Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

This paper proposes 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 Zolésio [11], we consider the characteristic functions of the subsets of R2 and their distance functions. The L2 norm of the difference of characteristic functions, the L∞ and the W 1,2 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 Date received: May 7, 2003. Final version received: March 18, 2004. Date accepted: March 24, 2004. Communicated by Peter Olver. Online publication: July 6, 2004. AMS classification: 35Q80, 49Q10, 60D05, 62P30, 68T45.