Analysis of the Orthogonal Matching Pursuit Algorithm with Prior Information

Compressed sensing uses a small amount of compressed data to represent high dimensional data, where the reconstruction algorithm is one of the main research topics. Among various algorithms, orthogonal matching pursuit (OMP) recovers the original signals in a greedy manner. Recently, the performance bound of OMP algorithm has been widely investigated. In this paper, we study OMP algorithm in the scenario that the decoder has the support probability vector, which can be used as prior information for recovery. We develop the relationship between the restricted isometry property (RIP) constant $$\delta _{K + 1}$$δK+1 and prior information. Based on the RIP results, some special cases that are further discussed to provide a deeper understanding of the relationship. The derived results show that the effective prior information is useful for relaxing the performance bound of the RIP isometry $$\delta _{K + 1}$$δK+1.

[1]  Wei Lu,et al.  Modified compressive sensing for real-time dynamic MR imaging , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[2]  Olgica Milenkovic,et al.  Subspace Pursuit for Compressive Sensing Signal Reconstruction , 2008, IEEE Transactions on Information Theory.

[3]  Hassan Mansour,et al.  Recovering Compressively Sampled Signals Using Partial Support Information , 2010, IEEE Transactions on Information Theory.

[4]  Michael Rabbat,et al.  Compressed Network Monitoring , 2007, 2007 IEEE/SP 14th Workshop on Statistical Signal Processing.

[5]  Yonina C. Eldar,et al.  Block-Sparse Signals: Uncertainty Relations and Efficient Recovery , 2009, IEEE Transactions on Signal Processing.

[6]  Michael B. Wakin,et al.  Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property , 2009, IEEE Transactions on Information Theory.

[7]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[8]  Babak Hassibi,et al.  Recovering Sparse Signals Using Sparse Measurement Matrices in Compressed DNA Microarrays , 2008, IEEE Journal of Selected Topics in Signal Processing.

[9]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[10]  Jiaru Lin,et al.  Partial segmented compressed sampling for analog-to-information conversion , 2013 .

[11]  Entao Liu,et al.  Orthogonal Super Greedy Algorithm and Applications in Compressed Sensing ∗ , 2010 .

[12]  Seung Jun Baek,et al.  Sufficient Conditions on Stable Recovery of Sparse Signals With Partial Support Information , 2013, IEEE Signal Processing Letters.

[13]  Abdulhakem Y. Elezzabi,et al.  Maximum Frame Rate Video Acquisition Using Adaptive Compressed Sensing , 2011, IEEE Transactions on Circuits and Systems for Video Technology.

[14]  Jubo Zhu,et al.  Recovery of sparse signals using OMP and its variants: convergence analysis based on RIP , 2011 .

[15]  Olgica Milenkovic,et al.  Subspace Pursuit for Compressive Sensing: Closing the Gap Between Performance and Complexity , 2008, ArXiv.

[16]  Ray Maleh,et al.  Improved RIP Analysis of Orthogonal Matching Pursuit , 2011, ArXiv.

[17]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[18]  E. F. Kaasschieter,et al.  Preconditioned conjugate gradients for solving singular systems , 1988 .

[19]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[20]  Weiyu Xu,et al.  Improved sparse recovery thresholds with two-step reweighted ℓ1 minimization , 2010, 2010 IEEE International Symposium on Information Theory.

[21]  A. Ravi Kumar,et al.  Compressed-Sensing-Enabled Video Streaming for Wireless Multimedia Sensor Networks , 2014 .

[22]  Cristiano Jacques Miosso,et al.  Compressive Sensing With Prior Information: Requirements and Probabilities of Reconstruction in ${\mbi \ell}_{\bf 1}$-Minimization , 2013, IEEE Transactions on Signal Processing.

[23]  Yanjun Han,et al.  Time-varying channel estimation based on dynamic compressive sensing for OFDM systems , 2014, 2014 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting.

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

[25]  Ling-Hua Chang,et al.  An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit , 2014, IEEE Transactions on Information Theory.

[26]  Weiyu Xu,et al.  Analyzing Weighted $\ell_1$ Minimization for Sparse Recovery With Nonuniform Sparse Models , 2010, IEEE Transactions on Signal Processing.

[27]  Yi Shen,et al.  A Remark on the Restricted Isometry Property in Orthogonal Matching Pursuit , 2012, IEEE Transactions on Information Theory.

[28]  Chiara Sabatti,et al.  Empirical Bayes Estimation of a Sparse Vector of Gene Expression Changes , 2005, Statistical applications in genetics and molecular biology.

[29]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[30]  Yue Wang,et al.  Performance analysis of partial support recovery and signal reconstruction of compressed sensing , 2014, IET Signal Process..

[31]  Jamie S. Evans,et al.  Compressed Sensing With Prior Information: Information-Theoretic Limits and Practical Decoders , 2013, IEEE Transactions on Signal Processing.

[32]  Deanna Needell,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, ArXiv.