Do Uncertainty Minimizers Attain Minimal Uncertainty?

Maaß, Peter; Sagiv, Chen; Sochen, Nir; Stark, Hans‐Georg

doi:10.1007/s00041-009-9099-4

Cited by 13 publications

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References 12 publications

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“…Theorem 3.4). (iii) Corollary 3.5 is rather general: The non-existence of global uncertainty minimizers for the affine group and the affine Weyl-Heisenberg group shown in [11] follow directly from this statement (see also below).…”

Section: Discussionmentioning

confidence: 94%

“…Given the infinitesimal operators T 1 and T 2 according to [11] the phase space is a plane, whose coordinate axes represent the spectra σ (T 1 ) and σ (T 2 ), respectively (cf. Fig.…”

Section: Phase Spacementioning

confidence: 99%

“…1. This results from the fact that after proper transformations the joint localization properties of ψ with respect to T 1 and T 2 may be visualized by such a rectangle [11]. We shall investigate in this note, how this rectangle is shifted and/or dilated, if the state changes from ψ to U(g)ψ for g ∈ G. This question is motivated by the connection to frames.…”

Section: Phase Spacementioning

confidence: 99%

“…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. In fact, it was shown in [11] for the one-dimensional affine group, related to the one-dimensional continuous wavelet transform, that this localization measure can be made arbitrarily small.…”

Section: Introductionmentioning

confidence: 99%

“…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. In fact, it was shown in [11] for the one-dimensional affine group, related to the one-dimensional continuous wavelet transform, that this localization measure can be made arbitrarily small.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Square Integrable Group Representations and the Uncertainty Principle

Stark

Sochen

2010

J Fourier Anal Appl

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Let U be a square integrable representation of a Lie group G of transformations in a Hilbert space H, and let ψ ∈ H be an admissible state. We call the product of variances in the state ψ, associated to two non-commuting infinitesimal operators T 1 and T 2 , uncertainty measure.We investigate, in this note, how uncertainty measures of admissible states ψ are changing when ψ is transformed according to U(g)ψ for g ∈ G. We derive these transform laws for certain types of the associated Lie-algebra and apply them to relevant transform groups (affine group, similitude group, shearlets), investigated recently in signal processing.By doing so, we are able to show that the associated uncertainty measures can be made arbitrarily small in many cases. This implies that the corresponding "uncertainty minimizers", which are traditionally defined via equality in the famous uncertainty inequality and which have been investigated recently for above mentioned groups, actually do not minimize the uncertainty measure.In cases, where the uncertainty measure of U(g)ψ cannot be made arbitrarily small, when g varies over G, the transform laws derived in this paper indicate how to pick group elements g such that the uncertainty measure of U(g )ψ attains a local minimum. This method potentially enables discretizations of square integrable group representations with "low uncertainty", or equivalently "high locality" of the features of interest. This is of great importance for connecting uncertainty principles with frame constructions, and for maximum accuracy of feature measurements in signals and images.Communicated by Stephan Dahlke.

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Section: Discussionmentioning

confidence: 94%

“…Given the infinitesimal operators T 1 and T 2 according to [11] the phase space is a plane, whose coordinate axes represent the spectra σ (T 1 ) and σ (T 2 ), respectively (cf. Fig.…”

Section: Phase Spacementioning

confidence: 99%

Section: Phase Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Square Integrable Group Representations and the Uncertainty Principle

Stark

Sochen

2010

J Fourier Anal Appl

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Existence of uncertainty minimizers for the continuous wavelet transform

Simon

Olsen

Sochen

et al. 2023

Mathematische Nachrichten

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Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals.

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A survey of uncertainty principles and some signal processing applications

Ricaud

Torrésani

2013

Adv Comput Math

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The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, highlight their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized.

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Do Uncertainty Minimizers Attain Minimal Uncertainty?

Cited by 13 publications

References 12 publications

Square Integrable Group Representations and the Uncertainty Principle

Square Integrable Group Representations and the Uncertainty Principle

Existence of uncertainty minimizers for the continuous wavelet transform

A survey of uncertainty principles and some signal processing applications

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