Compressive Sensing for Multimedia Communications

in Wireless Sensor Networks

By:Wael Barakat Rabih Saliba

EE381K-14 MDDSPLiterary Survey Presentation

March 4th, 2008

2

Introduction to Data Acquisition

Shannon/Nyquist Sampling Theorem Must sample more than twice the signal

bandwidth,

Might end up with a huge number of samples Need to Compress!

Doing more work than needed?

Sample Compressx N KTransmit/Store

N > KTransformEncoder

3

What is Compressive Sensing?

Combines sampling & compression into one non-adaptive linear measurement process.

Measure inner products between signal and a set of functions: Measurements no longer point samples, but… Random sums of samples taken across entire

signal.

4

Compressive Sensing (CS)

Consider an N-length, 1-D, DT signal x in

Can represent x in terms of a basis of vectors

or

where s is the vector of weighing coefficients and is the basis matrix.

CS exploits signal sparsity: x is a linear combination of just K basis vectors with K < N (Transform coding)

Key Paper #1

NR

Nii 1

N

iiisx

1

sx

1N

NN

5

Compressive Sensing

Measurement process computes M < N inner products between x and as in . So:

is a random matrix whose elements are i.i.d Gaussian random variables with zero-mean and 1/N variance.

Use norm reconstruction to recover sparsest coefficients satisfying such that

Nii 1

[Baraniuk, 2005]

jj xy , ssxy

1

1minarg ss sy

6

Single-Pixel Imaging

New camera architecture based on Digital Micromirror Devices (DMD) and CS.

Optically computes random linear measurements of the scene under view.

Measures inner products between incident light x and 2-D basis functions

Employs only a single photon detector Single Pixel!

Key Paper #2

mxmy ,][ m

7

Single-Pixel Imaging

Each mirror corresponds to a pixel, can be oriented as1/0.

To compute CS measurements, set mirror orientations randomly using a pseudo-random number generator.

Original

10%

20%[Wakin et al.,

2006]

8

Distributed CS

Notion of an ensemble of signals being jointly sparse

3 Joint Sparsity Models: Signals are sparse and share common

component Signals are sparse and share same supports Signals are not sparse

Each sensor collects a set of measurements independently

Key Paper #3

9

Distributed CS

Each sensor acquires a signal and performs Mj measurements

Need a measurement matrix Use node ID as a seed for the random

generation Send measurement, timestamp, index and

node ID Build measurement matrix at receiver and

start reconstructing signal.

jjjj xy

NM j

10

Distributed CS

Advantages: Simple, universal encoding, Robustness, progressivity and resilience, Security, Fault tolerance and anomaly detection, Anti-symmetrical.

11

Conclusion

Implement CS on images and explore the quality to complexity tradeoff for different sizes and transforms.

Further explore other hardware architectures that directly acquire CS data

12

References

E. Candès, “Compressive Sampling,” Proc. International Congress of Mathematics, Madrid, Spain, Aug. 2006, pp. 1433-1452.

Baraniuk, R.G., "Compressive Sensing [Lecture Notes]," IEEE Signal Processing Magazine, vol. 24, no. 4, pp. 118-121, July 2007.

M. Duarte, M. Wakin, D. Baron, and R. Buraniak, “Universal Distributed Sensing via Random Projections”, Proc. Int. Conference on Information Processing in Sensor Network, Nashville, Tennessee, April 2006, pp. 177-185.

R. Baraniuk, J. Romberg, and M. Wakin, “Tutorial on Compressive Sensing”, 2008 Information Theory and Applications Workshop, San Diego, California, February 2008.

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly and R. Baraniuk, “An Architecture for Compressive Imaging”, Proc. Int. Conference on Image Processing, Atlanta, Georgia, October 2006, pp. 1273-1276.

M. Duarte, M. Davenport, D. Takhar, J. Laska, T. Sun, K. Kelly and R. Baraniuk, “Single-Pixel Imaging via Compressive Sampling”, IEEE Signal Processing Magazine [To appear].