La mediació en salut: experiències rellevants

Before proceeding to details, the organization of this thesis is given here for an overview: chapter 2 introduces the theory of active contours/surfaces and several applications of this model to both 2-D and 3-D images; a novel registration technique is then proposed in chapter 3 with its feasibility to solve this problem validated; chapter 4 investigates into PCA modelling of lungs, performing analyses of the lung shape class and proposes a novel hybrid modelling method; With essential techniques ready, chapter5describes a hierarchical scheme incorporating these techniques to achieve a satisfactory segmentation of volumetric images in presence of large amount of noise and various types of occlusions; in addition to the prior shape, a volumetric texture segmentation method to employ texture information as prior knowledge is given in chapter6; Finally, chapter7concludes the thesis and proposes several future directions for this research topic.

Active Contours/Surfaces

2.1

Previous Works

Kass et al. (1988) first introduced the idea of active contours (snakes) into the field

of image segmentation. Classical active contours are explicitly presented in an Euler- Lagrange equation deduced from an energy functional that basically consists of two parts: internal energy and external energy. The internal energy serves to control the geometric properties of the contours and the external energy are mostly exerted by the target image. In an attempt to minimize the energy functional, the associated Euler-Lagrange equation which controls the motion of the contours is derived. When the contours achieve its steady state, local minima of the image, mostly lines and edges, are detected. Its 3-D version is proposed byTerzopoulos et al.(1988) and this methodology was later applied effectively into medical image segmentation in the forms of topology-adaptive snakes (t-snakes) (Montagnat and Delingette,2000b;McInerney and Terzopoulos,2000), Eigen-snakes (Toledo et al., 2000a,b) , 3-D parametric active surface (Montagnat and

Delingette,2000a), deformable tubular model (Yim et al.,2001).

Novel models of a similar style were proposed by Caselles et al. (1993) and Malladi

et al.(1994,1995) based on the theory of curve evolution and geometric flows. In these

models, the contours evolve in a velocity that constrains two terms, one relates to the regularity of the contour and the other pushes the contours towards the boundary of the image. However, stopping condition has to be chosen properly, for the geometric models are given in the form of curve evolution equation, but not an energy minimization one. The numerical implementation of the geometric models is hugely facilitated by the level- set based method (Osher and Sethian,1988), in which 2-D curves are embedded into a 3-D surface (often a signed distance function). Thus, automatic handling of the curve topologies makes it possible for several objects to be detected simultaneously without prior knowledge.

Caselles et al. (1997a) generalized a particular case of classical snakes into finding a geodesic curve in a Riemannian space with a metric derived from the image. This means that boundary detection in this sense, becomes finding a geodesic curve that has the minimal length in Riemannian space. The framework is similar to the geometric active contours, however, the geodesic active contour model is intrinsic and non-parametric, also, unlike geometric active contours, no stopping term has to be chosen, for the curve becomes static when it reaches the steepest area of the boundary. Moreover, it is capable of detecting objects with sharper corners compared to geometric active contours. While the above models are more or less dependent on image gradient to detect bound- aries, Chan and Vese(2001) presents a region feature based image segmentation model. The contours in this model are described as the interface of two regions: inside and outside on a finite domain of the image. The image force that drives the contour is derived from minimizing the respective variance of a specified term, say pixel intensity, inside and outside the contour, therefore the steady state of contour indicates that both the areas inside and outside of the contour achieve maximal similarity with respect to the specified term. This model is a reduced form of Mumford-Shah (MS) functional

(Mumford and Shah,1989), referred to as the minimal partition problem. However, the

implementation of this model is different from previous ones. The embedment of the contours into the signed distance function (SDF) is achieved along with derivation of the associated Euler-Lagrange equation, or in other words, the curve evolution equation of this model is in essence an SDF evolution equation and thus achieves better numerical stability.

Kimmel and Bruckstein (2003) put both geodesic active contours and active contours

without edges into the one curve evolution equation along with an optimal edge integra- tion term which is generalized as an equivalence with the zeros crossings of the image Laplacian. Later, this unified work was referred to as integrated active contours bySagiv

et al.(2006), in which, the methodology appeared to be capable of detecting objects with

interior boundaries in a background of gradually attenuating illumination, which may be the case in medical images. This was also applied into the segmentation of thin and tree like structure such as blood vessels, in the volumetric CT images (Holtzman-Gazit et al.,2006).