A hybrid active contour model for ultrasound image segmentation

Abstract

Abundant noise, low contrast, intensity heterogeneity, shadows, and blurry boundaries exist in most medical images, especially for 2D ultrasound (US) image. In this paper, we propose a semiautomatic hybrid active contour model for 2D US image segmentation. The proposed method mainly uses a local bias correction function and probability score. It is well known that most region-based active contour models are based on the assumption of intensity homogeneity. It is very difficult to define a region descriptor for US images with intensity heterogeneity. Here, a bias field can account for the intensity heterogeneity of the US image. Therefore, the proposed local bias correction function is considered to integrate with respect to the neighborhood center of the US image. Besides, to segment complex ultrasound images more accurately, a probability score is constructed from the edge-based operator. Based on the estimation of the bias field and an interleaved process of probability score, minimization of the proposed energy functional is achieved. The proposed method is validated on synthetic images and real US images, with satisfactory performance in the presence of noise, intensity heterogeneity, and blurry boundaries.

Appendices

Appendix A. Modeling of the proposed energy term and its solution

The proposed energy functional term is defined as:

$$ E\left( \phi \right) = \int_{\varOmega } {p\left( s \right)F\left( y \right)} {\text{d}}y $$

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The parameters used here are defined:

For pixel $y \in \varOmega$, an energy functional is

$$ F\left( y \right) = \left| {I\left( y \right) - I_{f} \left( y \right)} \right|^{2} $$

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Assume the evolution curve $C$ is divided $\varOmega$ into two regions:

$$ \begin{aligned} & \Omega _{1} {\text{ = }}\left\{ {y \in \Omega :\phi \left( y \right) > 0} \right\}\;{\text{and}} \\ & \Omega _{2} = \left\{ {y \in \Omega :\phi \left( y \right) < 0} \right\},\;{\text{then}}\;{\text{area}}\;{\text{in}} \\ & \Omega _{1} :\;\;A_{1} = \int_{\Omega } {\Theta ^{r} \left( {x,y} \right)H\left( {\phi \left( y \right)} \right)} {\text{d}}y,\;\;{\text{and}} \\ & \Omega _{2} :\;\;A_{2} = \int_{\Omega } {\Theta ^{r} \left( {x,y} \right)\left( {1 - H\left( {\phi \left( y \right)} \right)} \right)} {\text{d}}y. \\ \end{aligned} $$

The sum of intensity is

$$ \begin{aligned} & \varOmega_{1} :\;\;S_{1} = \int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)I\left( y \right)H\left( {\phi \left( y \right)} \right)} {\text{d}}y ,\;\;{\text{and}} \\ & \varOmega_{2} :\;\;S_{2} = \int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)I\left( y \right)\left( {1 - H\left( {\phi \left( y \right)} \right)} \right)} {\text{d}}y. \\ \end{aligned} $$

The average intensity is

$$ \begin{aligned} & \varOmega_{1} :\;\;c_{p1} = \frac{{S_{1} }}{{A_{1} }} = \frac{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)I\left( y \right)H\left( {\phi \left( y \right)} \right)} {\text{d}}y}}{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)H\left( {\phi \left( y \right)} \right)} {\text{d}}y}} = \frac{{\varTheta^{r} \left( {x,y} \right) * \left[ {I\left( y \right)H\left( {\phi \left( y \right)} \right)} \right]}}{{\varTheta^{r} \left( {x,y} \right) * H\left( {\phi \left( y \right)} \right)}},\;\;{\text{and}} \\ & \varOmega_{2} :\;\;c_{p2} = \frac{{S_{2} }}{{A_{2} }} = \frac{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)I\left( y \right)\left( {1 - H\left( {\phi \left( y \right)} \right)} \right)} {\text{d}}y}}{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)\left( {1 - H\left( {\phi \left( y \right)} \right)} \right)} {\text{d}}y}} = \frac{{\varTheta^{r} \left( {x,y} \right) * \left[ {I\left( y \right)\left( {1 - H\left( {\phi \left( y \right)} \right)} \right)} \right]}}{{\varTheta^{r} \left( {x,y} \right) * \left( {1 - H\left( {\phi \left( y \right)} \right)} \right)}}. \\ \end{aligned} $$

The optimal level set $\phi$ of the proposed segmentation problem is transformed into the minimization of Eq. (17):

$$ \phi^{ * } { = }\arg \mathop {\hbox{min} }\limits_{\phi } \left\{ {E\left( \phi \right)} \right\}. $$

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Appendix B. Solution of the proposed energy term

To solve the above equation, the following partial derivatives are calculated firstly:

$$ \begin{aligned} & \frac{{\partial c_{p1} }}{\partial \phi } = \frac{{S_{1}^{\prime } A_{1} - S_{1} A_{1}^{\prime } }}{{A_{1}^{2} }} = \frac{{S_{1}^{\prime } - c_{p1} A_{1}^{\prime } }}{{A_{1} }} \\ & \quad = \frac{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)I\left( y \right)\delta \left( {\phi \left( y \right)} \right)} dy - c_{p1} \int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)\delta \left( {\phi \left( y \right)} \right)} {\text{d}}y}}{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)H\left( {\phi \left( y \right)} \right)} {\text{d}}y}} \\ & \quad = \frac{{\varTheta^{r} \left( {x,y} \right) * \left[ {I\left( y \right)\delta \left( {\phi \left( y \right)} \right)} \right] - c_{p1} \left[ {\varTheta^{r} \left( {x,y} \right) * \delta \left( {\phi \left( y \right)} \right)} \right]}}{{\varTheta^{r} \left( {x,y} \right) * H\left( {\phi \left( y \right)} \right)}} \\ \end{aligned} $$

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and

$$ \begin{aligned} & \frac{{\partial c_{p2} }}{\partial \phi } = \frac{{S_{2}^{\prime } A_{2} - S_{2} A_{2}^{\prime } }}{{A_{2}^{2} }} = \frac{{S_{2}^{\prime } - c_{p2} A_{2}^{\prime } }}{{A_{2} }} = \frac{{ - \int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)I\left( y \right)\delta \left( {\phi \left( y \right)} \right)} dy + c_{p2} \int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)\delta \left( {\phi \left( y \right)} \right)} {\text{d}}y}}{{\int_{\varOmega } {\varTheta^{r} \left( {x,y} \right)\left( {1 - H\left( {\phi \left( y \right)} \right)} \right)} {\text{d}}y}} \\ & \quad = - \frac{{\varTheta^{r} \left( {x,y} \right) * \left[ {I\left( y \right)\delta \left( {\phi \left( y \right)} \right)} \right] - c_{p2} \left[ {\varTheta^{r} \left( {x,y} \right) * \delta \left( {\phi \left( y \right)} \right)} \right]}}{{\varTheta^{r} \left( {x,y} \right) * \left( {{\mathbf{1}} - H\left( {\phi \left( y \right)} \right)} \right)}} \\ \end{aligned} $$

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where ${\mathbf{1}}$ denotes a constant matrix with value 1.

For convenience, we suppose

$$ \frac{{\partial c_{p1} }}{\partial \phi } = \frac{{\varTheta^{r} * \left( {I\delta } \right) - c_{p1} \left( {\varTheta^{r} * \delta } \right)}}{{\varTheta^{r} * H}}\;{\text{and}}\;\frac{{\partial c_{p2} }}{\partial \phi } = - \frac{{\varTheta^{r} * \left( {I\delta } \right) - c_{p2} \left( {\varTheta^{r} * \delta } \right)}}{{\varTheta^{r} * \left( {{\mathbf{1}} - H} \right)}}. $$

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So

$$ \frac{\partial F}{\partial \phi } = 2\left( {I\left( y \right) - I_{f} \left( y \right)} \right) \cdot \left( {\frac{{\partial c_{p1} }}{\partial \phi } - \frac{{\partial c_{p2} }}{\partial \phi }} \right) = 2\left( {I\left( y \right) - I_{f} \left( y \right)} \right) \cdot \left( {\frac{{\varTheta^{r} * \left( {I\delta } \right) - c_{p1} \left( {\varTheta^{r} * \delta } \right)}}{{\varTheta^{r} * H}} + \frac{{\varTheta^{r} * \left( {I\delta } \right) - c_{p2} \left( {\varTheta^{r} * \delta } \right)}}{{\varTheta^{r} * \left( {1 - H} \right)}}} \right). $$

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The final curve evolution is yielded:

$$ \frac{\partial \phi }{\partial t} = - \frac{\partial E}{\partial \phi } = - 2\rho \left( s \right)\left( {I\left( y \right) - I_{f} \left( y \right)} \right) \cdot \left( {\frac{{\varTheta^{r} * \left( {I\delta } \right) - c_{p1} \left( {\varTheta^{r} * \delta } \right)}}{{\varTheta^{r} * H}}{ + }\frac{{\varTheta^{r} * \left( {I\delta } \right) - c_{p2} \left( {\varTheta^{r} * \delta } \right)}}{{\varTheta^{r} * \left( {{\mathbf{1}} - H} \right)}}} \right). $$

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