Table 1.

Proposed boundary-overlap-based similarity metrics: directional and symmetric variants.

Boundary Dice metrics	Boundary Jaccard metrics
${\mathrm{DBD}}_G=\frac{\sum _{x\in \partial G}\mathrm{DSC}(N_x)}{|\partial G|}$	${\mathrm{DBJ}}_G=\frac{\sum _{x\in \partial G}\mathrm{JSC}(N_x)}{|\partial G|}$
${\mathrm{DBD}}_M=\frac{\sum _{y\in \partial M}\mathrm{DSC}(N_y)}{|\partial M|}$	${\mathrm{DBJ}}_M=\frac{\sum _{y\in \partial M}\mathrm{JSC}(N_y)}{|\partial M|}$
$\mathrm{SBD}=\frac{\sum _{x\in \partial G}\mathrm{DSC}(N_x)+\sum _{y\in \partial M}\mathrm{DSC}(N_y)}{|\partial G|+|\partial M|}$	$\mathrm{SBJ}=\frac{\sum _{x\in \partial G}\mathrm{JSC}(N_x)+\sum _{y\in \partial M}\mathrm{JSC}(N_y)}{|\partial G|+|\partial M|}$
Boundary TPVF metrics	Boundary TNVF metrics
${\mathrm{DBTP}}_G=\frac{\sum _{x\in \partial G}\mathrm{TPVF}(N_x)}{|\partial G|}$	${\mathrm{DBTN}}_G=\frac{\sum _{x\in \partial G}\mathrm{TNVF}(N_x)}{|\partial G|}$
${\mathrm{DBTP}}_M=\frac{\sum _{y\in \partial M}\mathrm{TPVF}(N_y)}{|\partial M|}$	${\mathrm{DBTN}}_M=\frac{\sum _{y\in \partial M}\mathrm{TNVF}(N_y)}{|\partial M|}$
$\mathrm{SBTP}=\frac{\sum _{x\in \partial G}\mathrm{TPVF}(N_x)+\sum _{y\in \partial M}\mathrm{TPVF}(N_y)}{|\partial G|+|\partial M|}$	$\mathrm{SBTN}=\frac{\sum _{x\in \partial G}\mathrm{TNVF}(N_x)+\sum _{y\in \partial M}\mathrm{TNVF}(N_y)}{|\partial G|+|\partial M|}$
Boundary Prec metrics
${\mathrm{DBP}}_G=\frac{\sum _{x\in \partial G}\mathrm{Prec}(N_x)}{|\partial G|}$
${\mathrm{DBP}}_M=\frac{\sum _{y\in \partial M}\mathrm{Prec}(N_y)}{|\partial M|}$
$\mathrm{SBP}=\frac{\sum _{x\in \partial G}\mathrm{Prec}(N_x)+\sum _{y\in \partial M}\mathrm{Prec}(N_y)}{|\partial G|+|\partial M|}$