Table 1 Formulas of some variational image segmentation methods

Edge-based

GAC[23]

$\vartheta ||\nabla \mathrm{\Phi }||\mathrm{div}\left(\mathrm{g}\frac{\nabla \mathrm{\Phi }}{||\nabla \mathrm{\Phi }||}\right)+\alpha \mathrm{g}||\nabla \mathrm{\Phi }||$

PBLS[24]

$\vartheta \left|\right|\nabla \mathrm{\Phi }\left|\right|\mathrm{div}\left(\frac{\nabla \mathrm{\Phi }}{\left|\left|\nabla \mathrm{\Phi }\right|\right|}\right)-\alpha \left|\right|\nabla \mathrm{\Phi }\left|\right|\Re \left({\widehat{q}}_{\mathrm{PBLS}}\right)$

DRLSE[25]

$\mu \mathrm{div}\left(D\left(\left|\left|\nabla \mathrm{\Phi }\right|\right|\right)\nabla \mathrm{\Phi }\right)+\vartheta {\delta }_{ϵ}\left(\mathrm{\Phi }\right)\mathrm{div}\left(\mathrm{g}\frac{\nabla \mathrm{\Phi }}{\left|\left|\nabla \mathrm{\Phi }\right|\right|}\right)+\mathit{\alpha }\mathrm{g}{\delta }_{ϵ}\left(\mathrm{\Phi }\right)$

ARLS[28]

$\mu \left({\nabla }^2\mathrm{\Phi }-\mathrm{div}\left(\frac{\nabla \mathrm{\Phi }}{||\nabla \mathrm{\Phi }||}\right)\right)+\vartheta {\delta }_{ϵ}\left(\mathrm{\Phi }\right)\mathrm{div}\left({||\nabla \mathrm{\Phi }||}^{s\left(\nabla \left(I*G_{\sigma }\right)\right)-2}\nabla \mathrm{\Phi }\right)+\alpha {\delta }_{ϵ}\left(\mathrm{\Phi }\right){\nabla }^2\left(I*G_{\sigma }\right)$

Region-based

RBLSE[33]

$\mu \mathrm{div}\left(D\left(\left|\left|\nabla \mathrm{\Phi }\right|\right|\right)\nabla \mathrm{\Phi }\right)+\vartheta {\delta }_{ϵ}\left(\mathrm{\Phi }\right)\mathrm{div}\left(\frac{\nabla \mathrm{\Phi }}{\left|\left|\nabla \mathrm{\Phi }\right|\right|}\right)-\alpha {\delta }_{ϵ}\left(\mathrm{\Phi }\right)\left(e_1-e_2\right),$