Geodesic Active Contour And Topology-Perserving Image Segmentation

Digital image processing has a long history. The basic tools are Fourier anal-ysis with a long history and wavelet analysis which has become popular in the engineering community since 1980's. In the past few decades more sophisticated tools have been developed. Partial differential equations (PDEs) and variational approaches, traditionally applied in physics, have been successfully and widely transferred to computer vision over the last decade.Many mathematical models in image processing are often derived from mini-mization problems, which can be solved using the calculus of variations and PDEs. Active contour (including its variations) [3,4,12,20,21] is one of the most successful variational models for the segmentation problem. Active contour model attempts to minimize an energy functional associated to the current contour as a sum of an internal and external energy. The internal energy is used to impose a smoothness constraint, and the external energy attracts the curve toward the edges of the objects. Using gradient descent method and calculus of variations, we can get the corresponding curve evolution problem which is an ODE system.The level set method [5-7] is an effective numerical technique for curve evolu-tion. The level set method use an implicit representation to defines the interface as the isocontour of some higher dimensional level-set function. The level set presentation converts the original ODE problem to PDE problem. Initially, the implicit representation might seem wasteful, since the implicit function is defined on all of the space. However, a number of very powerful tools are readily available when we use this representation. And one can perform numerical computations involving curves without considering the curves parameterizations. Also, the level set method makes it very easy to follow shapes that change topology, for example when a shape splits in two, develops holes, or the reverse of these operations.However, the topological flexibility is not always desired in some applications [8-11]. For instance, when we have the prior knowledge of the topology of the object to be detected, or when the segmentation result must be homeomorphic to the initial contour. For example, in the human cortex reconstruction, it is known that the human cortex has a spherical topology so throughout the reconstruction process this topological feature must be preserved.In this paper, we mainly discuss the geodesic active contour model including its derivation and geometric meaning, describe the level set method which used to solve the model, and propose a topology-preserving method for the geodesic active contour model.Our main work consists of:·We describe the basic framework of active contour model [3], describes the geodesic active contour models which is derived from the geodesic active contour model [4] and derive the Euler-Lagrange equations of the problem in detail, and obtain the corresponding curve evolution problem.·We introduce the level set method [5-7] for solving the curve evolution prob-lem, and then obtain the level set evolution equation of geodesic active con-tour.·We discuss the process of re-initialization of the level evolution set in detail, and use a more precise method [13,14,18] of re-initialization in implementa-tion.·We propose a topology preserving method for the geodesic active contour model.·We discuss the numerical implementation [15,16] of the model and the dis-cretization of re-initialization equation in detail. ·We conclude the paper with experimental results on various 2 or 3 dimensions synthetic images.