Steerable filters and invariant recognition in spacetime

The groups which have received most attention in signal processing research are the affine groups and the Heisenberg-Weyl group related to wavelets and time-frequency methods. In low-level image processing the rotation-groups SO(2) and SO(3) were studied in detail. We argue that the Lorentz group SO(1,2) provides a natural framework in the study of dynamic processes like the analysis of image sequences. We summarize the connection between the group SO(1,2) and the groups SU(1,1) and SL(2,R) and give an overview over their representations. We show that their representation theory is in parts similar to the corresponding theory for the three-dimensional rotation group. The main differences between the compact groups (like SO(2) and SO(3)) is however that the Fourier transforms for these groups involves infinite-dimensional representations and that the finite-dimensional representations are no longer unitary. In the signal processing context this means that the filter vectors computed by finite-dimensional steerable filter systems no longer transform as unitary vector transformations under the symmetry operations in SO(1,2).