Construction of Target Controllable Image Segmentation Model Based on Homotopy Perturbation Technology

and Applied Analysis 3 curves of the objects should be embedded into the level set function as follows: C = {(x, y) | (x, y) ∈ Ω, φ (x, y) = 0} , Ω 1 = {(x, y) | (x, y) ∈ Ω, φ (x, y) > 0} , Ω 2 = {(x, y) | (x, y) ∈ Ω, φ (x, y) < 0} . (2) Then, the level set-based C-V model can be rewritten as follows: E (c 1 , c 2 , φ) = λ 1 ∫ Ω 󵄨󵄨󵄨󵄨I0 − c1 󵄨󵄨󵄨󵄨 2 H(φ) dxdy + λ 2 ∫ Ω 󵄨󵄨󵄨󵄨I0 − c2 󵄨󵄨󵄨󵄨 2 (1 − H (φ)) dxdy + ν∫ Ω 󵄨󵄨󵄨󵄨H (φ) 󵄨󵄨󵄨󵄨 dxdy, H (φ) = { 1, φ ≥ 0 , 0, φ < 0 , δ ε = ε π (ε + φ) . (3) Using the variational method, the PDEs with respect to the variable φ can be obtained as follows: ∂φ ∂t = δ ε (φ) [νdiv( ∇φ 󵄨󵄨󵄨󵄨∇φ 󵄨󵄨󵄨󵄨 ) − λ 1 󵄨󵄨󵄨󵄨I0 − c1 󵄨󵄨󵄨󵄨 2 + λ 2 󵄨󵄨󵄨󵄨I0 − c2 󵄨󵄨󵄨󵄨 2 ] . (4) Obviously, div(∇φ/|∇φ|) is the curvature of the level set function φ, and δ ε (φ) is used to constrain the growth of the level set function. The solution of (4) is the level set function φ(x, y, t) at time t. The zero level set is the object contour curve, which can be obtained by solving φ(x, y, t) = 0. In the following, what we are talking about is how to construct the target controllable image segmentation model based on the basic idea of HPM. It is easy to understand that the function of the curvature in C-V model is just to preserve the smoothness of the object contour. Neglecting the curvature in (4), the simplified model can be obtained as follows: ∂φ ∂t = δ ε (φ) [−λ 1 󵄨󵄨󵄨󵄨I0 − c1 󵄨󵄨󵄨󵄨 2 + λ 2 󵄨󵄨󵄨󵄨I0 − c2 󵄨󵄨󵄨󵄨 2 ] . (5) In solving the C-V model with HPM and iteration method, the average gray level values inside and outside of the contour curves c 1 and c 2 vary with the evolution of the level set function. This evolution will end up with that the contour curve coincides with the object boundary. Then, c 1 and c 2 become constants, and the right hand of (5) should equal zero; that is,