Compressive Sensing for Multimedia Communications in Wireless Sensor Networks

1. Compressive Sensing for Multimedia Communications in Wireless Sensor Networks EE381K-14 MDDSPLiterary Survey Presentation March 4th, 2008 By:Wael Barakat Rabih Saliba

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? N > K K N Sample Transform Encoder Transmit/Store Compress x

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. Key Paper #1 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)

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 [Baraniuk, 2005]

6. Key Paper #2 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!

7. Original 10% 20% 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. [Wakin et al., 2006]

8. Key Paper #3 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

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.

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].