TY - JOUR
T1 - Nonparametric estimation of means on Hilbert manifolds and extrinsic analysis of mean shapes of contours
AU - Ellingson, Leif
AU - Patrangenaru, Vic
AU - Ruymgaart, Frits
N1 - Funding Information:
The first and second author thank the National Science foundation for support from Grant DMS-0805977 . The second author thanks the National Science Foundation for support from Grant DMS-1106935 and the National Security Agency for support from Grant MSP-H98230-08-1-0058 . We are grateful to the organizers of the Analysis of Object Data program 2010/2011 at SAMSI, and, in particular, to Hans Georg Müller and to James O. Ramsey for useful conversations on infinite object data analysis regarded as data analysis on Hilbert manifolds. Thanks also to Rabi N. Bhattacharya and John T. Kent for discussions on the subject and to Ben Kimia and Shantanu H. Joshi for providing access to the silhouette data library. Finally, we wish to thank the anonymous referee for suggestions that have improved the manuscript.
PY - 2013/11
Y1 - 2013/11
N2 - Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider the analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert-Schmidt operators, we can define extrinsic mean shapes of random planar contours and their sample analogues. We then apply the general methods to this problem while considering the computational restrictions faced when utilizing digital imaging data. Comparisons of computational cost are provided to another method for analyzing shapes of contours.
AB - Motivated by the problem of nonparametric inference in high level digital image analysis, we introduce a general extrinsic approach for data analysis on Hilbert manifolds with a focus on means of probability distributions on such sample spaces. To perform inference on these means, we appeal to the concept of neighborhood hypotheses from functional data analysis and derive a one-sample test. We then consider the analysis of shapes of contours lying in the plane. By embedding the corresponding sample space of such shapes, which is a Hilbert manifold, into a space of Hilbert-Schmidt operators, we can define extrinsic mean shapes of random planar contours and their sample analogues. We then apply the general methods to this problem while considering the computational restrictions faced when utilizing digital imaging data. Comparisons of computational cost are provided to another method for analyzing shapes of contours.
KW - Automated randomized landmark selection
KW - Data analysis on Hilbert manifolds
KW - Digital image analysis
KW - Extrinsic mean
KW - Nonparametric bootstrap
KW - Planar contours
KW - Statistical shape analysis
UR - http://www.scopus.com/inward/record.url?scp=84883351413&partnerID=8YFLogxK
U2 - 10.1016/j.jmva.2013.08.010
DO - 10.1016/j.jmva.2013.08.010
M3 - Article
AN - SCOPUS:84883351413
VL - 122
SP - 317
EP - 333
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
SN - 0047-259X
ER -