Shape analysis and symmetry detection in gray-level objects using the analytical Fourier-Mellin representation

The analytical Fourier-Mellin transform is used in order to assess motion parameters between gray-level objects having the same shape with distinct scale and orientation. From results on commutative harmonic analysis, a functional is constructed in which the location of the minimum gives an estimation of the size and orientation parameters. Furthermore, when the set of geometrical transformations is restricted to the compact rotation group, we show that this minimum is exactly the Hausdorff distance between shapes represented in the Fourier-Mellin domain. This result is used for the detection and the estimation of both all rotation and reflection symmetric in objects.