影像分割往往在醫學影像處理問題上扮演十分重要的腳色。在過往利用能量泛函，並經由變分學推導出偏微分方程以進行最佳化過程的方法中，以Mumford和 Shah提出利用光滑函數，輔以邊界函數進行分割影像; 以及Chan與Vese利用水平集分割物件，為二主要方法。
在此篇論文中，將會介紹上述兩個模型且比較其優劣，並提出我們的方法：利用水平集以及像素與區域之間的歧異度所建立的模型，並且展示成果。
數值結果顯示，我們的方法在邊界上的表現比Chan和Vese方法來得好。總結而論，我們的方法有以下兩個優點： 較不受雜訊影響且刻劃更清楚的邊界。此方法不僅可以應用於醫學影像，也可以應用於一般影像。

Image segmentation plays an essential role in medical image processing. There are several methods to accomplish segmentation: one is proposed by Mumford and Shah that utilized the smoother and boundary detector, another was proposed by Chan and Vese that took level set to separate different regions. In this dissertation, my professor and I have proposed a way to distinguish the area of interested by minimizing the dissimilarity in each region. Moreover, we put a smoother into the functional to reduce the noise. The numerical results show that ours can characterize the boundary better than Chan and Vese method. In summary, the advantages of our approach are robust against noise, and give a more clear edge. Besides, it can not only apply to medical images but the general images adequately.

[1] Luigi Ambrosio, Nicola Fusco, and John E. Hutchinson. Higher integrability of
the gradient and dimension of the singular set for minimisers of the mumford–shah
functional. Calculus of Variations and Partial Differential Equations, 16(2):187–
215, 2003.
[2] Luigi Ambrosio and Vincenzo Maria Tortorelli. Approximation of functional de-
pending on jumps by elliptic functional via t-convergence. Communications on
Pure and Applied Mathematics, 43(8):999–1036, 1990.
[3] Gilles Aubert and Pierre Kornprobst. Mathematical problems in image processing:
Partial differential equations and the calculus of variations. 2006.
[4] Andrea Braides. Approximation of free-discontinuity problems. pages 27–38, 1998.
[5] Vicent Caselles, Ron Kimmel, and Guillermo Sapiro. Geodesic active contours.
International Journal of Computer Vision, 22(1):61–79, 1997.
[6] T. F. Chan and L. A. Vese. Active contours without edges. IEEE Transactions on
Image Processing, 10(2):266–277, 2001.
[7] Michael Kass, Andrew Witkin, and Demetri Terzopoulos. Snakes: Active contour
models. International Journal of Computer Vision, 1(4):321–331, 1988.
[8] David Mumford and Jayant Shah. Optimal approximations by piecewise smooth
functions and associated variational problems. Communications on Pure and Ap-
plied Mathematics, 42(5):577–685, 1989.
[9] Luminita A. Vese and Carole Le Guyader. Variational methods in image processing.
2015.