The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

Abstract

Abstract. The uncertainty principle is a fundamental concept in the context of signal and image processing, just as much as it has been in the framework of physics and more recently in harmonic analysis. Uncertainty principles can be derived by using a group theoretic approach. This approach yields also a formalism for finding functions which are the minimizers of the uncertainty principles. A general theorem which associates an uncertainty principle with a pair of self-adjoint operators is used in finding the minimizers of the uncertainty related to various groups. This study is concerned with the uncertainty principle in the context of the Weyl-Heisenberg, the SIM(2), the Affine and the Affine-WeylHeisenberg groups. We explore the relationship between the two-dimensional affine group and the SIM(2) group in terms of the uncertainty minimizers. The uncertainty principle is also extended to the Affine-Weyl-Heisenberg group in one dimension. Possible minimizers related to these groups are also presented and the scale-space properties of some of the minimizers are explored.

Square Integrable Group Representations and the Uncertainty Principle

“…Equation (5) deals with the case that the right hand side expansion contains one single element differing from the elements in the left-hand-side commutator. These equations cover the situation encountered for example in transform groups associated with 1-d-or 2-d-wavelets and shearlets, respectively, which have been analyzed recently [10,12,15].…”

“…Consequently, many authors were looking for "uncertainty minimizers", i.e., states ψ reaching equality in the above inequality, related to important groups in one-and multidimensional signal processing [1,3,10,12,15]. However, as pointed out in [11], obtaining minimizers in this sense does not imply minimization of the uncertainty (or maximization of the localization) measure v ψ (T 1 )v ψ (T 2 ), since the right-hand-side of the above inequality depends on ψ.…”

Do Uncertainty Minimizers Attain Minimal Uncertainty?

“…Here and for the rest of the paper A and B shall denote two linear self-adjoint operators on H. These operators in signal processing and quantum mechanics, respectively, usually are the generators of Lie-groups [12].…”

“…In information and signal theory, Gabor discovered that Gaussian-modulated complex exponentials provide the best trade-off for time resolution and frequency resolution. In computer vision-exploiting the potential of the affine Weyl-Heisenberg group-they are of interest in designing efficient filter banks for affine-invariant texture segmentation [12].…”

“…Specifically, it was extensively used for finding the minimizers related to infinitesimal generators of unitary groups, which are associated with important integral transforms. In this context the Weyl-Heisenberg group [6], the affine group in one dimension, the similitude group in two dimensions [1,2], and more recently the affine-WeylHeisenberg group [11][12][13][14] as well as the Shearlet group [13] have been investigated.…”