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

From: Some fast projection methods based on Chan-Vese model for image segmentation

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