State of the Art of Level Set Methods in Segmentation and Registration of Medical Imaging Modalities

Segmentation of medical images is an important step in various applications such as visualization, quantitative analysis and image-guided surgery. Numerous segmentation methods have been developed in the past two decades for extraction of organ contours on medical images. Low-level segmentation methods, such as pixel-based clustering, region growing, and filter-based edge detection, require additional pre-processing and post-processing as well as considerable amounts of expert intervention or information of the objects of interest. Furthermore the subsequent analysis of segmented objects is hampered by the primitive, pixel or voxel level representations from those region-based segmentation [1]. Deformable models, on the other hand, provide an explicit representation of the boundary and the shape of the object. They combine several desirable features such as inherent connectivity and smoothness, which counteract noise and boundary irregularities, as well as the ability to incorporate knowledge about the object of interest [2, 3] [4]. However, parametric deformable models have two main limitations. First, in situations where the initial model and desired object boundary differ greatly in size and shape, the model must be re-parameterized dynamically to faithfully recover the object boundary. The second limitation is that it has difficulty dealing with topological adaptation such as splitting or merging model parts, a useful property for recovering either multiple objects or objects with unknown topology. This difficulty is caused by the fact that a new parameterization must be constructed whenever topology change occurs, which requires sophisticated schemes [5, 6]. Level set deformable models [7, 8], also referred to as geometric deformable models, provide an elegant solution to address the primary limitations of parametric deformable models. These methods have drawn a great deal of attention since their introduction in 1988. Advantages of the contour implicit formulation of the deformable model over parametric formulation include: (1) no parameterization of the contour, (2) topological flexibility, (3) good numerical stability, (4) straightforward extension of the 2D formulation to n-D. Recent reviews on the subject include papers from Suri [9, 10]. In this chapter we give a general overview of the level set segmentation methods with emphasize on new frameworks recently introduced in the context of medical imaging problems. We then introduce novel approaches that aim at combining segmentation and registration in a level set formulation. Finally we review a selective set of clinical works with detailed validation of the level set methods for several clinical applications. A. Level set methods for Segmentation A recent paper from Montagnat, Delingette and Ayache review the general family of deformable models and surfaces with a classification based on their representation. This classification has been reproduced, to some extends in Figure 1. Level set deformable models appear in this classification diagram as part of continuous deformable models with implicit representation. Figure 1:Geometric representations of deformable surfaces. A. Level Set Framework Segmentation of an image I via active contours, also referred to as snakes [2], operates through an energy functional controlling the deformation of an initial contour curve under the influence of internal and external forces achieving a minimum energy state at high-gradient locations. The generic energy functional for active contour models is expressed as: ( ) [ ] , 0, C p p∈ 1 ( ) ( ) ( ) ( ) ( ) 1 1 1 2 2 ' '' 0 0 0 E C C s ds C s ds I C s α β λ = + − ∇ ∫ ∫ ∫ ds (1) where are positive parameters. The first two terms control the rigidity and elasticity of the contour (defining the internal energy of the deformable object) while the last term attracts the model to high-gradient locations in the image I (defining the external energy of the model). ( , , α β λ) Active contour segmentation via minimization of the energy functional in Equation (1) is typically implemented with a parametric framework in which the deformable model is explicitly formulated as a parameterized contour on a regular spatial grid, tracking its point positions in a Lagrangian framework [11]. In their original paper from 1988 [7], Osher and Sethian introduced the concept of geometric deformable models, which provide an implicit formulation of the deformable contour in a level set framework. To introduce the concept of the level set framework we focus on the boundaryvalue problem of a close contour C deforming with a speed V along its normal direction: 1, 0 C V V ∇ = > (2) Their fundamental idea is, instead of tracking in time the positions of the front on a regular grid as: ( , C x y) ( ) { ( ) ( , ) | , t x y C x y Γ = } t = (3) to embed the curve into a higher dimension function ( ) , , x y t φ such that: (1) at time zero the initial contour corresponds to the level zero of the function φ: 0 C . (4) ( ) ( ) { 0 , / , ,0 0 C x y x y φ = } = (2) the function φ evolves with the dynamic equation: V t φ φ ∂ = ∇ ∂ . (5) In this framework, at any time t, the front implicitly defined by: (6) ( ) ( ) ( ) { , / , , 0 t x y x y t φ Γ = } = corresponds to the solution of the initial boundary value problem defined parametrically in Equation (3). This result is illustrated in Figure 2. Figure 2: Correspondence between a parametric and implicit level-set formulation of the deformation of a contour with a speed term oriented along the normal direction. In their pioneer paper, Osher and Sethian focused on motion under mean curvature flow where the speed term is expressed as: V div φ φ ⎛ ⎞ ∇ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ∇ ⎝ ⎠ . (7) Since its introduction, the concept of deformable models for image segmentation defined in a level set framework has motivated the development of several families of method that include: geometric active contours based on mean curvature flow, gradient-based implicit active contours and geodesic active contours. B. Geometric Active Contours In their work introducing geometric active contours, Caselles et al. [12] proposed the following functional to segment a given image I: ( ) g I div t φ φ φ ⎛ ⎞ ⎛ ⎞ ∂ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ = ∇ ∇ ⎜ + ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ∂ ∇ ⎟ ⎝ ⎠ ⎝ ⎠ φ ν ∇ , (8)