﻿ 数值求解优化问题在活动轮廓模型上的应用
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1. 江南大学理学院, 江苏无锡 214122;
2. 江南大学物联网工程学院, 江苏无锡 214122

Application of a numerical solution to the optimization problem in the active contour model
LIAO Cuicui1 , LI Min2, LIANG Jiuzhen2, LIAO Zuhua1
1. Department of Information and Computaion Science, College of Science, Jiangnan University, Wuxi 214122, China;
2. Institute of Intelligent Systems and Network Computing, School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, China
Abstract: In this paper, we analyze numerical optimization procedures and propose high-order numerical methods to deal with the problems of slow convergence and low efficiency in the active contour model. First, we analyze the global information region-based active contour Chan-Vese(CV) model, the local information region-based local binary fitting(LBF) model, and the local image fitting(LIF) model. Then, we compare and analyze image segment results utilizing second-and third-order explicit Runge-Kutta methods, and the standard explicit Euler method. We also analyze the segment results of different sliding coefficient parameters and time steps of the LBF model. The experimental results for the intensity inhomogeneities and common images show that the proposed numerical methods can reduce the number of iterations, and improve convergence accuracy and computational efficiency. In addition, for different coefficients and time steps, the proposed methods yield greater stability.
Key words: CV model     LBF model     Runge-Kutta method     numerical optimization procedure     image segment

CV模型中水平集函数增加了其计算量，存在着耗时长、效率低以及易于陷入局部极小值等问题。 针对模型的改进，李春明等[4]提出的处理强度不均匀图像的局部二元拟合模型；此外，潘改等[5]结合了LBF模型和GAC活动轮廓模型这2种方法，能有效地处理弱边界图像的分割；张开华等[6]提出的局部图像拟合模型，基于高斯滤波的变分水平集方法来处理强度不均匀的图像分割；刘瑞娟等[7]提出的融合局部和全局信息的活动轮廓模型方法；王小峰等[8]提出了一种局部CV活动模型，将局部图像信息融入到模型中。对于数值求解方面的改进，如牛顿方法、与置信域相结合的一般牛顿方法[9]以及在此基础上的改进牛顿方法。此外还有二阶、三阶Runge-Kutta方法[10]在CV模型上的应用。这些算法都能有效地增加了模型的求解速度，并针对不同的问题都有所改进。目前，CV模型被广泛地应用于医学图像分割，并具有很好的发展前景。

1 活动轮廓模型 1.1 基于全局区域信息的CV模型

1.2 基于局部区域信息的LBF模型及LIF模型

σ的取值过大时，计算量增大；而σ的取值过小时，获取局部区域灰度变化信息的能力降低，一般σ取5。

2 数值求解方法

2.1 Euler 方法

2.2 Runge-Kutta 方法

Runge-Kutta方法是求解非线性微分方程的重要数值迭代方法，是Euler方法的一种推广。它提高了计算收敛精度，缩小截断误差，并且具有更好的稳定性。Runge-Kutta方法的导出基于Taylor展开，对所求问题的解具有较好的光滑度，可以使近似公式达到所需要的阶数，并且能够有效提高方法的精度。计算时使用Euler-Lagrange函数在若干点上函数值的线性组合来构造近似公式，因此会在时间复杂度上造成线性的倍数增加[14, 15]。本文采用的显式二阶三阶Runge-Kutta方法并不会在计算复杂度上造成过多的影响。

3 实验与结果分析

3.1 LBF模型的参数讨论

3.1.1 对光滑系数项nu值的讨论

 图 1 显式Euler方法的图像分割结果Fig. 1 Segment results by explicit Euler method
 图 2 RK-2方法的图像分割结果Fig. 2 Segment results by RK-2 method
 图 3 RK-3方法的图像分割结果Fig. 3 Segment results by RK-3 method
3.1.2 对光滑系数项nu值的讨论

 图 4 显式Euler方法的图像分割结果Fig. 4 Segment results by explicit Euler method
 图 5 RK-2方法的图像分割结果Fig. 5 Segment results by RK-2 method
 图 6 RK-3方法的图像分割结果Fig. 6 Segment results by RK-3 method
3.2 不同灰度图像分割实验结果分析

3.2.1 局部信息拟合项对强度不均匀图像的分析

 图 7 显式Euler方法的图像分割结果Fig. 7 Segment results by explicit Euler method
 图 8 RK-2方法的图像分割结果Fig. 8 Segment results by RK-2 method
 图 9 RK-3方法的图像分割结果Fig. 9 Segment results by RK-3 method

 方法 图3(a)119×78 图3 (b)111×110 图3 (c)103×131 图3 (d)128×128 Euler 0.142 0.435 0.556 0.122 RK-2 0.194 0.731 1.111 0.203 RK-3 0.312 1.100 1.613 0.310
3.2.2 全局信息拟合项对一般灰度图像的分析

 图 10 显式Euler方法的图像分割结果Fig. 10 Segment results by explicit Euler method
 图 11 RK-2方法的图像分割结果Fig. 11 Segment results by RK-2 method
 图 12 RK-3方法的图像分割结果Fig. 12 Segment results by RK-3 method

 方法 图4(e)84×84 图4 (f)128×128 图4(g)256×256 Euler 0.030 0.048 0.302 RK-2 0.034 0.039 0.444 RK-3 0.038 0.041 0.602
4 结束语

 [1] VESE L A, CHAN T F. A multiphase level set framework for image segmentation using the Mumford and Shah model[J]. International Journal of Computer Vision, 2002, 50(3):271-293. [2] CHAN T F, VESE L A. Active contours without edges[J]. IEEE Transactions on Image Processing, 2001, 10(2):266-277. [3] COHEN L D, COHEN I. Finite-element methods for active contour models and balloons for 2-D and 3-D images[J]. IEEE Transactionson on Pattern Analysis Machine Intelligence, 1993, 15(11):1131-1147. [4] LI Chunming, KAO C Y, GORE J C, et al. Implicit active contours driven by local binary fitting energy[J]. IEEE Conference on Computer Vision and Pattern Recognition, 2007:1-7. [5] 潘改, 高立群, 张萍. 基于LBF方法的测地线活动轮廓模型[J]. 模式识别与人工智能, 2013, 26(12):1179-1184. PAN Gai, GAO Liqun, ZHANG Ping. Geodesic active contour based on LBF method[J]. Pattern Recognition and Aitificial Intelligence, 2013, 26(12):1179-1184. [6] ZHANG Kaihua, SONG Huihui, ZHANG Lei. Active contours driven by local image fitting energy[J]. Pattern Recognition, 2010, 43(4):1199-1206. [7] 刘瑞娟, 何传江, 原野. 融合局部和全局图像信息的活动轮廓模型[J]. 计算机辅助设计与图形学报, 2012, 24(3):364-371. LIU Ruijuan, HE Chuanjiang, YUAN Ye. Active contours driven by local and global image fitting energy[J]. Journal of Computer-Aided Design & Computer Graphics, 2012, 24(3):364-371. [8] WANG Xiaofeng, HUANG Deshuang, XU Huan. An efficient local Chan-Vese model for image segmentation[J]. Pattern Recognition, 2010, 43(3):603-618. [9] BAR L, SAPIRO G. Generalized Newton-type methods for energy formulations in image processing[J]. SIAM Journal on Imaging Sciences, 2009, 2(2):508-531. [10] SCHEUERMANN B, ROSENHAHN B. Analysis of numerical methods for level set based image segmentation[J]. Advances in Visual Computing, 2009, 5876:196-207. [11] LI Chunming, KAO C Y, GORE J C, et al. Minimization of region-scalable fitting energy for image segmentation[J]. IEEE Transcations on Image Processing, 2008, 17(10):1940-1949. [12] GE Feng, WANG Song, LIU Tiecheng. New benchmark for image segmentation evaluation[J]. Journal of Electronic Imaging, 2007, 16(3):1010-1016. [13] 任守纲, 马超, 徐焕良. 基于改进主动轮廓模型的图像分割方法研究[J]. 计算机科学, 2013, 40(7):289-292, 296. REN Shougang, MA Chao, XU Huanliang. Improved skeleton extracton algorithm based active contour model research[J]. Computer Science, 2013, 40(7):289-292, 296. [14] CORMEN T H, LEISER C E, RIVEST R L, et al. Introduction to Algorithms[M]. 3rd ed. Cambridge, Mass:MIT press, 2009:350-400. [15] JOHNSON M L. Essential numerical computer methods[M]. Burlington, MA:Academic Press, 2010:230-275.
DOI: 10.11992/tis.201507037

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#### 文章信息

LIAO Cuicui, LI Min, LIANG Jiuzhen, LIAO Zuhua

Application of a numerical solution to the optimization problem in the active contour model

CAAI Transactions on Intelligent Systems, 2015, 10(6): 886-892.
DOI: 10.11992/tis.201507037