reconstruction algorithms for compressive sensing i
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ACCESS IC LAB
Graduate Institute of Electronics Engineering, NTU
Reconstruction Algorithms for Compressive Sensing I
Presenter: 黃乃珊Advisor: 吳安宇 教授
Date: 2014/03/25
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Schedule
19:30 @ EEII-225日期 內容 Lab & HW Speaker
3/11 Introduction to Compressive Sensing System Nhuang
3/25 Reconstruction Algorithm Nhuang
4/8 Reconstruction Algorithm Lab1 Nhuang
4/15 Break; 決定期末題目方向4/22 Sampling Algorithm: Yumin
4/29 Midterm Presentation (Tutorial, Survey)
5/6 Application: Single Pixel Camera Lab2 Yumin
5/13 ~ 6/10 期末報告討論6/24 Final Presentation
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
Outline
Review Compressive Sensing Reconstruction Algorithms for Compressive Sensing Basis Pursuit Orthogonal Matching Pursuit Reference
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ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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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
Sampling ReconstructionChannel
𝒚𝑴=𝚽𝑴×𝑵 𝒙𝑵
𝒙𝑵 �̂�𝑵
𝒚𝑴+𝒏𝒐𝒊𝒔𝒆
(1−𝛿) ∙𝑀𝑁 ∙‖𝑥‖22≤‖Φ 𝑥‖2
2≤ (1+𝛿) ∙𝑀𝑁 ∙‖𝑥‖22
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Reconstruction
Original underdetermined question
Linear programming question
Two condition Restricted Isometry property (RIP) Sparse signal
min𝑥
‖𝒙‖0 s .t .𝚽 𝒙=𝒚 ,‖𝒙‖0≔¿{𝑘:𝑥𝑘≠0 }
min𝑥
‖𝒙‖1 s . t .𝚽 𝒙=𝒚 ,‖𝒙‖1≔∑𝑖
¿ 𝑥 𝑖∨¿¿
NP-hard!
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Recovery Algorithms for Compressive Sensing
Linear Programming Basis Pursuit (BP)
Greedy Algorithm Matching Pursuit
Orthogonal Matching Pursuit (OMP) Stagewise Orthogonal 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)
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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
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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
basis
=Measurement Input
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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
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Orthogonal Matching Pursuit (OMP) [5]
Use greedy algorithm to iteratively recover sparse signal Procedure:
1. Initialize2. Find the column that is most correlated3. Set Union (add one col. every iter.)4. Solve the least squares 5. Update data and residual6. Back to step 2 or output
[14]
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Stagewise Orthogonal Matching Pursuit (StOMP) [6]
Derive from OMP, but with small fixed number of iteration Procedure:
1. Initialize2. Find the column that is most correlated3. Hard thresholding4. Set Union (add some col. every iter.)5. Find corresponding x by projection6. Update data and residual7. Back to step 2 or output
better global optimizationcorrelation
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Compressive Sampling Matching Pursuit (CoSaMP)[7]
Inspired by the RIP, the energy in proxy approximates the energy in target signal
Procedure:1. Initialize2. Proxy3. Set Union4. Signal estimation by projection5. Prune approximation6. Update data and residual7. Back to step 2 or output
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Subspace Pursuit (SP) [8]
Re-evaluate all candidates at each iteration Procedure:
1. Initialize2. Proxy3. Set Union4. Signal estimation by projection5. Prune approximation6. Update data and residual7. Back to step 2 or output
ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU
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Next Lecture Linear Programming
Basis Pursuit (BP) Greedy Algorithm
Matching Pursuit Orthogonal Matching Pursuit (OMP) Stagewise Orthogonal 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)
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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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