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Nonparametric Estimation for Extrinsic Mean Shapes of Planar Contours

In this paper we consider an extrinsic approach to analyzing direct similarity shapes of planar contours. We adapt Kendall’s definition of direct similarity shapes of planar k-ads to shapes of planar contours, under certain regularity conditions, and Ziezold’s nonparametric view on Fréchet mean shapes ([43]). The space of direct similarity shapes of planar simple closed curves is embedded in a space of Hilbert-Schmidt operators. Using Veronese-Whitney embeddings of Kendall’s Σ2 approximations of this embedding, for large k, one computes extrinsic sample means of direct similarity shapes of closed curves and their asymptotic distributions. To estimate these means we appeal to a functional data analysis technique. This methodology is also applied to a one-sample test for the extrinsic mean direct similarity shape of regular contours. In addition, we consider using nonparametric bootstrap to approximate confidence regions for such mean shapes. Computational examples are also included for Ben Kimia’s contour library. Using the contour data set, we demonstrate the computational efficiency of these techniques compared to other methods for the analysis of mean contours. For application of our technique on digital imaging data, specifically for cases when landmarks for the k-ad representation of the contours are not provided, we propose an automated randomized pseudo-landmark selection, that is useful for within population contour matching and is coherent with the underlying asymptotic theory.