Shape metrics, warping and statistics

Approximations of shape metrics, such as the Hausdorff distance, to define similarity measures between shapes are proposed. Our approximations being continuous and differentiable, they provide an obvious way to warp a shape onto another by solving a partial differential equation (PDE), in effect a curve flow, obtained from their first order variation. This first order variation defines a normal deformation field for a given curve. We use the normal deformation fields induced by several sample shape examples to define their mean, their covariance "operator", and the principal modes of variation. Our theory, which can be seen as a nonlinear generalization of the linear approaches proposed by several authors, is illustrated with numerous examples. Our approach being based upon the use of distance functions is characterized by the fact that it is intrinsic, i.e. independent of the shape parametrization.