Table 2.

Overview of the quantitative measures that are used to assess the registration performance. We report the number of outer iterations (steps for updating the control variable v^h) and the number of hyperbolic PDE solves (i.e., how often we have to solve one of the hyperbolic PDEs (4.1a), (4.1b), (4.3a), and (4.3b) that appear in the optimality systems) to assess the work load. We report the relative change of the L²-distance, the objective, and the reduced gradient to assess the quality of the inversion. We report values for the determinant of the deformation gradient to assess the regularity of the computed deformation map. We report the relative power spectrum of the coefficients ${\mathit{v}}_l^h$ to assess which of the coefficients of the expansion in (4.10) are significant.

Description	Definition
# of required outer iterations	k★
# of required hyperbolic PDE solves	nPDE
Relative change of L2-distance	${\Vert m_1^h-m_R^h\Vert }_{2,\text{rel}}:={\Vert m_1^h-m_R^h\Vert }_2^2/{\Vert m_T^h-m_R^h\Vert }_2^2$
Relative change of objective value	$\delta {\mathcal{J}}_{\text{rel}}^h:={\mathcal{J}}^h({\mathit{v}}_{k^{★}}^h)/{\mathcal{J}}^h({\mathit{v}}_0^h)$
Relative change of reduced gradient	${\Vert {\mathit{g}}^h\Vert }_{\infty ,\text{rel}}:={\Vert {\mathit{g}}_{k^{★}}^h\Vert }_{\infty }/{\Vert {\mathit{g}}_0^h\Vert }_{\infty }$
Determinant of deformation gradient	$det({\mathit{F}}_1^h)$
Relative power spectrum of ${\mathit{v}}_l^h$	${\Vert {\mathit{v}}_l^h\Vert }_{2,\text{rel}}:={\Vert {\mathit{v}}_l^h\Vert }_2/{\Vert {\{{\mathit{v}}_{l^{\prime }}^h\}}_{l^{\prime }=1}^{n_c}\Vert }_2$