Fuzzy energy based active contour model for multi-region image segmentation

Abstract

In this article, we present a new multi-phase pseudo 0.5 level set framework on fuzzy energy based active contour model to segment images into more than two regions. The proposed method is a generalization of fuzzy active contour based on 2-phase segmentation (object and background), developed by Krinidis and Chatzis. The proposed method needs only log₂n pseudo 0.5 level set functions for n phase piece-wise constant case. In piece-wise smooth case, only two pseudo 0.5 level set functions are sufficient to represent any partition based on ‘the four colo theorem. The proposed fuzzy active contour model can segment images into multiple regions instead of two regions (object and background) based on curve evolution. In this article, instead of solving the Euler-Lagrange equation, a multi-phase pseudo 0.5 level set based optimization is proposed to speed up the convergence. Finally, the proposed method is compared with state-of-the-art techniques on several images. Analysis (both qualitative and quantitative) of the results concludes that the proposed method segments images into multiple regions in a better way as compared to the existing ones.

Appendix A

Since, an image is discrete in nature, instead of integration, summation is considered here.

Let us assume four prototypes c₁, c₂, c₃ and c₄ correspond to four regions Ω₁ = {u₁ > 0.5,u₂ > 0.5}, Ω₂ = {u₁ > 0.5,u₂ < 0.5}, Ω₃ = {u₁ < 0.5,u₂ > 0.5} and Ω₄ = {u₁ < 0.5,u₂ < 0.5}, which approximate the image intensity within these regions. Thus, it can be written as

$$ \begin{array}{@{}rcl@{}} c_{1} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {u_{1} (x,y)} \right]}^{m} \left[ {u_{2} (x,y)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {u_{1} (x,y)} \right]}^{m} \left[ {u_{2} (x,y)} \right]^{m} }}, \end{array} $$

(15)

$$ \begin{array}{@{}rcl@{}} c_{2} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {u_{1} (x,y)} \right]}^{m} \left[ {1 - u_{2} (x,y)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {u_{1} (x,y)} \right]}^{m} \left[ {1 - u_{2} (x,y)} \right]^{m} }}, \end{array} $$

(16)

$$ \begin{array}{@{}rcl@{}} c_{3} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {1 - u_{1} (x,y)} \right]}^{m} \left[ {u_{2} (x,y)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {1 - u_{1} (x,y)} \right]}^{m} \left[ {u_{2} (x,y)} \right]^{m} }} \end{array} $$

(17)

and

$$ \begin{array}{@{}rcl@{}} c_{4} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {1 - u_{1} (x,y)} \right]}^{m} \left[ {1 - u_{2} (x,y)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {1 - u_{1} (x,y)} \right]}^{m} \left[ {1 - u_{2} (x,y)} \right]^{m} }}, \end{array} $$

(18)

where I(x,y) is the intensity value at pixel location (x,y), u₁(x,y) and u₂(x,y) are the degree of memberships of pixel (x,y) correspond to four regions, and m is the fuzzifier which determines the fuzziness present in the given image.

Therefore, total fuzzy energy for the whole image can be computed as

$$ \begin{array}{@{}rcl@{}} F &=& \lambda_{1} \sum\limits_{\varOmega} {\left[ {u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {u_{2} (x,y)} \right]^{m} \left\| {I(x,y) - c_{1} } \right\|^{2} \\ &&+ \lambda_{2} \sum\limits_{\varOmega} {\left[ {u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - u_{2} (x,y)} \right]^{m} \left\| {I(x,y) - c_{2} } \right\|^{2} \\ &&+ \lambda_{3} \sum\limits_{\varOmega} {\left[ {1 - u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {u_{2} (x,y)} \right]^{m} \left\| {I(x,y) - c_{3} } \right\|^{2} \\ &&+ \lambda_{4} \sum\limits_{\varOmega} {\left[ {1 - u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - u_{2} (x,y)} \right]^{m} \left\| {I(x,y) - c_{4} } \right\|^{2} \\ &=&F_{A} + F_{B} + F_{C} + F_{D}. \end{array} $$

(19)

Let us assume that a pixel (x₀,y₀) ∈ I with intensity I(x₀,y₀) and degree of memberships $u_{1}^{old}(x_{0},y_{0})$ and $u_{2}^{old}(x_{0},y_{0})$. If we change the degree of memberships of pixel (x₀,y₀) to the values $u_{1}^{new}(x_{0},y_{0})$ and $u_{2}^{new}(x_{0},y_{0})$, respectively using (9) and (10), then c₁, c₂, c₃ and c₄ will be changed to new values $\overline {c_{1}}$, $\overline {c_{2}}$, $\overline {c_{3}}$ and $\overline {c_{4}}$, respectively. The new values of c₁, c₂, c₃ and c₄ are calculated as

$$ \begin{array}{@{}rcl@{}} \bar c_{1} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {\bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {\bar u_{2} \left( {x,y} \right)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {\bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {\bar u_{2} \left( {x,y} \right)} \right]^{m} }}\ &=& \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {u_{2} \left( {x,y} \right)} \right]^{m} + I(x_{0},y_{0}) b_{1} }} {{\sum\limits_{\varOmega} {\left[ {u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {u_{2} \left( {x,y} \right)} \right]^{m} + b_{1}}}\\ &=& \frac{{c_{1} a_{1} + I(x_{0},y_{0}) b_{1} }}{{a_{1} + b_{1} }} = c_{1} + s_{1} \left\| {I(x_{0},y_{0}) - c_{1} } \right\|, \end{array} $$

(20)

where

$$ \begin{array}{@{}rcl@{}} a_{1} &=& \sum\limits_{\varOmega} {\left[ {u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {u_{2} \left( {x,y} \right)} \right]^{m},\\ b_{1} &=& \left\{ {\left[ {u_{1}^{new}(x_{0},y_{0}) } \right]^{m} - \left[ {u_{1}^{old}(x_{0},y_{0}) } \right]^{m} } \right\} \left\{ {\left[ {u_{2}^{new}(x_{0},y_{0}) } \right]^{m} - \left[ {u_{2}^{old}(x_{0},y_{0}) } \right]^{m} } \right\}\\ \text{and}\ s_{1} = \frac{{b_{1} }}{{a_{1} + b_{1} }}. \end{array} $$

Similarly,

$$ \begin{array}{@{}rcl@{}} \bar c_{2} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {\bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - \bar u_{2} \left( {x,y} \right)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {\bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - \bar u_{2} \left( {x,y} \right)} \right]^{m} }} = c_{2} + s_{2} \left\| {I(x_{0},y_{0}) - c_{2} } \right\| \end{array} $$

(21)

where

$$ \begin{array}{@{}rcl@{}} a_{2} &=& \sum\limits_{\varOmega} {\left[ {u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - u_{2} \left( {x,y} \right)} \right]^{m},\\ b_{2} &=& \left\{ {\left[ {u_{1}^{new}(x_{0},y_{0}) } \right]^{m} - \left[ {u_{1}^{old}(x_{0},y_{0}) } \right]^{m} } \right\} \left\{\left[{1 - u_{2}^{new}(x_{0},y_{0}) } \right]^{m}\right.\\ && \left. - \left[ {1 - u_{2}^{old}(x_{0},y_{0}) } \right]^{m} \right\}\\ \text{and}\ s_{2} = \frac{{b_{2}}}{{a_{2} + b_{2}}}. \end{array} $$

$$ \begin{array}{@{}rcl@{}} \bar c_{3} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {1 - \bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {\bar u_{2} \left( {x,y} \right)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {1 - \bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {\bar u_{2} \left( {x,y} \right)} \right]^{m} }} = c_{3} + s_{3} \left\| {I(x_{0},y_{0}) - c_{3} } \right\|, \end{array} $$

(22)

where

$$ \begin{array}{@{}rcl@{}} a_{3} &=& \sum\limits_{\varOmega} {\left[ {1 - u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {u_{2} \left( {x,y} \right)} \right]^{m},\\ b_{3} &=& \left\{ {\left[ {1 - u_{1}^{new}(x_{0},y_{0}) } \right]^{m} - \left[ {1 - u_{1}^{old}(x_{0},y_{0}) } \right]^{m} } \right\} \left\{\left[ {u_{2}^{new}(x_{0},y_{0}) } \right]^{m}\right.\\ && \left.- \left[ {u_{2}^{old}(x_{0},y_{0}) } \right]^{m}\right\}\\ \text{and}\ s_{3} = \frac{{b_{3} }}{{a_{3} + b_{3} }}. \end{array} $$

$$ \begin{array}{@{}rcl@{}} \bar c_{4} = \frac{{\sum\limits_{\varOmega} {I\left( {x,y} \right)\left[ {1 - \bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - \bar u_{2} \left( {x,y} \right)} \right]^{m} }}{{\sum\limits_{\varOmega} {\left[ {1 - \bar u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - \bar u_{2} \left( {x,y} \right)} \right]^{m} }} = c_{4} + s_{4} \left\| {I(x_{0},y_{0}) - c_{4} } \right\|, \end{array} $$

(23)

where

$$ \begin{array}{@{}rcl@{}} a_{4} &=& \sum\limits_{\varOmega} {\left[ {1 - u_{1} \left( {x,y} \right)} \right]}^{m} \left[ {1 - u_{2} \left( {x,y} \right)} \right]^{m},\\ b_{4} &=& \left\{ {\left[ {1 - u_{1}^{new}(x_{0},y_{0}) } \right]^{m} - \left[ {1 - u_{1}^{old}(x_{0},y_{0}) } \right]^{m} } \right\} \left\{\left[ {1 - u_{2}^{new}(x_{0},y_{0}) } \right]^{m}\right.\\ && \left.- \left[ {1 - u_{2}^{old}(x_{0},y_{0}) } \right]^{m} \right\}\\ \text{and}\ s_{4} = \frac{{b_{4} }}{{a_{4} + b_{4} }}. \end{array} $$

From (19), it is seen that if degree of memberships u₁(x,y) and u₂(x,y) are changed, then the energy of the model will also be changed. If F denotes the old energy and $\overline {F}$ denotes the new energy due to changing of degree of memberships of the point (x₀,y₀), then

$$ \begin{array}{@{}rcl@{}} \overline F = \overline F_{A} + \overline F_{B} + \overline F_{C} + \overline F_{D}, \end{array} $$

(24)

with

$$ \begin{array}{@{}rcl@{}} \bar F_{A} &=& \lambda_{1} \sum\limits_{\varOmega} {\left\| {I(x,y) - \bar c_{1} } \right\|}^{2} \left[ {\bar u_{1} (x,y)} \right]^{m} \left[ {\bar u_{2} (x,y)} \right]^{m}\\ &&+ \lambda_{1} \sum\limits_{\varOmega} {\left\| {I(x,y) - c_{1} } \right\|}^{2} \left[ {u_{1} (x,y)} \right]^{m} \left[ {u_{2} (x,y)} \right]^{m} + \\ &&\left\{ {\left[ {u_{1}^{new}(x_{0},y_{0}) } \right]^{m} - \left[ {u_{1}^{old}(x_{0},y_{0}) } \right]^{m} } \right\} \left\{ \left[ {u_{2}^{new}(x_{0},y_{0}) } \right]^{m}\right.\\ &&\left.- \left[ {u_{2}^{old}(x_{0},y_{0}) } \right]^{m} \right\}\left\| {I_(x_{0},y_{0}) - c_{1} } \right\|^{2}\\ &=& F_{A} + \lambda_{1} a_{1} s_{1} \left\| {I(x_{0},y_{0}) - c_{1} } \right\|^{2} . \end{array} $$

(25)

Similarly,

$$ \begin{array}{@{}rcl@{}} \bar F_{B} &=& \lambda_{2} \sum\limits_{\varOmega} {\left\| {I(x,y) - \bar c_{2} } \right\|}^{2} \left[ {\bar u_{1} (x,y)} \right]^{m} \left[ {1 - \bar u_{2} (x,y)} \right]^{m}\\ &=& F_{B} + \lambda_{2} a_{2} s_{2} \left\| {I(x_{0},y_{0}) - c_{2} } \right\|^{2}. \end{array} $$

(26)

$$ \begin{array}{@{}rcl@{}} \bar F_{C} &=& \lambda_{3} \sum\limits_{\varOmega} {\left\| {I(x,y) - \bar c_{3} } \right\|}^{2} \left[ {1 - \bar u_{1} (x,y)} \right]^{m} \left[ {\bar u_{2} (x,y)} \right]^{m} \\ &=& F_{C} + \lambda_{3} a_{3} s_{3} \left\| {I(x_{0},y_{0}) - c_{3} } \right\|^{2}. \end{array} $$

(27)

$$ \begin{array}{@{}rcl@{}} \bar F_{D} &=& \lambda_{4} \sum\limits_{\varOmega} {\left\| {I(x,y) - \bar c_{4} } \right\|}^{2} \left[ {1 - \bar u_{1} (x,y)} \right]^{m} \left[ {1 - \bar u_{2} (x,y)} \right]^{m} \\ &=& F_{D} + \lambda_{4} a_{4} s_{4} \left\| {I(x_{0},y_{0}) - c_{4} } \right\|^{2}. \end{array} $$

(28)

Then

$$ \begin{array}{@{}rcl@{}} \bar F &=& F_{A} + \lambda_{1} a_{1} s_{1} \left\| {I(x_{0},y_{0}) - c_{1} } \right\|^{2} + F_{B} + \lambda_{2} a_{2} s_{2} \left\| {I(x_{0},y_{0}) - c_{2} } \right\|^{2} \\ &&+ F_{C} + \lambda_{3} a_{3} s_{3} \left\| {I(x_{0},y_{0}) - c_{3} } \right\|^{2} + F_{D} + \lambda_{4} a_{4} s_{4} \left\| {I(x_{0},y_{0}) - c_{4} } \right\|^{2} \\ &=& F + \lambda_{1} a_{1} s_{1} \left\| {I(x_{0},y_{0}) - c_{1} } \right\|^{2} + \lambda_{2} a_{2} s_{2} \left\| {I(x_{0},y_{0}) - c_{2} } \right\|^{2} \\ &&+ \lambda_{3} a_{3} s_{3} \left\| {I(x_{0},y_{0}) - c_{3} } \right\|^{2} + \lambda_{4} a_{4} s_{4} \left\| {I(x_{0},y_{0}) - c_{4} } \right\|^{2} \\ {\varDelta} F &=& \lambda_{1} a_{1} s_{1} \left\| {I(x_{0},y_{0}) - c_{1} } \right\|^{2} + \lambda_{2} a_{2} s_{2} \left\| {I(x_{0},y_{0}) - c_{2} } \right\|^{2} \\ &&+ \lambda_{3} a_{3} s_{3} \left\| {I(x_{0},y_{0}) - c_{3} } \right\|^{2} + \lambda_{4} a_{4} s_{4} \left\| {I(x_{0},y_{0}) - c_{4} } \right\|^{2} \end{array} $$

(29)