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1. 14] Ωext where dp = dxdy is deﬁned and ds corresponds to the parametrization of the arc length. In this expression, the ﬁrst term is a contour integral involving K c which corresponds to a contour descriptor of the object to be detected. Thus, this descriptor is frequently based upon the gradient of the image. The last two integrals of E(Γ) correspond, however, to region terms, and involve a descriptor of the properties of the inner region K int and of the outer region K ext more often based on the statistical characteristics of the image.
As we will see in the next section, a DT can integrate, besides a geometric representation relating to the location and to the shape, information related to the texture or the material. 4. Deformable templates DTs are parameterized models that exhibit a real signiﬁcance when an a priori geometric shape of the object to segment is available. They have shown some success for tasks of shape detection and recognition [JAI 96, YUI 91]. A DT consists of a reference template that describes the most standard geometric shape of the object, of a transformation function that drives the variation of the template reference (geometric shape) and of a matching measure of the deformed template to the image information.
Model based on region characteristics In their seminal article, Chan and Vese [CHA 01] focused on the segmentation of an object showing no sharp edges, making use of information based on the gradient difﬁcult. The functional that they proposed to solve this problem involves terms using the characteristics of the interior (Ωint ) and exterior regions (Ωext ) deﬁned by the active contour. 20] 24 Multi-modality Cardiac Imaging where cint and cext correspond to the average of the image intensities of the inner and outer regions, respectively.