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Directions and projective shapes. (English) Zbl 1078.62068

Summary: 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 \(k-m-2\) copies of axial spaces \(\mathbb{R} P^m\). 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 Fréchet 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.

MSC:

62H35 Image analysis in multivariate analysis
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H11 Directional data; spatial statistics
62H10 Multivariate distribution of statistics
62E20 Asymptotic distribution theory in statistics
68U10 Computing methodologies for image processing
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