In this chapter, we focus on statistical region-based active contour models where the region descriptor is chosen as the probability density function of an image feature (e.g. intensity) inside the region. Image features are then considered as random variables whose distribution may be either parametric, and then belongs to the exponential family, or non parametric and is then estimated through a Parzen window. In the proposed framework, we consider the optimization of divergences between such PDFs as a general tool for segmentation or tracking in medical images. The optimization is performed using a shape gradient descent through the evolution of an active region. Using shape derivative tools, our work is directed towards the construction of a general expression for the derivative of the energy (with respect to a domain), and the differentiation of the corresponding evolution speed for both parametric and non parametric PDFs. Experimental results on medical images (brain MRI, contrast echocardiography, perfusion MRI) confirm the availability of this general setting for medical structures segmentation or tracking in 2D or 3D.
Top1 Introduction
Medical structures segmentation or tracking is a key issue to improve medical diagnosis. These steps become crucial to cope with the increasing amount of medical data encountered in medicine. We focus here on active contours or surfaces (Kass, Witkin, & Terzopoulos, 1988; Caselles, Kimmel, & Sapiro, 1997) that are particularly well adapted to the treatment of medical structures because they provide a compact and analytical representation of object shape. The general idea behind active contours model is to apply partial differential equations (PDEs) to deform a curve (in 2D) or a surface (in 3D) towards the boundaries of the objects of interest. Snakes (Kass, Witkin, & Terzopoulos, 1988), balloons (Cohen, 1991) and geodesic active contours (Caselles, Kimmel, & Sapiro, 1997) were pioneering works on active contour models. In these methods, the contour is driven towards image edges. More recently, region-based active contours (i.e. RBAC) were proposed (Cohen, Bardinet, & Ayache, 1993; Ronfard, 1994; Zhu & Yuille, 1996; Chakraborty, Staib, & Duncan, 1996; Paragios & Deriche, 2000; Chan & Vese, 2001). In these approaches, region-based terms can be advantageously combined with boundary-based ones. The evolution equation is generally deduced from a general criterion to minimize that includes both region integrals and boundary integrals. The combination of those two terms in the energy functional allows the use of photometric image properties, such as texture (Paragios & Deriche, 2002; Aujol, Aubert, & Blanc-Féraud, 2003; Rousson, Lenglet, & Deriche, 2004; Karoui, Fablet, Boucher, & Augustin, 2006) and noise (Martin, Réfrégier, Goudail, & Guérault, 2004; Lecellier, Jehan-Besson, Fadili, Aubert, & Revenu, 2006; Galland, Bertaux, & Réfrégier, 2005), as well as geometric properties such as the prior shape of the object to be segmented (Leventon, 2000; Cremers, Tischhäuser, Weickert, & Schnörr, 2002; Tsai, Yezzi, & Wells, 2003; Gastaud, Barlaud, & Aubert, 2003; Foulonneau, Charbonnier, & Heitz, 2003; Lecellier, Jehan-Besson, Fadili, Aubert, Revenu, & Saloux, 2006), see also the review in (Cremers, Rousson, & Deriche, 2007). RBACs have proven their efficiency for a wide range of applications and are widely used in medical image segmentation (see for example Lau & Ozawa, 2004; Cheng, Yang, Fan, 2005; Paragios, 2002; Dydenko, Jamal, Bernard, D’Hooge, Magnin & Friboulet 2006).