The Annals of Statistics

Directions and projective shapes

Kanti V. Mardia and Vic Patrangenaru

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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

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

First available in Project Euclid: 5 August 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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