Statistical Estimation under Projective Transformations: Theory and Application in Multi-view Estimation ProblemsAswin C. Sankaranarayanan, and Rama Chellappa |
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Abstract: 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 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 prove 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.
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| IEEE Transactions on Pattern Analysis and Machine Intelligence (under review) |
| A preliminary version of this idea appeared in the Motion Workshop. A. C. Sankaranarayanan and R. Chellappa, "Optimal Multi-view Fusion of Object Locations", IEEE Workshop on Motion and Video Computing (MOTION), Copper Mountain, Jan 9, 2008 (pdf) |