1. Introduction to Compressive Sensing
Richard Baraniuk, Compressive sensing. IEEE Signal Processing Magazine, 24(4), pp , July 2007)
Emmanuel Candès and Michael Wakin, An introduction to compressive sampling. IEEE Signal Processing Magazine, 25(2), pp , March 2008
A course on compressive sensing,

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

3. Introduction Compressive sensing First CS theory 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
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’
Inverse
Quantization
human coding y’ s
 : transform basis (e.g., DCT basis)

6. Theory and Core Technology compression
K-sparse
most of the energy is at low frequencies
K non-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
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 Signal Representation for Compressive Sensing in Wireless 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 SettingX
x11
x12
x13
x14
x21
x22
x23
x24 … ..
Irregular network setting [4]
Graph wavelet
Diffusion wavelet
Example of the considered multi-hop topology.

22. Measurement matrix  Built on 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 Error pre-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 , Mar
[5] M. Rabbat, J. Haupt, A. Singh, and R. Novak, “Decentralized Compression and Predistribution via Randomized Gossiping,” in IPSN, 2006.