﻿ 基于变分水平集理论的水下图像分割方法
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 应用科技  2019, Vol. 46 Issue (2): 53-58  DOI: 10.11991/yykj.201807004 0

引用本文

XI Zhihong, ZHAO Chunmei. An underwater image segmentation method based on the variational level set theory[J]. Applied Science and Technology, 2019, 46(2), 53-58. DOI: 10.11991/yykj.201807004.

文章历史

An underwater image segmentation method based on the variational level set theory
XI Zhihong, ZHAO Chunmei
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: In order to solve the problem of underwater image segmentation, the segmentation methods for a designated target and a multi-grayscale target are proposed respectively based on the Li Chunming model (Lee’s model) and the Chan−Vese (C−V) model. For the segmentation method of a specified grayscale object, a small range of distance restriction item is added on the basis of the C−V model, making it have locality characteristic, and segment the desired target from the multi-gray target. The multi-gray target segmentation method is to join the edge detection function as the item of internal energy based on Li Chunming method. The results of multi-gray target segmentation are good, and the anti-noise characteristic is better. Finally, the effectiveness of the proposed two methods is verified.
Keywords: underwater image segmentation    active contour model    Lee's model    C−V model    variational method    horizontal set theory    grayscale image    underwater images

1 水下图像分割

2 2种典型水平集图像分割方法

2.1 C−V模型

Mumford−Shah（M−S）模型是基于能量最小化的分割模型。对于给定的目标图像 ${I_0}\left( {x, y} \right)$ ，根据轮廓曲线 $C$ 不断对其进行分段光滑逼近，最终得到分割图像 $I\left( {x, y} \right)$ 。Chan和Vese根据水平集方法提出了经典的C−V模型去简化求解M−S模型。假设待分割的目标是由闭合曲线内外两同质区域组成，则有如下的能量函数：

 \begin{aligned} {E^{CV}}({c_0}, {c_1}, C) = &\mu L(C) + \nu A{(}i(C){) +} {\lambda _1}\int\limits_{i\left( C \right)} {{{\left| {I(x, y) - {c_0}} \right|}^2}{\rm{d}}x{\rm{d}}y} + \\ &{\lambda _2}\int\limits_{o\left( C \right)} {{{\left| {I(x, y) - {c_b}} \right|}^2}{\rm{d}}x{\rm{d}}y} \\ \end{aligned}

 \begin{aligned} {E^{CV}}({c_0}, {c_1}, \phi ) =& \mu \int\limits_\varOmega {{\delta _\varepsilon }(\phi )} \left| {\nabla \phi } \right|{\rm{d}}x{\rm{d}}y + \nu \int\limits_\varOmega {{H_\varepsilon }(\phi )} {\rm{d}}x{\rm{d}}y + \\ &{\lambda _1}\int\limits_{i(C)} {{{\left| {I(x, y) - {c_0}} \right|}^2}{H_\varepsilon }(\phi ){\rm{d}}x{\rm{d}}y} + \\ & {\lambda _2}\int\limits_{o(C)} {{{\left| {I(x, y) - {c_b}} \right|}^2}{(}1 - {H_\varepsilon }(\phi ){\rm{)d}}x{\rm{d}}y} \\ \end{aligned}

2.2 李纯明模型

 \begin{aligned} E = &\mu {E_p}(\phi ) + {E_{{\rm{in}}}}(\phi ) = \mu {E_p}(\phi ) + \lambda {L_g}(\phi ) + v{A_g}(\phi ) = \\ &\mu \iint\limits_\varOmega {\frac{1}{2}}{\left( {\left| {\nabla \phi } \right| - 1} \right)^2}{\rm{d}}x{\rm{d}}y + \lambda \iint\limits_\varOmega g\delta (\phi )\left| {\nabla \phi} \right|{\rm{d}}x{\rm{d}}y + \\ & v\iint\limits_\varOmega gH(\phi ){\rm{d}}x{\rm{d}}y \\ \end{aligned}

 $g\left( {\left| \varGamma \right|} \right) = \frac{1}{{1 + \beta {{\left| \varGamma \right|}^n}}}$

3 基于变分水平集理论的水下图像分割方法 3.1 指定目标分割算法

 \begin{aligned} E\left( {{S_1}, {S_2}} \right) =& \mu L\left( {{S_2}} \right) + \nu A\left( {i\left( {{S_2}} \right)} \right) + {\lambda _1}{\int\limits_{i\left( {{S_2}} \right)} {\left| {I\left( {{S_2}} \right) - {c_1}} \right|} ^2}{\rm d}\left( {{S_2}} \right) + \\ &{\lambda _1}{\int\limits_{d\left( {{S_2}} \right)} {\left| {I\left( {{S_2}} \right) - {c_1}} \right|} ^2}{\rm d}\left( {{S_2}} \right) \\ \end{aligned}

 \left\{\begin{aligned} & {S_1} = \left\{ {{\rm SDF}|\phi \left( {x, y} \right) = \pm d} \right\}\\ & {S_2} = \left\{ {\rm idx|\left| {S{\rm DF}} \right| \leqslant 1.2} \right\} \end{aligned}\right.

 \begin{aligned} \frac{{\partial E}}{{\partial \phi }} =& - \delta \left( \phi \right)[v{\rm{div}}\left( {\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}}} \right) - \mu - {\lambda _1}{\left( {I\left( {{\rm{idx}}} \right) - {c_1}} \right)^2} + \\ &{\lambda _2}{\left( {I\left( {{\rm{idx}}} \right) - {c_2}} \right)^2}] \\ \end{aligned}

 \begin{aligned} \frac{{\partial \phi }}{{\partial t}} =& \delta \left( \phi \right)[v{\rm{div}}\left( {\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}}} \right) - \mu - {\lambda _1}{\left( {I\left( {{\rm{idx}}} \right) - {c_1}} \right)^2} + \\ &{\lambda _2}{\left( {I\left( {{\rm{idx}}} \right) - {c_2}} \right)^2}] \\ \end{aligned}

3.2 多灰度目标分割算法

 $\begin{split} E = &\mu {E_p}\left( C \right) \!+\! {E_{{\rm{in}}}}\left( C \right) \!+\! {E_{{\rm{out}}}}\left( C \right) \!=\! \mu {E_p}\left( C \right) \!+\! \lambda {L_g}\left( C \right) + \\ & v{A_g}\left( C \right)\!+\! {\lambda _1}\iint\limits_{i\left( C \right)} {{{\left| {g \!-\! {c_0}} \right|}^2}{\rm{d}}x{\rm{d}}y} \!+\! {\lambda _2}\iint\limits_{o\left( C \right)} {{{\left| {g \!-\! {c_1}} \right|}^2}{\rm{d}}x{\rm{d}}y} \end{split}$ (1)

 ${c_1} = \frac{{\iint\limits_\varOmega {g{H_\varepsilon }\left( \phi \right){\rm{d}}x{\rm{d}}y}}}{{\iint\limits_\varOmega {{H_\varepsilon }\left( \phi \right){\rm{d}}x{\rm{d}}y}}}$
 ${c_2} = \frac{{\iint\limits_\varOmega {g\left( {1 - {H_\varepsilon }\left( \phi \right)} \right){\rm{d}}x{\rm{d}}y}}}{{\iint\limits_\varOmega {\left( {1 - {H_\varepsilon }\left( \phi \right)} \right){\rm{d}}x{\rm{d}}y}}}$
 $g\left( {\left| \varGamma \right|} \right) = \frac{1}{{1 + \beta {{\left| \varGamma \right|}^n}}} = \frac{1}{{1 + \beta {{\left| {\nabla {G_\sigma }I\left( {x, y} \right)} \right|}^n}}}$

 ${G_\sigma }\left( {x, y} \right) = \frac{1}{{\sqrt {2 \text{π} } \sigma }}\exp \left\{ { - \frac{{{x^2} + {y^2}}}{{2{\sigma ^2}}}} \right\}$
 $\phi \left( {x, y, 0} \right) = {\phi _0}\left( {x, y} \right) = \left\{\!\!\! \begin{array}{l} c, \quad \left( {x, y} \right) \in i\left( c \right) \\ 0, \quad \left( {x, y} \right) \in on\left( c \right) \\ - c, \quad \!\!\!\! \left( {x, y} \right) \in o\left( c \right) \end{array} \right.$
 $\delta \left( x \right) = \left\{ \!\!\! \begin{array}{l} 0, \quad \left| x \right| > \varepsilon \\ \displaystyle\frac{1}{{2\varepsilon }}\left[ {1 + \cos \left( {\frac{{\text{π} x}}{\varepsilon }} \right)} \right], \quad \left| x \right| \leqslant \varepsilon \end{array} \right.$

 \begin{aligned} \frac{{\partial \phi }}{{\partial t}}=&\mu \left[ {\Delta \phi - {\rm{div}}\left( {\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}}} \right)} \right] + \lambda \delta \left( \phi \right){\rm{div}}\left( {g\frac{{\nabla \phi }}{{\left| {\nabla \phi } \right|}}} \right) + \\ & vg\delta \left( \phi \right) - \delta \left( \phi \right){\lambda _1}\left( {{{\left| {g - {c_1}} \right|}^2}} \right) + {\lambda _2}\delta \left( \phi \right)\left( {{{\left| {g - {c_2}} \right|}^2}} \right) \\ \end{aligned}