Here are some simulations I did for image segmentation

	This model is proposed in T. F. Chan and L. A. Vese, "Active Contours without Edges", IEEE Trans. Image Processing, Vol. 10, pp. 266-277, 2001
	It is a model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets.
	The initial guess contour (I choose circle in the following examples) is represented by the zero level set of a Lipschitz function s.t.
	One tries to find as the minimizer of the energy
	The associated Euler-Lagrange equation can be derived as
	is the length parameter. need to be small in order to larger objects while need to be large in order to detect all or as many objects as possible.

 

	This model is proposed in Xiao Han, Chenyang Xu
	It achieves topology preservation by applying the simple point concept from digital topology.
	Two examples are shown here. Segmentation with topology preserving is used in the first example. The second one is the standard Chan-Vese model. We can see that the standard Chan-Vese model can have topological change while the TGDM model keeps the topology.