1 / 15

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.

joie
Télécharger la présentation

Reconstruction Algorithms for Compressive Sensing I

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related