10.0	z	−4.8742	−0.3023	10.0866	0.0002	0.0013	−0.0008
	20.0	z	−3.7203	0.7381	20.0894	−0.0000	0.0001	−0.0008
3plane	0.5	z	0.0023	−0.12880	0.3746	−0.0005	0.0003	0.0000
	1.0	z	−0.0470	−0.2840	−0.8252	−0.0060	0.0003	0.0001
	10.0	z	−8.9059	4.1357	11.6053	0.0083	−0.0080	−0.0024
	20.0	z	1.0816	−4.9472	18.1816	−0.0095	0.0014	−0.0021
	10.0	y	−2.1412	13.7773	1.6485	0.0079	0.0060	0.0009
	−10.0	y	−0.1596	−6.6095	1.3820	0.0062	0.0004	0.0006
	10.0	x	8.4082	2.8181	1.2289	0.0064	−0.0053	−0.0034
Bowl	0.5	z	0.0818	−0.0440	0.2600	0.0009	0.0004	−0.0004
	1.0	z	0.0527	−0.0803	0.5211	0.0016	0.0003	−0.0004
	10.0	z	−0.1868	−0.0644	10.2913	−0.0002	−0.1752	0.0064
	20.0	z	0.6199	−0.0374	20.2321	0.0092	−0.1716	0.0172
	10.0	y	0.0319	10.0246	−0.4303	0.0058	−0.1689	0.0073
	−10.0	y	0.4347	−9.4345	0.9152	−0.0312	−0.1553	0.0238
	10.0	x	9.9406	−2.1170	0.6590	0.0190	0.0004	0.0001
	−10.0	x	−10.3737	−1.6094	0.3040	0.0133	0.0020	−0.0003
Torso	0.5	z	0.3383	−2.5946	0.8076	0.0004	0.0001	−0.0128
	1.0	z	0.1906	−2.6231	0.9682	−0.0000	0.0000	−0.0121
	10.0	z	0.0014	0.2854	10.4856	0.0040	0.0018	−0.0006
	20.0	z	−0.4539	−0.3521	19.4238	0.0008	0.0023	0.0150
	10.0	y	0.5342	8.9866	0.3959	0.0007	−0.0020	−0.0169
	−10.0	y	−0.1569	−8.6553	1.2477	−0.0070	0.0020	0.0012
	10.0	x	10.4504	−2.5372	0.4949	0.0003	0.0003	−0.0131
	−10.0	x	−10.2027	2.5363	0.3930	−0.0003	−0.0002	0.0120
AlignRT algorithm
Plane	0.5	z	0.1	0.3	0.0	0.0	0.0	0.0
	1.0	z	0.2	0.1	0.5	0.0	0.0	0.1
	10.0	z	0.3	0.0	10.1	0.0	0.0	0.1
	20.0	z	0.2	1.2	20.1	0.0	0.0	0.2
3plane	0.5	z	0.0	0.1	0.5	0.0	0.0	0.0
	1.0	z	0.1	0.1	1.1	0.0	0.0	0.0
	10.0	z	0.0	0.2	10.2	0.0	0.0	0.0
	20.0	z	0.0	0.1	20.1	0.0	0.0	0.1
	10.0	y	0.2	10.2	0.2	0.0	0.0	0.0
	−10.0	y	0.1	−9.8	0.1	0.0	0.0	0.0
	10.0	x	9.8	0.2	0.1	0.0	0.0	0.0
Bowl	0.5	z	0.1	0.1	0.5	0.0	0.0	0.0
	1.0	z	0.1	0.2	1.1	0.0	0.0	0.0
	10.0	z	0.1	0.2	10.0	0.1	0.0	1.2
	20.0	z	0.3	0.3	19.9	0.1	0.0	1.6
	10.0	y	0.2	10.3	0.2	0.0	0.1	1.6
	−10.0	y	0.7	−9.8	0.2	0.1	0.1	2.7
	10.0	x	10.1	0.2	0.2	0.0	0.1	0.1
	−10.0	x	−10.0	0.5	0.2	0.0	0.0	1.7
Torso	0.5	z	0.1	0.0	0.6	0.0	0.0	0.0
	1.0	z	0.1	0.0	1.1	0.0	0.0	0.0
	10.0	z	0.1	0.0	9.9	0.1	0.0	0.1
	20.0	z	0.2	0.1	19.8	0.0	0.1	0.1
	10.0	y	0.2	10.1	0.2	0.0	0.0	0.0
	−10.0	y	0.2	−9.8	0.1	0.1	0.0	0.0
	10.0	x	10.0	0.2	0.1	0.0	0.0	0.0
	−10.0	x	−9.9	0.1	0.0	0.0	0.0	0.0