A variational level set model with closed-form solution for bimodal image segmentation

Abstract

In this work, we present a variational level set model with closed–form solution via combining with the fuzzy clustering method for robust and efficient image segmentation. For the designed energy functional, the two region parameters are first quickly pre–computed by means of the fuzzy c–means method and then embedded into a variational binary level set framework. Unlike the traditional variational level set models and optimization algorithms, our proposed model could directly obtain an exact closed–form solution of the level set function without using any iterative calculations and it is thus the globally optimal solution. Furthermore, we investigate the closed–form formula and achieve a significant property of the solution. As a byproduct, the manual initialization of the level set function and the sophisticated setting of time step in the process of numerical implementation are completely eliminated and thus leads to more robust segmentation results. Numerical experiments on both synthetic and real images verify the theoretical analysis of the proposed model and confirm the segmentation performance of the proposed method in terms of efficiency, accuracy and insensitiveness to parameters tuning.

Appendix A

Proof of Theorem 1

Proof

∀m,n > 0 with m + n = 1. For given ${\phi _{1}}, {\phi _{2}} \in {L^{2}}\left ({\varOmega } \right )$ and ϕ₁≠ϕ₂, we have

$$ \begin{array}{@{}rcl@{}} {\left( {m{\phi_{1}} + n{\phi_{2}} + 1} \right)^{2}} &=& {\left( {m\left( {{\phi_{1}} + 1} \right) + n\left( {{\phi_{2}} + 1} \right)} \right)^{2}} \\ &=& {m^{2}}{\left( {{\phi_{1}} + 1} \right)^{2}} + {n^{2}}{\left( {{\phi_{2}} + 1} \right)^{2}} + 2mn\left( {{\phi_{1}} + 1} \right)\left( {{\phi_{2}} + 1} \right) \\ &<& {m^{2}}{\left( {{\phi_{1}} + 1} \right)^{2}} + {n^{2}}{\left( {{\phi_{2}} + 1} \right)^{2}} + mn\left( {{{\left( {{\phi_{1}} + 1} \right)}^{2}} + {{\left( {{\phi_{2}} + 1} \right)}^{2}}} \right) \\ &=& m\left( {m + n} \right){\left( {{\phi_{1}} + 1} \right)^{2}} + n\left( {n + m} \right){\left( {{\phi_{2}} + 1} \right)^{2}} \\ &=& m{\left( {{\phi_{1}} + 1} \right)^{2}} + n{\left( {{\phi_{2}} + 1} \right)^{2}}. \end{array} $$

(20)

Analogous with (20), we obtain

$$ \begin{array}{@{}rcl@{}} {\left( {m{\phi_{1}} + n{\phi_{2}} - 1} \right)^{2}}& =& {\left( {m\left( {{\phi_{1}} - 1} \right) + n\left( {{\phi_{2}} - 1} \right)} \right)^{2}} \\ &<& m{\left( {{\phi_{1}} - 1} \right)^{2}} + n{\left( {{\phi_{2}} - 1} \right)^{2}}. \end{array} $$

(21)

combining (20) and (21), we get the relation

$$ E\left( {m{\phi_{1}} + n{\phi_{2}}} \right) < mE\left( {{\phi_{1}}} \right) + nE\left( {{\phi_{2}}} \right). $$

Hence, $E\left (\phi \right )$ is strictly convex in ${L^{2}}\left ({\varOmega } \right )$. The proof is completed. □

Proof of Theorem 2

Proof

According to Theorem 1, we know that the proposed energy functional (13) has unique global minimum ϕ^∗. Therefore, we from the expression (16) can directly derive the following relation:

$$ {\lambda_{1}}{\left( {I - {m_{1}}} \right)^{2}}\left( {{\phi^{*}} + 1} \right) + {\lambda_{2}}{\left( {I - {m_{2}}} \right)^{2}}\left( {{\phi^{*}} - 1} \right) = 0. $$

(22)

Then we get the closed–form solution by explicitly solving the (22) as follows:

$$ {\phi^{*}} = - \frac{{{\lambda_{1}}{{\left( {I - {m_{1}}} \right)}^{2}} - {\lambda_{2}}{{\left( {I - {m_{2}}} \right)}^{2}}}}{{{\lambda_{1}}{{\left( {I - {m_{1}}} \right)}^{2}} + {\lambda_{2}}{{\left( {I - {m_{2}}} \right)}^{2}}}}. $$

(23)

At the meantime, we can obtain the following relation

$$ \begin{array}{@{}rcl@{}} \left| {{\phi^{*}}} \right| &=& \left| { - \frac{{{\lambda_{1}}{{\left( {I - {m_{1}}} \right)}^{2}} - {\lambda_{2}}{{\left( {I - {m_{2}}} \right)}^{2}}}}{{{\lambda_{1}}{{\left( {I - {m_{1}}} \right)}^{2}} + {\lambda_{2}}{{\left( {I - {m_{2}}} \right)}^{2}}}}} \right| \\ &\le & \frac{{{\lambda_{1}}\left| {{{\left( {I - {m_{1}}} \right)}^{2}}} \right| + {\lambda_{2}}\left| {{{\left( {I - {m_{2}}} \right)}^{2}}} \right|}}{{{\lambda_{1}}\left| {{{\left( {I - {m_{1}}} \right)}^{2}}} \right| + {\lambda_{2}}\left| {{{\left( {I - {m_{2}}} \right)}^{2}}} \right|}} \\ &= & 1. \end{array} $$

(24)

This completes the proof. □