Scale-space generation via uncertainty principles

Abstract

This study is concerned with the uncertainty principles which are related to the Weyl-Heisenberg, the SIM (2) and the Affine groups. A general theorem which associates an uncertainty principle to a pair of self-adjoint operators was previously used in finding the minimizers of the uncertainty principles related to various groups, e.g., the one and two-dimensional Weyl-Heisenberg groups, the one-dimensional Affine group, and the two-dimensional similitude group of ℝ2, SIM (2) = ℝ2 × (ℝ+ × SO(2)). In this study the relationship between the affine group in two dimensions and the SIM (2) group is investigated in terms of the uncertainty minimizers. Moreover, we present scale space properties of a minimizer of the SIM (2) group. © Springer-Verlag Berlin Heidelberg 2005.