Nonlinear Diffusion Models In Image Segmentation

Segmentation is a fundamental task of image processing because of its many areas of application. When acquiring an image, it might be interesting to know what parts of the image do belong to each other. It could be a satellite image, where we wish to quantify and locate different types of vegetation, or a medical MRI image where the doctors are interested to know how much of each tissue type is present, and where it is located. Image segmentation techniques offer a method to perform these tasks, and can be regarded as the process of dividing an image into groups of pixels which from a preset property are connected to each other. Segmentation techniques locate objects consisting of pixels having something in common. Commonly this means that pixels with almost the same intensity values are grouped together, or pixels with the same color or texture etc. We will review some PDE-based image segmentation methods raised in recent years in this paper.In section 1,we briefly review some traditional image segmentation methods.In section 2,we introduce the variational method solving a kind of integral functional and the level set method proposed by Osher and Sethian (1988) [11]. Level set method is to implicitly represent evolving curves and surfaces by a level set function, it has many applications, because it allows for automatic change of topology, such as merging and breaking, and the calculations are made on a fixed rectangular grid.This paper will focus on energy-based segmentation methods, which completethe segmentation via minimizing energy functionals. They define optimal segmentation as a minimizer of objective functional. The Euler-Lagrange equa- tions associated to energy functionals of these models can often be described using a partial differential equation, which is iterated until it reaches steady state. There are basically two different class energy functionals hence two classes of models. The first is edge-based(gradient-based),it is to extract some "significant"contours on which the magnitude of gradient of the image reaches its local maximum. The second is region-based, the image is segmented into different homogeneous regions sharing similar trait.we discuss edge-based segmentation models in section 3. The idea of this approach is that contours are characterized by sharp variation of the image intensity,and so by gradients. We begin by describing the Snake model[5]. In this model the energy to be minimized is defined on piecewise regular parametric curvesE_s(C) =(∫_a^b|c'(q)|²dq+β∫_a^b|c"(q)|²dq)+λ∫_a^bg²(|â–½I(c(q))|)dq,(1)where g(|â–½I(x)|) is called edge detection function cause it reach its local minimum at object contours. Snake model is the earliest active contour model, it is based on deforming an initial contour C₀ towards the boundary of the object to be detected. The deformation is obtained by trying to minimize the energy so that its (local) minimum is obtained at the boundary of the object. The main drawback of the Snake model is its energy functional is not intrinsic, since it depend on the parametrization of curve C. So this model can not handle changes of topology of the contours, it is impossible to detect more than one object.Caselles et al. [17] introduced an intrinsic energyE_GAC(C) =∫₀¹g(|â–½I(C(q))|)C'(q)|dq.(2)This model is called geodesic active contour(GAC), it find a minimal distance curve in a Riemann space, according to the metric defined by the image content.This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. It allows automatic changes in the topology when implemented using the level-sets based numerical algorithm. We also show that the solution to the geodesic flow exists in the viscosity framework, and is unique and stable. Consistency of the model is presented as well, showing that the geodesic curve converges to the right solution in the case of ideal objects.In section 4.1,we first present the formulation introduced by Mumford and Shah[1, 24]. It is an energy-based segmentation. For a given u₀(x)(the initial image),we search for the pair (u, K) minimizingE_ms[u,K]=∫_Ω(u-u₀)²dx+α∫_Ω\k|â–½u|²dx+βH¹(K), (3)where function u :Ω→Ris a piecewise smooth approximation of u₀ and K (?)Ωis the discontinuity set associate to u(x). An object in this formulation is characterizedby having relatively uniform feature intensity. This type of functional forms part of a wider class of problems called "free discontinuity problems." We present some theoretical results of existence of minimizers of (3) ,and optimality conditions of this minimizing problem.The main computational challenge arises from the free-boundary nature of the MS model. We discuss the approach by elliptic approximation, as proposed by Ambrosio and Tortorelli[31, 34]. In section 4.2, we discuss the "active contours without edges" model(CV model) proposed by Chan and Vese[44] which is a reduced case of MS model. The level set method was first introduced for numerical computation of MS model by Chan and Vese [44] and Tsai et al. [45]. The energy being minimized in CV model isIn the level set formulation, the problem becomes a "mean-curvature flow" -like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. However, the CV model and piecewise constant MS model involve minimizingfunctionals over characteristic functions of sets, which is a non-convex collection; this feature is responsible for the presence of local minima.Many solution techniques for variational models are based on gradient descent,and are therefore prone to getting stuck in such local minima. This makes initial guess for gradient descent based algorithms sometimes critically important for obtaining satisfactory results. In section 4.3, we present some results concerningfinding global minimizers of active models. Mainly we study Chan, Esedoglu and Nikolova's approach [51]. They proposed a new approach to deal with global minimum and overcome the limitation of local minima.Theorem 2.1 [51] For any given fixed c₁,c₂∈R, a global minimize of E_cv(·, c₁, c₂) can be found by carrying out the following convex minimization:and then setting∑= {x : u(x)≥μ} for a.e.μ∈[0,1].Thus non-convex minimization problems that arise in CV model can equivalently be restated as convex minimization problems. This allows, in particular, finding global minimizers via standard convex minimization schemes.We finish our paper with some numerical method to implement PDE equationsraised in those models.