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Reconstruction Algorithms for Compressive Sensing I

Reconstruction Algorithms for Compressive Sensing I. Presenter: 黃乃珊 Advisor: 吳安宇 教授 Date: 2014/03/25. Schedule. 19:30 @ EEII-225. Outline. Review Compressive Sensing Reconstruction Algorithms for Compressive Sensing Basis Pursuit Orthogonal Matching Pursuit Reference.

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Reconstruction Algorithms for Compressive Sensing I

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  1. Reconstruction Algorithms for Compressive Sensing I Presenter: 黃乃珊 Advisor: 吳安宇 教授 Date:2014/03/25

  2. Schedule • 19:30 @ EEII-225

  3. Outline • Review Compressive Sensing • Reconstruction Algorithms for Compressive Sensing • Basis Pursuit • Orthogonal Matching Pursuit • Reference

  4. Compressive Sensing in Mathematics • Sampling matrices should satisfy restricted isometry property (RIP) • Ex. Random Gaussian matrices • Reconstruction solves an underdetermined question • Linear Programming (ex. Basis Pursuit) • Greedy Algorithm (ex. Orthogonal Matching Pursuit) • Iterative Thresholding Channel Sampling Reconstruction

  5. Reconstruction • Original underdetermined question • Linear programming question • Two condition • Restricted Isometry property (RIP) • Sparse signal NP-hard!

  6. Recovery Algorithms for Compressive Sensing • Linear Programming • Basis Pursuit (BP) • Greedy Algorithm • Matching Pursuit • Orthogonal Matching Pursuit (OMP) • StagewiseOrthogonal Matching Pursuit (StOMP) • Compressive Sampling Matching Pursuit (CoSaMP) • Subspace Pursuit (SP) • Iterative Thresholding • Iterative Hard Thresholding (IHT) • Iterative Soft Thresholding (IST) • Bayesian Compressive Sensing (BCS) • Approximate Message Passing(AMP)

  7. Basis Pursuit (BP) [3][4] • Find signal representation in overcomplete dictionaries by convex optimization • BP-simplex • Optimize by swapping element • BP-interior • Optimize by modifying coefficient • More common ↑BP-simplex ↑BP-interior

  8. Compressive Sensing in Linear Algebra • Reconstruction is composed of two parts: • Localize nonzero terms • Approximate nonzero value • Do correlation to find the location of non-zero terms • Solve least square problem to find the value • Projection (pseudo-inverse) coefficient = Measurement Input basis

  9. Matrix Inverse • Matrix inverse for invertible square matrix • A square matrix with nonzero determinant • Non-square matrix has enough rank • To find inverse matrix • Gauss-Jordan elimination, • LU decomposition • QR decomposition • Pseudo inverse • To find least square solution

  10. Orthogonal Matching Pursuit (OMP) [5] • Use greedy algorithm to iteratively recover sparse signal • Procedure: • Initialize • Find the column that is most correlated • Set Union (add one col. every iter.) • Solve the least squares • Update data and residual • Back to step 2 or output [14]

  11. Stagewise Orthogonal Matching Pursuit (StOMP) [6] • Derive from OMP, but with small fixed number of iteration • Procedure: • Initialize • Find the column that is most correlated • Hard thresholding • Set Union (add some col. every iter.) • Find corresponding x by projection • Update data and residual • Back to step 2 or output better global optimization correlation

  12. Compressive Sampling Matching Pursuit (CoSaMP)[7] • Inspired by the RIP, the energy in proxy approximates the energy in target signal • Procedure: • Initialize • Proxy • Set Union • Signal estimation by projection • Prune approximation • Update data and residual • Back to step 2 or output

  13. Subspace Pursuit (SP) [8] • Re-evaluate all candidates at each iteration • Procedure: • Initialize • Proxy • Set Union • Signal estimation by projection • Prune approximation • Update data and residual • Back to step 2 or output

  14. Next Lecture • Linear Programming • Basis Pursuit (BP) • Greedy Algorithm • Matching Pursuit • Orthogonal Matching Pursuit (OMP) • StagewiseOrthogonal Matching Pursuit (StOMP) • Compressive Sampling Matching Pursuit (CoSaMP) • Subspace Pursuit (SP) • Iterative Thresholding • Iterative Hard Thresholding (IHT) • Iterative Soft Thresholding (IST) • Bayesian Compressive Sensing (BCS) • Approximate Matching Pursuit (AMP)

  15. Reference [1] E. J. Candes, and M. B. Wakin, "An Introduction To Compressive Sampling," Signal Processing Magazine, IEEE , vol.25, no.2, pp.21-30, March 2008 [2] G. Pope, “Compressive Sensing – A Summary of Reconstruction Algorithm”, Swiss Federal Instituute of Technology Zurich [3] E. J. Candes, and T. Tao, "Decoding by linear programming," IEEE Transactions on  Information Theory, vol.51, no.12, pp. 4203- 4215, Dec. 2005 [4] S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci Comp., vol. 20, no. 1, pp. 33–61, 1999. [5] J. A. Tropp, A. C. Gilbert, “Signal Recovery from Random Measurements via Orthogonal Matching Pursuit,” IEEE Transactions on  Information Theory, vol.53, no.12, pp. 4655-4666, Dec. 2007 [6] D. L. Donoho, Y. Tsaig, I. Drori, and J.-L. Starck, “Sparse solution of underdetermined linear equations by stagewise Orthogonal Matching Pursuit (StOMP),” Information Theory, IEEE Transactions on , vol.58, no.2, pp.1094,1121, Feb. 2012 [7] D. Needell, and J. A. Tropp, "CoSaMP: Iterative signal recovery from incomplete and inaccurate samples." Applied and Computational Harmonic Analysis 26.3 (2009): 301-321. [8]W. Dai, and O. Milenkovic, "Subspace Pursuit for Compressive Sensing Signal Reconstruction," Information Theory, IEEE Transactions on , vol.55, no.5, pp.2230,2249, May 2009

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