Group Actions, Homeomorphisms, and Matching: A General Framework
International Journal of Computer Vision - Special issue on statistical and computational theories of vision: Part II
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We consider the problem of object and image registration, where the observation is assumed to be the result of a geometric deformation of a known template. More specifically, in the current framework we are interested in the case where the overall deformation is a result of a sequence of deformations the object has undergone and one is interested in estimating only part of the sequence, in a way that is invariant to the existence of the other deformations. The simplest example is the case where the deformation is affine, but one is interested only in estimating the scaling and rotation, irrespectively of the possible translation. In this paper we present a general framework allowing one to "focus" only on the parts of interest in the deformation, being invariant to "nuisance" parts of the overall deformation. This is achieved by transforming the original problem in which the deformation parameters appear implicitly rather than explicitly, into an equivalent problem which is explicit and linear in the deformation parameters. The representation of the original problem in the "classic" framework of linear spaces enables the adoption of well known linear methods.