在這篇論文中我們提出了一個利用多個等位階函數(multiphase level set)與多個動態形狀模型的方法來分割多個有特定形狀的物體，這個方法主要在三個方面改進原本Chan和Vese所提出的方法。第一個是利用多個等位階函數來同時分割多個物體，第二個就是利用形狀模型來輔助分割，第三個是利用動態形狀模型來解決物體形狀差異的問題。
由於斷層掃瞄影像本身的特性，使得在斷層掃瞄影上腎臟分割面臨到下面這些問題。第一個是，腎臟的邊界是模糊的甚至與肝臟相連。第二個是，腎臟中有一個區域其灰階值與背景相似，使得他難以被一起分割出來。第三個是，左右邊的腎臟通常會有不同的大小與方向，因此難以用一個固定的形狀模型來表達它。我們提出的方法可以同時解決這三個問題。
為了評估分割結果的好壞，我們將我們提出的方法應用在不同斷層掃描影像的腎臟分割上，並將結果與Chan-Vese方法和另一個方法的結果做比較。最後證明我們的方法能比另外兩個方法更完整的分割出腎臟。

In this thesis, a multiphase level set method with multiple dynamic shape models is proposed to segment multiple objects with specific shape. Comparing with the original Chan-Vese model, three changes are made. The first is using multiphase level set to simultaneously segment multiple objects. The second is using shape model to help the segmentation. The third is using dynamic shape model to fit objects which are similar to the given shape.
Due to the characteristic of CT images, the kidney segmentation on them needs to face several challenges. First, the edges of the kidneys are blurred and may even connect to the liver. Second, the calyx region within the kidney is difficult to be segmented due to its similar gray level with the background. Third, the left kidney and right kidney reveal different orientation and size which are difficult to model. Our approach can simultaneously apply the shape model to conquer the above problems, while also resolve the difficulty in applying the shape model.
For evaluation, several CT image series are used for the test. The proposed method is compared with the Chan-Vese model and another model to prove that the proposed method can achieve better performance.

[1] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: active contour models,” Intl. Journal of Computer Vision, vol. 1, pp. 321-331, 1988
[2] V Caselles, R Kimmel, and G Sapiro, “Geodesic active contours,” Intl. Journal of Computer Vision, vol. 22, no. 1, pp. 61-79, 1997
[3] N. Paragios and R. Deriche, “Geodesic active contours and level sets for the detection and tracking of moving objects,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 22, pp. 266-280, 2000
[4] Y. Chen, H.D. Tagare, S. Thiruvenkadam, F Huang, and D. Wilson, “Using prior shapes in geometric active contours in a variational framework,” Intl. Journal of Computer Vision, vol. 50, no. 3, pp. 315-328, 2002
[5] D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Communications on Pure and Appl. Math., vol. 42, pp. 577-684, 1989
[6] L.A. Vese, “Multiphase object detection and image segmentation,” Geometric Level Set Methods in Imaging, Vision and Graphics, pp. 175-194, 2003
[7] J. Piovano, M. Rousson and T. Papadopoulo, “Efficient segmentation of piecewise smooth images,” Scale Space and Variational Methods in Computer Vision, pp. 709-720, 2007
[8] Y. Zhang and L. Ge, “Edge preserving regularization for the piecewise smooth Mumford-Shah model,” The Intl. Conference on Computers as a Tool, pp. 282-288, Sept. 2007
[9] T.F. Chan and L.A. Vese, “Active contours without edges,” IEEE Transactions on Image Processing, vol. 10, no. 2, pp. 266-277, February 2001
[10] R. Xia, W. Liu, Y. Wang, and X. Wu, “Fast initialization of level set method and an improvement to Chan-Vese model,” The Fourth International Conference on Computer and Information Technology, pp. 18-23, 2004
[11] L.J. Zhang, X.J. Wu, and Z. Sheng, “A fast image segmentation approach based on level set method,” Intl. Conference on Signal Processing, vol. 2, 2006
[12] Y.S. Sun, P. Li, and B.Y. Wu, “An improved approach to image segmentation based on Mumford-Shah model,” International Conference on Machine Learning and Cybernetics, pp. 3996-4001, 2006
[13] D. Cremers, T. Kohlberger, and C. Schnorr, “Nonlinear shape statistics in Mumford-Shah based segmentation,” European Conf. on Computer Vision, vol. 2351, pp. 516-518, 2002
[14] D. Cremers, N. Sochen, and C. Schnorr, “Towards recognition-based variational segmentation using shape priors and dynamic labeling,” Intl. Conf. on Scale-Space Theories in Computer Vision, pp. 388-400, June 2003
[15] T. Riklin-Raviv, N. Kiryati, and N. Sochen, “Unlevel-sets: geometry and prior-based segmentation, “European Conf. on Computer Vision, pp. 50-61, 2004
[16] T.F. Chan and W. Zhu, “Level set based shape prior segmentation,” IEEE Computer Society Conf. on CVPR, vol. 2, pp. 1164-1170, June 2005
[17] K. Cheng, L. Gu, and J. Xu, “A novel shape prior based level set method for liver segmentation from MR images,” International Conference on Technology and Applications in Biomedicine, pp. 144-147, May 2008
[18] T.F. Chan and L.A. Vese, “A level set algorithm for minimizing the Mumford-Shah functional in image processing,” Workshop on Variational and Level Set Methods in Computer Vision, pp. 161-168, 2001
[19] L.A. Vese and T.F. Chan, “A multiphase level set framework for image segmentation using the mumford and shah model,” Intl. Journal of Computer Vision, vol. 50, no. 3, pp. 271-293, 2002
[20] J. Lie, M. Lysaker, and X.C. Tai “A binary level set model and some applications to Mumford-Shah image segmentation,” IEEE Transactions on Image Processing, vol. 15, pp. 1171-1181, 2006
[21] P. Yan, A.A. Kassim, W. Shen, and M. Shah, “Modeling interaction for segmentation of neighboring structures,” IEEE Transactions on Information Technology in Biomedicine, vol. 13, no.2, March 2009
[22] S. Kim, “A hybrid level set approach for efficient and reliable image segmentation,” Proceeding of the Fifth IEEE International Symposium on Signal Processing and information Technology, pp. 743-748, 2005
[23] S. Kim and H. Lim, “A hybrid level set segmentation for medical imagery,” IEEE Nuclear Science Symposium Conf. Record, vol. 3, pp. 1790-1794, 2005