Approximating shape metrics and application to shape warping and empirical shape statistics
Olivier Faugeras
INRIA Sophia Antipolis
We propose a framework for addressing several problems related to the analysis of shapes. Two related problems are the definition of the relevant set of shapes and the definition of a metric on that set. Following a recent research monograph by Delfour and Zolesio, we consider the characteristic functions of subsets of Euclidean space and their distance functions.
The L² norm of the difference of characteristic functions, the L8 norm, and the W¹,² norm of the difference of distance functions define interesting topologies, including the well-known Hausdorff distance. Because of practical considerations arising from working with image shapes defined on finite pixel grids, we restrict our attention to subsets of Euclidean space with positive reach in the sense of Federer, and with smooth boundaries of bounded curvature. For this particular class of shapes, we show that the three previously mentioned topologies are equivalent.
We then consider the problem of warping one 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 that are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. These derivatives define deformation flows that can be used to warp one shape onto another by solving an initial value problem.
We present several examples of this warping and prove properties of our approximations related to the existence of local minima. We then use this framework to produce computational definitions of the empirical mean and covariance of a set of shape examples, yielding an analogue of the notion of principal modes of variation. These ideas are illustrated on examples such as computing the average of a set of eight cross-sections of corpus callosa and the corresponding principal modes of deformation.
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