Slice-to-Volume Nonrigid Registration of Histological Sections to MR Images of the Human Brain

Table 1

Single-stage full resolution registration results for six optimizers (NEWUOA, Powell’s Direction Set, Simplex, Genetic, Differential Evolution, and Conjugate Gradient). Rigid and nonrigid registration results are given in successive rows. Each registration result is presented by the cumulative similarity measure SM, the bending energy 𝐸_𝑇 (nonrigid only), and the running time 𝑇 in minutes. Columns 1–8 are for eight slices from eight distinct blocks of the brain sample. The last column is the sum of 1–8. Values referenced in the text are in boldface.

Optimizer	Reg	Var⧵Blk#	1	2	3	4	5	6	7	8	∑^81
NEWUOA	Rigid	SM	5.321	5.583	5.265	4.613	4.234	4.188	4.980	4.917	39.099
		𝑇 (min)	0.1	0.1	0.1	0.2	0.1	0.1	0.1	0.1	1.1
	TPS	SM	6.031	6.129	5.345	5.764	5.330	5.244	5.708	5.682	45.233
		𝐸𝑇	4.4	2.5	15.3	9.6	8.1	11.8	4.2	5.0	60.9
		𝑇 (min)	0.66	7.83	40	14.3	25	25	23	3.6	139.4
PDS	Rigid	SM	4.336	4.431	3.944	3.457	3.533	4.177	4.100	4.055	32.034
		𝑇 (min)	0.2	0.2	0.6	0.5	0.5	0.6	0.4	0.2	3.3
	TPS	SM	6.026	5.867	5.628	5.218	4.984	5.414	5.253	5.425	43.814
		𝐸𝑇	4	5.6	11.1	22.0	14.6	9.4	12.2	6.4	85.4
		𝑇 (min)	15.5	164.3	486.7	363	473	490.5	334.5	82.8	2410.3
Simplex	Rigid	SM	4.705	4.389	4.223	3.358	3.552	4.134	3.800	4.432	32.592
		𝑇 (min)	0.1	0.2	0.2	0.3	0.3	0.3	0.3	0.4	2.2
	TPS	SM	5.740	—	—	—	—	—	—	5.138	10.877
		𝐸𝑇	9.3	—	—	—	—	—	—	7.3	16.6
		𝑇 (min)	95.3	3000+	3000+	3000+	3000+	3000+	3000+	1977	20000+
Genetic	Rigid	SM	3.926	3.492	3.104	3.370	3.372	4.146	4.099	4.397	29.907
		𝑇 (min)	3.5	8.5	12.6	15.9	17.5	19.3	17.8	6.4	101.4
	TPS	SM	3.904	3.268	2.847	3.350	2.729	3.029	3.266	3.188	25.581
		𝐸𝑇	141.8	1019.9	2396.0	649.5	908.2	913.8	670.3	579.2	7278.7
		𝑇 (min)	6.7	28.3	51.1	55	64	64.5	55.7	18.3	343.6
DE	Rigid	SM	5.320	5.580	5.266	4.616	4.233	4.189	4.980	5.061	39.245
		𝑇 (min)	4.2	10.5	15.0	19.9	22.6	23.1	22.0	8.6	125.9
	TPS	SM	2.318	1.733	2.353	2.581	2.610	2.605	2.724	2.318	19.241
		𝐸𝑇	859.4	4104.2	7101.0	2522.9	3006.5	2909.6	2429.8	2628.8	25562.0
		𝑇 (min)	8.7	35.6	64.5	64.5	82	83	71.3	24.3	433.9
CG	Rigid	SM	3.587	4.259	4.523	3.451	3.774	4.163	4.117	3.990	31.863
		𝑇 (min)	0.1	0.2	0.2	0.9	0.3	0.1	0.2	0.2	2.2
	B-SPL	SM	4.042	4.181	4.520	3.417	3.799	4.201	4.136	4.067	32.363
		𝐸𝑇	2.7	0.2	0.3	0.0	0.2	0.0	0.1	0.4	3.9
		𝑇 (min)	0.2	0.4	0.2	0.7	0.5	0.5	0.3	0.6	3.3