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

No. Abbreviations Full name Energy functional
1 GDEWRM Gradient descent equation with re-initialization [16] E(ϕ) = ∫  Ω Q12H(ϕ)dx + γ ∫  Ω |H(ϕ)|dx and$ϕ t + Sign ϕ 0 ∇ ϕ − 1 = 0 ϕ x , 0 = ϕ 0$
2 GDEWORM Gradient descent equation without re-initialization [30] $E ϕ = ∫ Ω Q 12 H ϵ ϕ dx + γ ∫ Ω ∇ H ϵ ϕ dx + μ 2 ∫ Ω ∇ ϕ − 1 2 dx$
3 ALM Augmented Lagrangian method [34] $E ϕ , λ = ∫ Ω Q 12 H ϕ dx + γ ∫ Ω ∇ H ϕ dx + ∫ Ω λ ∇ ϕ − 1 dx + μ 2 ∫ Ω ∇ ϕ − 1 2 dx$
4 PLM Projection Lagrangian method [34] $E ϕ , w → = ∫ Ω Q 12 H ϕ dx + γ ∫ Ω ∇ H ϕ dx + ∫ Ω λ w → − 1 dx + μ 2 ∫ Ω w → − ∇ ϕ 2 dx$
5 CALM Completely augmented Lagrangian method [36] $E ϕ , φ , s , v → , w → = ∫ Ω Q 12 sdx + γ ∫ Ω v → dx + ∫ Ω λ 2 s − H ϵ φ dx + μ 2 2 ∫ Ω s − H ϵ φ 2 dx + ∫ Ω λ → 3 ⋅ v → − ∇ s dx + μ 3 2 ∫ Ω v → − ∇ s 2 dx + ∫ Ω λ 1 φ − ϕ dx + μ 1 2 ∫ Ω φ − ϕ 2 dx + ∫ Ω λ → 4 ⋅ w → − ∇ ϕ dx + μ 4 2 ∫ Ω w → − ∇ ϕ 2 dx s . t . w → = 1$
6 SBPM Split Bregman projection method $E ϕ , w → = ∫ Ω Q 12 H ϵ ϕ dx + γ ∫ Ω | w → | δ ϵ ϕ dx + θ 2 ∫ Ω w − ∇ ϕ − b → k + 1 dx s . t . w → = 1$
7 ALPM Augmented Lagrangian projection method $E ϕ , w → , λ → = ∫ Ω Q 12 H ϵ ( ϕ ) dx + γ ∫ Ω | w → | δ ϵ ϕ dx + ∫ Ω λ → ⋅ w → − ∇ ϕ dx + θ 2 ∫ Ω w → − ∇ ϕ 2 dx s . t . w → = 1$
8 DSBPM Dual Split Bregman projection method $E ϕ , p → , w → = ∫ Ω Q 12 H ϵ ϕ dx + γ ∫ Ω H ϵ ϕ ∇ ⋅ p → dx + θ 2 ∫ Ω w → − ∇ ϕ − b → k + 1 2 dx s . t . w → = 1$
9 DALPM Dual augmented Lagrangian method $E ϕ , p → , w → , λ → = ∫ Ω Q 12 H ϵ ( ϕ ) dx + γ ∫ Ω H ϵ ϕ ∇ ⋅ p → dx + ∫ Ω λ → ⋅ w → − ∇ ϕ dx + θ 2 ∫ Ω w → − ∇ ϕ 2 dx s . t . w → = 1$
10 FMM Fuzzy membership method [38] $E ϕ , w → = ∫ Ω Q 12 ϕdx + γ ∫ Ω | w → | dx + θ 2 ∫ Ω w − ∇ ϕ − b → k + 1 dx s . t . ϕ ∈ 0 , 1$