Table 1 Abbreviations, full names, and their corresponding energy functionals of all methods for comparison

1

GDEWRM

Gradient descent equation with re-initialization [16]

E(ϕ) = ∫  _Ω Q₁₂H(ϕ)dx + γ ∫  _Ω |∇H(ϕ)|dx and$\left\{\begin{array}{l}{\varphi }_t+\mathrm{Sign}\left({\varphi }_0\right)\left(\left|\nabla \varphi \right|-1\right)=0\\ \varphi \left(x,0\right)={\varphi }_0\end{array}\right.$

2

GDEWORM

Gradient descent equation without re-initialization [30]

$E\left(\varphi \right)={\displaystyle {\int }_{\Omega }{Q}_{12}{H}_{ϵ}\left(\varphi \right)\mathit{dx}}+\gamma {\displaystyle {\int }_{\Omega }\left|\nabla H_{ϵ}\left(\varphi \right)\right|\mathit{dx}}+\frac{\mu }{2}{\displaystyle {\int }_{\Omega }{\left(\left|\nabla \varphi \right|-1\right)}^2\mathit{dx}}$

3

ALM

Augmented Lagrangian method [34]

$E\left(\varphi ,\lambda \right)={\displaystyle {\int }_{\Omega }{Q}_{12}H\left(\varphi \right)\mathit{dx}}+\gamma {\displaystyle {\int }_{\Omega }\left|\nabla H\left(\varphi \right)\right|\mathit{dx}}+{\displaystyle {\int }_{\Omega }\lambda \left(\left|\nabla \varphi \right|-1\right)\mathit{dx}}+\frac{\mu }{2}{\displaystyle {\int }_{\Omega }{\left(\left|\nabla \varphi \right|-1\right)}^2\mathit{dx}}$

4

PLM

Projection Lagrangian method [34]

$E\left(\varphi ,\overrightarrow{w}\right)={\displaystyle {\int }_{\Omega }{Q}_{12}H\left(\varphi \right)\mathit{dx}}+\gamma {\displaystyle {\int }_{\Omega }\left|\nabla H\left(\varphi \right)\right|\mathit{dx}}+{\displaystyle {\int }_{\Omega }\lambda \left(\left|\overrightarrow{w}\right|-1\right)\mathit{dx}}+\frac{\mu }{2}{\displaystyle {\int }_{\Omega }{\left(\overrightarrow{w}-\nabla \varphi \right)}^2\mathit{dx}}$

5

CALM

Completely augmented Lagrangian method [36]

$\begin{array}{cc}\hfill \begin{array}{c}E\left(\varphi ,\phi ,s,\overrightarrow{v},\overrightarrow{w}\right)={\displaystyle {\int }_{\Omega }{Q}_{12}\mathit{sdx}}+\gamma {\displaystyle {\int }_{\Omega }\left|\overrightarrow{v}\right|\mathit{dx}}+{\displaystyle {\int }_{\Omega }{\lambda }_2\left(s-H_{ϵ}\left(\phi \right)\right)\mathit{dx}}\\ +\frac{{\mu }_2}{2}{\displaystyle {\int }_{\Omega }{\left(s-H_{ϵ}\left(\phi \right)\right)}^2\mathit{dx}}\\ +{\displaystyle {\int }_{\Omega }{\overrightarrow{\lambda }}_3\cdot \left(\overrightarrow{v}-\nabla s\right)\mathit{dx}}+\frac{{\mu }_3}{2}{\displaystyle {\int }_{\Omega }{\left(\overrightarrow{v}-\nabla s\right)}^2\mathit{dx}}\\ +{\displaystyle {\int }_{\Omega }{\lambda }_1\left(\phi -\varphi \right)\mathit{dx}}+\frac{{\mu }_1}{2}{\displaystyle {\int }_{\Omega }{\left(\phi -\varphi \right)}^2\mathit{dx}}\\ +{\displaystyle {\int }_{\Omega }{\overrightarrow{\lambda }}_4\cdot \left(\overrightarrow{w}-\nabla \varphi \right)\mathit{dx}}+\frac{{\mu }_4}{2}{\displaystyle {\int }_{\Omega }{\left(\overrightarrow{w}-\nabla \varphi \right)}^2\mathit{dx}}\end{array}\hfill & \hfill \mathrm{s}.\mathrm{t}.\left|\overrightarrow{w}\right|=1\hfill \end{array}$

6

SBPM

Split Bregman projection method

$\begin{array}{cc}\hfill E\left(\varphi ,\overrightarrow{w}\right)={\displaystyle {\int }_{\Omega }{Q}_{12}}{H}_{ϵ}\left(\varphi \right)\mathit{dx}+\gamma {\displaystyle {\int }_{\Omega }\left|\overrightarrow{w}\right|{\delta }_{ϵ}\left(\varphi \right)\mathit{dx}}+\frac{\theta }{2}{\displaystyle {\int }_{\Omega }\left(w-\nabla \varphi -{\overrightarrow{b}}^{k+1}\right)\mathit{dx}}\hfill & \hfill \mathrm{s}.\mathrm{t}.\left|\overrightarrow{w}\right|=1\hfill \end{array}$

7

ALPM

Augmented Lagrangian projection method

$\begin{array}{cc}\hfill E\left(\varphi ,\overrightarrow{w},\overrightarrow{\lambda }\right)={\displaystyle {\int }_{\Omega }{Q}_{12}}{H}_{ϵ}\left(\varphi \right)\mathit{dx}+\gamma {\displaystyle {\int }_{\Omega }\left|\overrightarrow{w}\right|{\delta }_{ϵ}\left(\varphi \right)\mathit{dx}}+{\displaystyle {\int }_{\Omega }\overrightarrow{\lambda }\cdot \left(\overrightarrow{w}-\nabla \varphi \right)\mathit{dx}}+\frac{\theta }{2}{\displaystyle {\int }_{\Omega }{\left(\overrightarrow{w}-\nabla \varphi \right)}^2\mathit{dx}}\hfill & \hfill \mathrm{s}.\mathrm{t}.\left|\overrightarrow{w}\right|=1\hfill \end{array}$

8

DSBPM

Dual Split Bregman projection method

$\begin{array}{cc}\hfill E\left(\varphi ,\overrightarrow{p},\overrightarrow{w}\right)={\displaystyle {\int }_{\Omega }{Q}_{12}}{H}_{ϵ}\left(\varphi \right)\mathit{dx}+\gamma {\displaystyle {\int }_{\Omega }{H}_{ϵ}\left(\varphi \right)\nabla \cdot \overrightarrow{p}\mathit{dx}}+\frac{\theta }{2}{\displaystyle {\int }_{\Omega }{\left(\overrightarrow{w}-\nabla \varphi -{\overrightarrow{b}}^{k+1}\right)}^2}\mathit{dx}\hfill & \hfill \mathrm{s}.\mathrm{t}.\left|\overrightarrow{w}\right|=1\hfill \end{array}$

9

DALPM

Dual augmented Lagrangian method

$\begin{array}{cc}\hfill E\left(\varphi ,\overrightarrow{p},\overrightarrow{w},\overrightarrow{\lambda }\right)={\displaystyle {\int }_{\Omega }{Q}_{12}}{H}_{ϵ}\left(\varphi \right)\mathit{dx}+\gamma {\displaystyle {\int }_{\Omega }{H}_{ϵ}\left(\varphi \right)\nabla \cdot \overrightarrow{p}\mathit{dx}}+{\displaystyle {\int }_{\Omega }\overrightarrow{\lambda }\cdot \left(\overrightarrow{w}-\nabla \varphi \right)}\mathit{dx}+\frac{\theta }{2}{\displaystyle {\int }_{\Omega }{\left(\overrightarrow{w}-\nabla \varphi \right)}^2}\mathit{dx}\hfill & \hfill \mathrm{s}.\mathrm{t}.\left|\overrightarrow{w}\right|=1\hfill \end{array}$

10

FMM

Fuzzy membership method [38]

$E\left(\varphi ,\overrightarrow{w}\right)={\displaystyle {\int }_{\Omega }{Q}_{12}}\mathit{\varphi dx}+\gamma {\displaystyle {\int }_{\Omega }\left|\overrightarrow{w}\right|\mathit{dx}}+\frac{\theta }{2}{\displaystyle {\int }_{\Omega }\left(w-\nabla \varphi -{\overrightarrow{b}}^{k+1}\right)\mathit{dx}}\mathrm{s}.\mathrm{t}.\varphi \in \left[0,1\right]$