Shape Metrics, Warping and Statistics
O. Faugeras, G. Charpiat and R. Keriven (INRIA Sophia-Antipolis, France)
We propose to use approximations of shape metrics, such as the Hausdorff
distance, to define similarity measures between shapes. 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 parameterization.