Introduction to Compressive Sensing

1. Introduction to Compressive Sensing Richard Baraniuk, Compressive sensing. IEEE Signal Processing Magazine, 24(4), pp. 118-121, July 2007) Emmanuel Candès and Michael Wakin, An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), pp. 21 - 30, March 2008 A course on compressive sensing, http://w3.impa.br/~aschulz/CS/course.html

2. Outline • Introduction to compressive sensing (CS) • First CS theory • Concepts and applications • Theory • Compression • Reconstruction

3. Introduction • Compressive sensing • Compressed sensing • Compressive sampling • First CS theory • E. Cand`es, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. Cand`es Romberg Tao

4. Compressive Sensing: concept and applications

5. Compression/Reconstruction Transmit X RNx1 CS sampling yRMx1 Quantization human coding   RMxN Measurement matrix CS Reconstruction Optimization Inverse transform (e.g., IDCT) X’ s Inverse Quantization human coding y’ : transform basis (e.g., DCT basis)

6. Theory and Core Technologycompression • K-sparse • most of the energy is at low frequencies • Knon-zero wavelet (DCT) coefficients

7. Compression Measurement matrix

8. Compression transform basis coefficient

9. Compression transform basis coefficient

10. Reconstruction

11. Reconstruction: optimization (1) NP-hard problem (2) Minimum energy ≠ k-sparse (3) Linear programming [1][2] Orthogonal matching pursuit (OMP) (4) Greedy algorithm [3]

12. Compressive sensing: significant parameters • What measurement matrix  should we use? • How many measurements? (M=?) • K-sparse?

13. Measurement Matrix  Incoherence (1) Correlation between  and 

14. Examples = noiselet, = Haar wavelet  (,)=2 = noiselet, = Daubechies D4 (,)=2.2 = noiselet, = Daubechies D8 (,)=2.9 • Noiselets are also maximally incoherent with spikes and incoherent with the Fourier basis = White noise (random Gaussian)

15. Restricted Isometry Property (RIP)preserving length • RIP: For each integer k = 1, 2, …, define the isometry constant k of a matrix A as the smallest number such that • A approximately preserves the Euclidean length of k-sparse signals (2) Imply that k-sparse vectors cannot be in the nullspace of A (3) All subsets of s columns taken from A are in fact nearly orthogonal • To design a sensing matrix , so that any subset of columns of size k be approximately orthogonal.

16. How many measurements ?

17. Single-Pixel CS Camera[Baraniuk and Kelly, et al.]

20. On the Interplay Between Routing and SignalRepresentation for Compressive Sensing inWireless Sensor Networks G. Quer, R. Masiero, D. Munaretto, M. Rossi, J. Widmer and M. Zorzi University of Padova, Italy. DoCoMo Euro-Labs, Germany Information Theory and Applications Workshop (ITA 2009)

21. Network Scenario Setting X Irregular network setting [4] Graph wavelet Diffusion wavelet Example of the considered multi-hop topology.

22. Measurement matrix Built on routing path Routing path …………………… ……………… …………………… ……………………

23. Measurement matrix  • R1:  is built according to routing protocol, •  randomly selected from {+1, -1} • R2:  is built according to routing protocol •  randomly selected from (0, 1] • R3: has all coefficients in  randomly selected from {+1, -1} • R4: has all coefficients in  randomly selected from(0, 1]

24. Transform basis • T1: DCT • T2: Haar Wavelet • T3: Horizontal difference • T4: Vertical difference + Horizontal difference

25. Degree of sparsity H-diff VH-diff Haar DCT

26. Incoherence DCT Haar H-diff VH-diff

27. Performance Comparison • Random sampling (RS) • each node sends its data with probability P= M/N, the data packets are not processed at internal nodes but simply forwarded. • RS-CS • the data values are combined with that of any other node encountered along the path. Routing path

28. Reconstruction Error

29. Reconstruction Errorpre-distribution for T3 and T4 [5]

30. Research issues when applying CS in Sensor Networks • How to construct measurement matrix  • Incoherent with transform basis  • Distributed • M=? • How to choose transformation basis  • Sparsity • Incoherent with measurement matrix  • Irregular sensor deployment • Graph wavelet • Diffusion wavelet

31. References [1] Bloomfield, P., Steiger, W., Least Absolute Deviations: Theory, Applications, and Algorithms. Progr. Probab. Statist. 6, Birkhäuser, Boston, MA, 1983. [2] Chen, S. S., Donoho, D. L., Saunders, M. A, Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20 (1999), 33–61. [3] J. Tropp and A. C. Gilbert, “Signal recovery from partial information via orthogonal matching pursuit,” Apr. 2005, Preprint. [4] J. Haupt, W.U. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Processing Mag., vol. 25, no. 2, pp. 92-101, Mar. 2008. [5] M. Rabbat, J. Haupt, A. Singh, and R. Novak, “Decentralized Compression and Predistribution via Randomized Gossiping,” in IPSN, 2006.