Approximating shape metrics and application to shape warping and empirical shape statistics

Olivier Faugeras
INRIA Sophia Antipolis

We propose a framework for dealing with several problems related to the
analysis of shapes. Two related such problems are the definition of the relevant
set of shapes and that of defining a metric on it. Following a recent research
monograph by Delfour and Zolesio, we consider the characteristic functions of
the subsets of

and their distance functions. The L2 norm of the difference of
characteristic functions, the L and the W1,2
norms of the difference of distance functions define interesting topologies, in
particular the well-known Hausdorff distance. Because of practical
considerations arising from the fact that we deal with image shapes defined on
finite grids of pixels we restrict our attention to subsets of
positive reach in the sense of Federer, with smooth boundaries of bounded
curvature. For this particular set of shapes we show that the three previous
topologies are equivalent. The next problem we consider is that of warping a
shape onto another by infinitesimal gradient descent, minimizing the
corresponding distance. Because the distance function involves an infimum, it is
not differentiable with respect to the shape. We propose a family of smooth
approximations of the distance function which are continuous with respect to the
Hausdorff topology, and hence with respect to the other two topologies. We
compute the corresponding Gâteaux derivatives. They define deformation flows
that can be used to warp a shape onto another by solving an initial value
problem. We show several examples of this warping and prove properties of our
approximations that relate to the existence of local minima. We then use this
tool to produce computational definitions of the empirical mean and covariance
of a set of shape examples. They yield an analog of the notion of principal
modes of variation. We illustrate them on a variety of examples such as the one
below where we compute the average of a set of eight cross-sections of corpus
callosi (left) and the principal modes of deformation (right).

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