Table 9.

Quantitative analysis of the parameter continuation in β. We report results for the hand images and the brain images for different regularization schemes (see Figure 1). The spatial grid size for the images is n_x = (512, 512)^⊤. The number of the time points n_t is chosen adaptively (see section 4.3.5 for details). We use the full set of stopping conditions (see section 4.2.4) with a tolerance of τ_𝒥 = 1E–3. We report results for plain H¹- and H²-regularization (γ = 0; top block) as well as for the Stokes regularization scheme (γ = 1; H¹-regularization; bottom block). We invert for a stationary velocity field (i.e., n_c = 1). We report (i) the considered lower bound on the deformation gradient (ε_F) or the grid angle (ε_θ), (ii) the number of the required estimation steps, (iii) the minimal value for the deformation gradient for the optimal regularization parameter, (iv) the computed optimal value for β(β^★), (v) the minimal change in β(δβ_min), as well as (vi) the relative change in the L²-distance ( ${\Vert m_R^h-m_1^h\Vert }_{2,\mathit{rel}}$).

Smoothness regularization
Data	𝒮	εF	steps	$min(det({\mathit{F}}_1^h))$	β★	δβmin	${\Vert m_R^h-m_1^h\Vert }_{2,\text{rel}}$
Brain images	H1	5E–2	9	5.09E–2	3.11E–1	5.00E–3	7.01E–1
	H2	5E–2	10	5.11E–2	2.13E–2	5.00E–4	5.52E–1
Hand images	H1	1E–1	10	1.13E–1	3.39E–2	5.00E–4	8.74E–2
	H2	1E–1	12	1.05E–1	2.69E–4	5.00E–6	6.83E–2
Stokes regularization
Data	𝒮	εθ	steps	$min(det({\mathit{F}}_1^h))$	β★	δβmin	${\Vert m_R^h-m_1^h\Vert }_{2,\text{rel}}$
Hand images	H1	π/16	10	10.00E–1	2.13E–2	5.00E–4	1.17E–1