Image "group-registration" based on representation theory

The general principle of a matching algorithm is to optimize a criterion that furnishes a measure of the similarity between two images for a given space of geometrical transformations. In this work, we propose a framework based on a similarity measure – the generalized correlation – built in a systematic way from the links between a features space and a group of transformations modeled by an action group. Using results from representation theory, we can extend the correlation function to any homogeneous space with a transitively acting group. When the generalized Fourier transform exists, the group-based correlation can be expressed in a spectral space and it becomes possible to implement fast algorithms for its computation. Two important examples in the field of image processing are then detailed: the similarity group (rotation and scaling) on gray-level shapes from 2D images and the 3D rigid motion group (rotation and translation) followed by a plan projection.

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