In this paper, we analyze the geometric active contour models discussed previously from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based 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 efficiently to the desired feature. Moreover, we consider some 3-D active surface models based on these ideas.

We employ the new geometric active contour models, previously formulated, 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.

The problem of existence of solitary wave solutions to some higher-order model evolution equations arising from water wave theory is discussed. A simple direct method for finding monotone solitary wave solutions is introduced, and by exhibiting explicit necessary and sufficient conditions, it is illustrated that a model admit exact ${\text{sech}}^2 $ solitary wave solutions. Moreover, it is proven that the only fifth-order perturbations of the Korteweg–deVries equation that admit solitary wave solutions reducing to the usual one-soliton solutions in the limit are those admitting families of explicit ${\text{sech}}^2 $ solutions.

The Perona--Malik equation is a formally ill-posed parabolic equation for which simple discretizations are nevertheless numerically found to be stable. After discussing the background of this paradox in computer vision, this paper shows the nonexistence of weak solutions even in those cases where computations are successful, and introduces a notion of generalized solutions for this equation, which do evolve smoothly and possess many of the features of numerical calculations.