Nonparametric Density Estimation on Homogeneous Spaces in High Level Image Analysis.

The landmark data reduction approach in high level image analysis has led to significant progress to scene recognition via statistical shape analysis (Dryden and Mardia, 1998). While a number of families of similarity shape densities have proven useful in data analysis, only a few parametric models have been considered only recently in the context of projective shape ( Mardia and Patrangenaru, 2004 ), or affine shape. Shape spaces of interest have the geometric structure of symmetric spaces: planar similarity shape spaces are complex projective spaces ( Kendall, 1984 ), affine shape spaces are real Grassmann manifolds ( Sparr, 1992), and spaces of planar projective shapes of configurations of points in general position are products of real projective spaces ( Mardia and Patrangenaru, 2004 ). Therefore, data driven density estimation of shapes, regarded as points on symmetric spaces and arising from digitizing landmarks in images, is necessary. Recently, Pelletier (2004) considered kernel density estimation on “general” Riemannian manifolds; his results however hold only in homogeneous spaces. This is sufficient for image analysis, since any symmetric space is homogeneous. Pelletier estimators generalize the density estimators on certain homogeneous spaces introduced by Ruymgaart (1989), by H. Hendriks, J. H. M. Janssen and Ruymgaart (1993), and by Lee and Ruymgaart (1998). In this paper, we propose a class of adjusted Pelletier density estimators, on homogeneous spaces, that converge uniformly and almost surely at the same rate as naive kernel density estimators on Euclidean spaces. A concrete example of projective shape density estimation of 6-ads arising from digitized images of the “actor” data set in Wayne et.al. (2001).