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Distributions of projective invariants and model-based machine vision

Part of: Sampling theory, sample surveys

Published online by Cambridge University Press:  01 July 2016

K. V. Mardia ,
Colin Goodall  and
Alistair Walder
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.
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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.

Keywords

MACHINE VISIONPROJECTIVE INVARIANTS

MSC classification

Primary: 62D05: Sampling theory, sample surveys
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 641 - 661
Copyright
Copyright © Applied Probability Trust 1996 

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