In this paper, we analyze the geometric active contour models discussed in [6, 181 from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new featurebased Riemannian metrics. This leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus the snake is attracted very naturally and eficiently to the desired feature. Moreover, we consider some 3 -0 active surface models based on these ideas.

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Abstract-In this note, we employ the new geometric active contour models formulated in [25] and [26] for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery. Our method is based on defining feature-based metrics on a given image which in turn leads to a novel snake paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus, the snake is attracted very quickly and efficiently to the desired feature.

We use the Fuchsian algorithm to construct singular solutions of Einstein's equations which belong to the class of Gowdy spacetimes. The solutions have the maximum number of arbitrary functions. Special cases correspond to polarized, or other known solutions. The method provides precise asymptotics at the singularity, which is Kasner-like. All of these solutions are asymptotically velocity-dominated. The results account for the fact that solutions with velocity parameter uniformly greater than one are not observed numerically. They also provide a justification of formal expansions proposed by Grubišić and Moncrief.

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In this second part, we prove that the equation Uu = eU has solutions blowing up near a point of any analytic, space-like hypersurface in Rn, without any additional condition; if ( d ( x , t ) = 0) is the equation of the surface, u -lr1(2/4~) is not necessarily analytic, and generally contains logarithmic terms. We then construct singular solutions of general semilinear equations which blow-up on a non-characteristic surface, provided that the first term of an expansion of such solutions can be found.We finally list a few other simple nonlinear evolution equations to which our methods apply; in particular, formal solutions of soliton equations given by a number of authors can be shown to be convergent by this procedure.