Sharper uncertainty principles associated with Lp-norm

Motivated by the well-established phase derivative embedded technique, this study devotes to sharper uncertainty principles related to the L^p-norm type of uncertainty product, giving rise to two kinds of uncertainty inequalities that improve the classical result through providing tighter lower bounds. The conditions that truly reach these better estimates are obtained. Examples and simulations are carried out to verify the correctness of the derived results, and finally, possible applications in time-frequency analysis are also given.     

Stronger uncertainty principles for hypercomplex signals

In this paper, for real para-vector-valued signals, we obtain stronger uncertainty principles in terms of covariance and absolute covariance based on Fourier transform in both directional and the spatial cases. We provide certain conditions that give rise to the equal relation between the two uncertainty principles. Examples are presented to verify the results.     

Adjoint translation,adjoint observable and uncertainty principles

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.     

Signal representation,uncertainty principles and localization measures

The following collection of articles addresses one of the most basic problems in signal and image processing, namly the search for function systems (basis, frames, dictionaries) which allow efficient representations of certain classes of signals/images. Such representations are essential for decomposition and synthesis of signals, hence they are at the core of almost any application (coding, compression, pattern matching, feature extraction, classification, etc.) in this field. Accordingly, this is one of the best-studied topics in data analysis and a multitude of different concepts also addressing discretization/algorithmic issues has been investigated in this context. The starting point for reviving activities in this field was a recently rediscovered inconsistency in the concept of constructing optimally localized basis functions by minimizing uncertainty principles. In this short introductory note, we shortly sketch the basic dilemma, which was the starting point for this research approximately three years ago. However, the subsequent investigations presented in this collection of papers cover a much wider range of more general localization measures, discretization concepts as well as discussing algorithmic efficiency and stability.     

Sharper uncertainty principles in quaternionic Hilbert spaces

The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.     

Variational principles with integrands defined through a minimum

We revisit a systematic approach for the computation and analysis of the convex hull of non-convex integrands defined through the minimum of convex densities. It consists in reformulating the non-convex variational problem as a double minimization and exploiting appropriately the nature of optimality of the inner minimization. This requires gradient Young measures in the vector case, even if the initial problem was scalar, as the full problem is recast through the computation of a certain quasiconvexification. We illustrate this strategy by looking at two typical non-convex scalar problems. We hope to address vector problems in the near future.     

On uncertainty principles in the finite dimensional setting

The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known “qualitative” uncertainty principles into more quantitative estimates. We then show how to transfer this result to the discrete version of the short time Fourier transform.     

Qualitative uncertainty principles for the hypergeometric Fourier transform

We prove an L ^p version of the Donoho–Stark’s uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\). Next, using the ultracontractive properties of the semigroups generated by the Heckman–Opdam Laplacian operator, we obtain an L ^p Heisenberg–Pauli–Weyl uncertainty principle for the hypergeometric Fourier transform on \({\mathbb{R}^d}\).     

A survey of uncertainty principles and some signal processing applications

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.     

The robust minimum spanning tree problem: Compact and convex uncertainty

We consider the robust minimum spanning tree problem where edges costs are on a compact and convex subset of Rⁿ. We give the location of the robust deviation scenarios for a tree and characterizations of strictly strong edges and non-weak edges leading to recognition algorithms.     

Estimating maxima of generalized cross ambiguity functions,and uncertainty principles

In certain signal processing problems, it is customary to estimate parameters in distorted signals by approximating what is termed a cross ambiguity function and estimating where it attains its maximum modulus. To unify and generalize these procedures, we consider a generalized form of the cross ambiguity function and give error bounds for estimating the parameters, showing that these bounds are lower if we maximize the real part rather than the modulus. We also reveal a connection between these bounds and certain uncertainty principles, which leads to a new type of uncertainty principle.     

Novel uncertainty principles associated with 2D quaternion Fourier transforms

The quaternion Fourier transform has been widely employed in the colour image processing. The use of quaternions allow the analysis of colour images as vector fields. In this paper, the right-sided quaternion Fourier transform and its properties are reviewed. Using the polar form of quaternions, two novel uncertainty principles associated with covariance are established. They prescribe the lower bounds with covariances on the products of the effective widths of quaternionic signals in the space and frequency domains. The results generalize the Heisenberg's uncertainty principle to the 2D quaternionic space.     

Tighter uncertainty principles for periodic signals in terms of frequency

In this study, we propose some new uncertainty principles for periodic signals with sharper lower bounds than those in the existing ones. The improved lower bounds, in particular, are related to the frequency of the signal. Three examples are employed to demonstrate sharpness of the new uncertainty principles. Copyright © 2014 John Wiley & Sons, Ltd.     

Qualitative uncertainty principles for groups with finite dimensional irreducible representations

Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations.     

Discrete minimum and maximum principles for finite element approximations of non-monotone elliptic equations

Summary. Uniform lower and upper bounds for positive finite-element approximations to semilinear elliptic equations in several space dimensions subject to mixed Dirichlet-Neumann boundary conditions are derived. The main feature is that the non-linearity may be non-monotone and unbounded. The discrete minimum principle provides a positivity-preserving approximation if the discretization parameter is small enough and if some structure conditions on the non-linearity and the triangulation are assumed. The discrete maximum principle also holds for degenerate diffusion coefficients. The proofs are based on Stampacchias truncation technique and on a variational formulation. Both methods are settled on careful estimates on the truncation operator.Mathematics Subject Classification (2000): 65N30, 65N12