Statistical Estimation under Projective Transformations: Theory and Applications in Multi-view problems

Projective transformations are a fundamental part of imaging with a camera. While the geometric properties of the transformation have been well studied, statistical estimation under the transformation has not been sufficiently addressed. In particular, the properties of random variables transformed by a projective transformation have not been adequately explored. In this paper, we study the effect of projective transformations on Gaussian random variables. The transformed random variable has a mixture density with the Cauchy density being one of the mixture components. We highlight the implications of the particular form of this density. For example, we show that the properties of the transformed random variable are tied closely to a geometric relationship between the random variable and the line at infinity. These properties allow for approximations using which we show that a Gaussian random variable projectively transforms to a Gaussian random variable. We demonstrate the usefulness of this result in applications such as Euclidean tracking or metrology. Furthermore, in the context of estimation using multi-view data, we motivate the need to treat estimates from each view differently given that the projective transformation linking each view and the scene can be remarkably different. Finally, we present efficient multi-view fusion methodologies for detection and tracking of objects. Index Terms Statistics, Geometry, Plane, Multi-camera, tracking, metrology, Ratio distributions, Ratio of Normals A. C. Sankaranarayanan and R. Chellappa are with Centre for Automation Research and Electrical and Computer Engineering Department, 4417 A. V. Williams Building, University of Maryland, College Park, MD 20742. Email: {aswch,rama}@cfar.umd.edu September 25, 2008 DRAFT 2