Table 1.

Trading off performance for invertibility.

Set	Width	Method	Opt. λ	Dice (↑)	Dice30 (↑)	% Folds (↓)	Sdlog|Jφ|(↓)
A	64	NGF [11]	0.0	0.696 ± 0.023	0.686	0.141 ± 0.043	0.072
	64	MI [38]	0.175	0.748 ± 0.021	0.739	0.461 ± 0.100	0.089
	64	LocalMI [8]	0.125	0.745 ± 0.023	0.737	0.402 ± 0.076	0.083
	64	MIND [14]	0.3	0.726 ± 0.023	0.716	0.258 ± 0.051	0.079
	64	CR (proposed)	0.05	0.776 ± 0.020	0.768	0.451 ± 0.074	0.083
	64	mCR (proposed)	0.125	0.781 ± 0.020	0.774	0.475 ± 0.070	0.084
B	256	SM-brains [15]	-	0.755 ± 0.020	0.749	0.023 ± 0.008	0.048
	256	SM-shapes [15]	-	0.721 ± 0.021	0.715	0.017 ± 0.011	0.056
	256	MI [38]	0.2	0.759 ± 0.021	0.750	0.487 ± 0.099	0.090
	256	CR (proposed)	0.075	0.774 ± 0.020	0.765	0.315 ± 0.0576	0.078
	256	mCR (proposed)	0.15	0.780 ± 0.021	0.773	0.416 ± 0.065	0.082
C	64	CR+MI	0.3	0.751 ± 0.021	0.742	0.246 ± 0.059	0.080
	64	CR+ExtNegs	0.05	0.764 ± 0.020	0.756	0.489 ± 0.073	0.085
	64	CR+MI+ExtNegs	0.3	0.747 ± 0.021	0.739	0.214 ± 0.056	0.078
	64	CR+SupPretrain	0.025	0.778 ± 0.020	0.770	0.465 ± 0.075	0.084
	64	mCR+SupPretrain	0.075	0.778 ± 0.020	0.770	0.406 ± 0.067	0.081
	64	mCR+RandAE	0.1	0.778 ± 0.020	0.770	0.393 ± 0.070	0.80
D	256	CR (10 int. steps)	0.075	0.773 ± 0.021	0.764	0.341 ± 0.058	0.079
	256	CR (16 int. steps)	0.05	0.779 ± 0.020	0.772	0.462 ± 0.071	0.083
	256	CR (32 int. steps)	0.075	0.774 ± 0.020	0.765	0.315 ± 0.0576	0.078

Registration accuracy (Dice), robustness (Dice30), and characteristics (% Folds, stddev. $log\left|J_{\phi }\right|$) for all benchmarked methods at values of $\lambda$ that maintains the percentage of folding voxels at less than $0.5\%$ of all voxels, as in [30], s.t. high performance is achieved alongside negligible singularities. This table is best interpreted in conjunction with figure 3, where results from all $\lambda$ values are visualized. A. CR and mCR obtain improved accuracy and robustness (A5–6) with similar deformation characteristics to baseline losses (A1–4). B. At larger model sizes, mCR and CR still obtain higher registration accuracy and robustness (B4–5), albeit at the cost of more irregular deformations in comparison to SM (B1). C. Further adding external losses, negative samples, or both to CR harms performance (C1–3), supervised pretraining (C4–5) very marginally improves results over training from scratch (A5–6), and random feature extraction only slightly reduces Dice while smoothening displacements (C6). D. At a given $\lambda$, increasing integration steps yields marginal Dice and smoothness improvements.