The Support Uncertainty Principle and the Graph Rihaczek Distribution: Revisited and Improved

Stanković, Ljubiša

doi:10.1109/lsp.2020.3000686

Cited by 5 publications

(5 citation statements)

References 13 publications

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“…An approach to improve the coherence index-based bound has been proposed in [27] by analyzing the initial estimate and the support uncertainty principle as in [28,29]. The approach presented in [27] guarantees unique reconstruction of a sparse signal using the orthogonal matching pursuit approach.…”

Section: Introductionmentioning

confidence: 99%

On the sparsity bound for the existence of a unique solution in compressive sensing by the Gershgorin theorem

Stanković

2022

Signal Processing

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Section: Introductionmentioning

confidence: 99%

On the sparsity bound for the existence of a unique solution in compressive sensing by the Gershgorin theorem

Stanković

2022

Signal Processing

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“…, k K } and are summed in phase. If we take into account the notation for sorted values in (12), then this accumulated disturbance value becomes…”

Section: Improved Bound Derivationmentioning

confidence: 99%

“…Assume, as in [2], [10], that the signal is sparse in these bases with sparsities ||X|| 0 = K and ||Y|| 0 = L, and that the Parseval's theorem holds in both bases, ||X|| 2 2 = 1 and ||Y|| 2 2 = 1. Form a function L(n, k, l) = X(k)Y * (l)u k (n)v * l (n) as in [12] such that…”

Section: Numerical Examplesmentioning

confidence: 99%

Improved Coherence Index-Based Bound in Compressive Sensing

Stanković

Brajović

Mandic

et al. 2021

IEEE Signal Process. Lett.

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Within the compressive sensing paradigm, sparse signals can be reconstructed based on a reduced set of measurements. The reliability of the solution is determined by its uniqueness. With its mathematically tractable and feasible calculation, the coherence index is one of very few CS metrics with considerable practical importance. In this paper, we propose an improvement of the coherence-based uniqueness relation for the matching pursuit algorithms. Starting from a simple and intuitive derivation of the standard uniqueness condition, based on the coherence index, we derive a less conservative coherence indexbased lower bound for signal sparsity. The results are generalized to the uniqueness condition of the l0-norm minimization for a signal represented in two orthonormal bases.

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“…Recall that functions H k (λ) satisfy ∑ K−1 k=0 H k (λ) = 1. In this example, we consider approximations of these functions using LS approximation (42), as well as the approximations based on Legendre polynomial (44). For convenience, we also consider the approximation based on Chebyshev polynomial.…”

Section: Legendre Polynomialmentioning

confidence: 99%

“…The support uncertainty principle in the LGFT can be derived in the same way as in the case of the graph Fourier transform. A simple derivation procedure, as in [44,45], will be followed. If a function…”

Section: Support Uncertainty Principle In the Lgftmentioning

confidence: 99%

From Time–Frequency to Vertex–Frequency and Back

et al. 2021

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The paper presents an analysis and overview of vertex–frequency analysis, an emerging area in graph signal processing. A strong formal link of this area to classical time–frequency analysis is provided. Vertex–frequency localization-based approaches to analyzing signals on the graph emerged as a response to challenges of analysis of big data on irregular domains. Graph signals are either localized in the vertex domain before the spectral analysis is performed or are localized in the spectral domain prior to the inverse graph Fourier transform is applied. The latter approach is the spectral form of the vertex–frequency analysis, and it will be considered in this paper since the spectral domain for signal localization is well ordered and thus simpler for application to the graph signals. The localized graph Fourier transform is defined based on its counterpart, the short-time Fourier transform, in classical signal analysis. We consider various spectral window forms based on which these transforms can tackle the localized signal behavior. Conditions for the signal reconstruction, known as the overlap-and-add (OLA) and weighted overlap-and-add (WOLA) methods, are also considered. Since the graphs can be very large, the realizations of vertex–frequency representations using polynomial form localization have a particular significance. These forms use only very localized vertex domains, and do not require either the graph Fourier transform or the inverse graph Fourier transform, are computationally efficient. These kinds of implementations are then applied to classical time–frequency analysis since their simplicity can be very attractive for the implementation in the case of large time-domain signals. Spectral varying forms of the localization functions are presented as well. These spectral varying forms are related to the wavelet transform. For completeness, the inversion and signal reconstruction are discussed as well. The presented theory is illustrated and demonstrated on numerical examples.

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The Support Uncertainty Principle and the Graph Rihaczek Distribution: Revisited and Improved

Cited by 5 publications

References 13 publications

On the sparsity bound for the existence of a unique solution in compressive sensing by the Gershgorin theorem

On the sparsity bound for the existence of a unique solution in compressive sensing by the Gershgorin theorem

Improved Coherence Index-Based Bound in Compressive Sensing

From Time–Frequency to Vertex–Frequency and Back

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