Improved Coherence Index-Based Bound in Compressive Sensing

Within the compressive sensing (CS) paradigm, sparse signals can be reconstructed based on a reduced set of measurements, whereby 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 uniqueness metrics with considerable practical importance. 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 index-based lower bound for signal sparsity. The results are generalized to the uniqueness condition of the $l_0$-norm minimization for a signal represented in two orthonormal bases.

[1]  R. B. Deshmukh,et al.  A Systematic Review of Compressive Sensing: Concepts, Implementations and Applications , 2018, IEEE Access.

[2]  R. Varga.  Geršgorin And His Circles , 2004 .

[3]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[4]  Ljubisa Stankovic,et al.  Quantization in Compressive Sensing: A Signal Processing Approach , 2019, IEEE Access.

[5]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[6]  Michael Elad,et al.  A generalized uncertainty principle and sparse representation in pairs of bases , 2002, IEEE Trans. Inf. Theory.

[7]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[8]  Danilo P. Mandic,et al.  Demystifying the Coherence Index in Compressive Sensing , 2019 .

[9]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[10]  Irena Orovic,et al.  A Tutorial on Sparse Signal Reconstruction and Its Applications in Signal Processing , 2018, Circuits Syst. Signal Process..

[11]  Lie Wang,et al.  Stable Recovery of Sparse Signals and an Oracle Inequality , 2010, IEEE Transactions on Information Theory.

[12]  Anthony G. Constantinides,et al.  Data Analytics on Graphs Part I: Graphs and Spectra on Graphs , 2020, Found. Trends Mach. Learn..

[13]  Ljubiša Stanković,et al.  Analysis of the Reconstruction of Sparse Signals in the DCT Domain Applied to Audio Signals , 2018, IEEE/ACM Transactions on Audio, Speech, and Language Processing.

[14]  Ljubisa Stankovic.  The Support Uncertainty Principle and the Graph Rihaczek Distribution: Revisited and Improved , 2020, IEEE Signal Processing Letters.

[15]  Jian Wang,et al.  On the Recovery Limit of Sparse Signals Using Orthogonal Matching Pursuit , 2012, IEEE Transactions on Signal Processing.

[16]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[17]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[18]  Ljubisa Stankovic,et al.  On the Sparsity Bound for the Existence of a Unique Solution in Compressive Sensing by the Gershgorin Theorem , 2021, Signal Process..

[19]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.