Demystifying the Coherence Index in Compressive Sensing [Lecture Notes]

Stanković, Ljubiša; Mandic, Danilo P.; Daković, Miloš; Kisil, Ilya

doi:10.1109/msp.2019.2945080

Cited by 18 publications

(25 citation statements)

References 20 publications

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“…The condition in (6) guarantees that the solutions obtained by minimizing the 0 -norm and 1 -norm produce the same common unique solution [7]. This condition guaranties unique solution produced by the OMP algorithm [4], [5], [9], [11], [13].…”

Section: Uniqueness Of the Omp Reconstructionmentioning

confidence: 88%

“…Although this criterion is commonly derived based on the support uncertainty principle [4], [7] or Gershogorin disk theorem [13], the coherence index condition (5) follows also as a result of the analysis in the process of detection of positions of non-zero values in original vector X [11].…”

Section: Uniqueness Of the Omp Reconstructionmentioning

confidence: 99%

“…This implicit inequality is easily solved by direct check, starting from K = 1, and then increasing the value of K by one, until the inequality holds [11]. The procedure is stopped for the smallest K when (11) does not hold.…”

Section: Improved Bound Derivationmentioning

confidence: 99%

“…In all cases, for any measurement matrix A, condition α A (2K −1) ≤ µ holds. This means that a more optimistic bound for K is obtained by (11) than the conventional CS bound in (6).…”

Section: Improved Bound Derivationmentioning

confidence: 99%

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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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Section: Uniqueness Of the Omp Reconstructionmentioning

confidence: 88%

Section: Uniqueness Of the Omp Reconstructionmentioning

confidence: 99%

Section: Improved Bound Derivationmentioning

confidence: 99%

“…In all cases, for any measurement matrix A, condition α A (2K −1) ≤ µ holds. This means that a more optimistic bound for K is obtained by (11) than the conventional CS bound in (6).…”

Section: Improved Bound Derivationmentioning

confidence: 99%

See 2 more Smart Citations

Improved Coherence Index-Based Bound in Compressive Sensing

Stanković

Brajović

Mandic

et al. 2021

IEEE Signal Process. Lett.

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“…The proposed uncertainty principle bound in (20) is always greater or equal to the bound in (17). If we used the equality condition in the Schwartz inequality, from (14) to (15), which reads |u k (n)||x(n)| = c |u k (n)||X(k)|, for all n, k, the unit energy signal and its GFT should be constant |x(n)| = 1/ √ M and |X(k)| = 1/ √ K, as in [8]. Then, relation (15) results in a tighter bound,…”

Section: (22)mentioning

confidence: 99%

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

Stanković

2020

IEEE Signal Process. Lett.

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The classical support uncertainty principle states that the signal and its discrete Fourier transform (DFT) cannot be localized simultaneously in an arbitrary small area in the time and the frequency domain. The product of the number of nonzero samples in the time domain and the frequency domain is greater or equal to the total number of signal samples. The support uncertainty principle has been extended to the arbitrary orthogonal pairs of signal basis and the graph signals, stating that the product of supports in the vertex domain and the spectral domain is greater than the reciprocal squared maximum absolute value of the basis functions. This form is then used in compressive sensing and sparse signal processing to define the reconstruction conditions. In this paper, we will revisit the graph signal uncertainty principle using the graph Rihaczek distribution as an analysis tool and derive an improved bound for the support uncertainty principle of graph signals.

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