Global Minimum for Curvature Penalized Minimal Path Method

L denotes the classical curve length, s is the arc-length parameter, and Γ : [0, L]→ Ω is a curve with non-vanishing velocity vector. κ is the curvature and α , β are two positively weighted functions computed by the optimally oriented flux filter [2]. Our first step is to cast the elastica energy (5) in the form of path length with respect to a degenerate Finsler metric. For that purpose, let S1 = [0, 2π[ be the space of angles, with periodic boundary conditions, and for each angle θ let~vθ = (cosθ ,sinθ) be the corresponding unit vector. For γ = (Γ,θ) ∈Ω×S1 and γ̇ = (Γ̇, θ̇) ∈ R2×R1 we can define