Adjoint translation, adjoint observable and uncertainty principles

Abstract

The motivation to this paper stems from signal/image processing where it is desired to measure various attributes or physical quantities such as position, scale, direction and frequency of a signal or an image. These physical quantities are measured via a signal transform, for example, the short time Fourier transform measures the content of a signal at different times and frequencies. There are well known obstructions for completely accurate measurements formulated as "uncertainty principles". It has been shown recently that "conventional" localization notions, based on variances associated with Lie-group generators and their corresponding uncertainty inequality might be misleading, if they are applied to transformation groups which differ from the Heisenberg group, the latter being prevailing in signal analysis and quantum mechanics. In this paper we describe a generic signal transform as a procedure of measuring the content of a signal at different values of a set of given physical quantities. This viewpoint sheds a light on the relationship between signal transforms and uncertainty principles. In particular we introduce the concepts of "adjoint translations" and "adjoint observables", respectively. We show that the fundamental issue of interest is the measurement of physical quantities via the appropriate localization operators termed "adjoint observables". It is shown how one can define, for each localization operator, a family of related "adjoint translation" operators that translate the spectrum of that localization operator. The adjoint translations in the examples of this paper correspond to well-known transformations in signal processing such as the short time Fourier transform (STFT), the continuous wavelet transform (CWT) and the shearlet transform. We show how the means and variances of states transform appropriately under the translation action and compute associated minimizers and equalizers for the uncertainty criterion. Finally, the concept of adjoint observables is used to estimate concentration properties of ambiguity functions, the latter being an alternative localization concept frequently used in signal analysis.