Curvature in image and shape processing

The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian operator. Such generalizations of the curvature vector and its numerical approximation as part of an image flow or triangulated shape representation, demonstrate the omnipresence of this operator and its usefulness in imaging sciences.