Variance based uncertainty principles and minimum uncertainty samplings

摘要

It has been shown recently that the uncertainty measure defined by variance products of subgroup generators, belonging to representations of popular signal transformation groups like, e.g., the wavelet group, can be arbitrarily small such that the search for a single function minimizing this measure becomes obsolete. Following an analysis of "local minima" of such an uncertainty measure, in this letter we introduce minimum uncertainty samplings. Denoting the transformation group with G, these samplings lead to lattices g _m(m∈M) with g _m∈G such that, starting from a function f, the total uncertainty of the function system f _(gm)m∈M has the lowest possible value under some given constraints (here f _(gm) denotes the function transformed with g _m). We apply this construction to, e.g., wavelet transforms and related techniques like shearlets.