The Annals of Statistics

Directions and projective shapes

Kanti V. Mardia and Vic Patrangenaru

Full-text: Open access

Abstract

This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space, as well as an appropriate coordinate system for this shape space. For generic configurations of k points in m dimensions, the resulting projective shape space is identified as a product of km−2 copies of axial spaces ℝPm. This identification leads to the need for developing multivariate directional and multivariate axial analysis and we propose parametric models, as well as nonparametric methods, for these areas. In particular, we investigate the Frećhet extrinsic mean for the multivariate axial case. Asymptotic distributions of the appropriate parametric and nonparametric tests are derived. We illustrate our methodology with examples from machine vision.

Article information

Source
Ann. Statist. Volume 33, Number 4 (2005), 1666-1699.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
http://projecteuclid.org/euclid.aos/1123250226

Digital Object Identifier
doi:10.1214/009053605000000273

Mathematical Reviews number (MathSciNet)
MR2166559

Zentralblatt MATH identifier
1078.62068

Subjects
Primary: 62H11: Directional data; spatial statistics
Secondary: 62H10: Distribution of statistics 62H35: Image analysis

Keywords
Projective transformations projective frame projective shape space equivariant embedding extrinsic means directional statistics tangent approximation bootstrapping shape analysis machine vision

Citation

Mardia, Kanti V.; Patrangenaru, Vic. Directions and projective shapes. Ann. Statist. 33 (2005), no. 4, 1666--1699. doi:10.1214/009053605000000273. http://projecteuclid.org/euclid.aos/1123250226.


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