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Distributions of projective invariants and model-based machine vision

Published online by Cambridge University Press:  01 July 2016

K. V. Mardia*
Affiliation:
University of Leeds
Colin Goodall*
Affiliation:
Pennsylvania State University
Alistair Walder*
Affiliation:
University of Leeds
*
Postal address: Department of Statistics, University of Leeds, Leeds LS2 9JT, UK.
∗∗ Postal address: Department of Statistics, Pennsylvania State University, University Park, PA 16802 USA.
Postal address: Department of Statistics, University of Leeds, Leeds LS2 9JT, UK.

Abstract

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

MSC classification

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 1996 

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