Related papers: Continuous Donoho-Elad Spark Uncertainty Principle
We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical…
The problem of sparse representation has significant applications in signal processing. The spark of a dictionary plays a crucial role in the study of sparse representation. Donoho and Elad initially explored the spark, and they provided a…
Breakthrough Sparsity Theorem, derived independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]}, Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform. Theory, 2004]}…
The aim of this paper is to establish a few uncertainty principles for the Fourier and the short-time Fourier transforms. Also, we discuss an analogue of Donoho--Stark uncertainty principle and provide some estimates for the size of the…
Celebrated breakthrough sparsity theorem obtained independently by Donoho and Elad \textit{[Proc. Natl. Acad. Sci. USA, 2003]} and Gribonval and Nielsen \textit{[IEEE Trans. Inform. Theory, 2003]} and Fuchs \textit{[IEEE Trans. Inform.…
Spark plays a great role in studying uniqueness of sparse solutions of the underdetermined linear equations. In this article, we derive a new lower bound of spark. As an application, we obtain a new criterion for the uniqueness of sparse…
Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $p=1$ or $p=\infty$. Let $(\{f_\alpha\}_{\alpha\in \Omega}, \{\tau_\alpha\}_{\alpha\in \Omega})$ and $(\{g_\beta\}_{\beta\in \Delta}, \{\omega_\beta\}_{\beta\in \Delta})$ be…
We present some forms of uncertainty principle which involve in a new way localization operators, the concept of $\varepsilon$-concentration and the standard deviation of $L^2$ functions. We show how our results improve the classical…
We show that one of the two important uncertainty principles derived by Maccone and Pati \textit{[Phys. Rev. Lett., 2014]} can be derived for arbitrary maps defined on subsets of $\mathcal{L}^p$ spaces for $1< p<\infty$. Our main tool is…
We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho--Stark uncertainty principle, and Meshulam's non-abelian uncertainty principle, have little to do…
In this paper we revisit under-determined linear systems of equations with sparse solutions. As is well known, these systems are among core mathematical problems of a very popular compressed sensing field. The popularity of the field as…
Let $(\{f_j\}_{j=1}^n, \{\tau_j\}_{j=1}^n)$ and $(\{g_k\}_{k=1}^m, \{\omega_k\}_{k=1}^m)$ be p-Schauder frames for a finite dimensional Banach space $\mathcal{X}$. Then for every $x \in \mathcal{X}\setminus\{0\}$, we show that \begin{align}…
We develop new discrete uncertainty principles in terms of numerical sparsity, which is a continuous proxy for the 0-norm. Unlike traditional sparsity, the continuity of numerical sparsity naturally accommodates functions which are nearly…
In an incoherent dictionary, most signals that admit a sparse representation admit a unique sparse representation. In other words, there is no way to express the signal without using strictly more atoms. This work demonstrates that sparse…
This chapter provides a principled introduction to uncertainty relations underlying sparse signal recovery. We start with the seminal work by Donoho and Stark, 1989, which defines uncertainty relations as upper bounds on the operator norm…
Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics…
We present a flexible framework for uncertainty principles in spectral graph theory. In this framework, general filter functions modeling the spatial and spectral localization of a graph signal can be incorporated. It merges several…
In this paper, we study forms of the uncertainty principle suggested by problems in control theory. We obtain a version of the classical Paneah-Logvinenko-Sereda theorem for the annulus. More precisely, we show that a function with spectrum…
The present work aims to search for an implementation of a new symmetry in the space-time by introducing the idea of an invariant minimum speed scale ($V$). Such a lowest limit $V$, being unattainable by the particles, represents a…
Radar SLAM is robust in challenging conditions, such as fog, dust, and smoke, but suffers from the sparsity and noisiness of radar sensing, including speckle noise and multipath effects. This study provides a performance-enhanced radar SLAM…