Volkan Cevher Richard Baraniuk Rice University dsp.rice/cs

1. Distributed CompressiveSensing Volkan Cevher Richard Baraniuk Rice University dsp.rice.edu/cs

2. Pressure is on Digital Sensors • Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support higher resolution / denser sampling • ADCs, cameras, imaging systems, microarrays, … x large numbers of sensors • image data bases, camera arrays, distributed wireless sensor networks, … xincreasing numbers of modalities • acoustic, RF, visual, IR, UV, x-ray, gamma ray, …

3. Pressure is on Digital Sensors • Success of digital data acquisition is placing increasing pressure on signal/image processing hardware and software to support higher resolution / denser sampling • ADCs, cameras, imaging systems, microarrays, … x large numbers of sensors • image data bases, camera arrays, distributed wireless sensor networks, … xincreasing numbers of modalities • acoustic, RF, visual, IR, UV = deluge of data • how to acquire, store, fuse, process efficiently?

4. Sensing by Sampling • Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) sample

5. Sensing by Sampling • Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) too much data! sample

6. Sensing by Sampling • Long-established paradigm for digital data acquisition • uniformly sampledata at Nyquist rate (2x Fourier bandwidth) • compress data sample compress transmit/store JPEG JPEG2000 … receive decompress

7. Sparsity / Compressibility largewaveletcoefficients (blue = 0) largeGabor (TF)coefficients pixels widebandsignalsamples frequency time

8. What’s Wrong with this Picture? • Long-established paradigm for digital data acquisition • uniformly sample data at Nyquist rate • compress data sample compress transmit/store sparse /compressiblewavelettransform receive decompress

9. What’s Wrong with this Picture? • Why go to all the work to acquire N samples only to discard all but K pieces of data? sample compress transmit/store sparse /compressiblewavelettransform receive decompress

10. What’s Wrong with this Picture? nonlinear processing nonlinear signal model (union of subspaces) linear processing linear signal model (bandlimited subspace) sample compress transmit/store sparse /compressiblewavelettransform receive decompress

11. Compressive Sensing • Directly acquire “compressed” data • Replace samples by more general “measurements” compressive sensing transmit/store receive reconstruct

12. Compressive Sampling • Dimensionality reductionvia random linear measurements • As long as can recover sparse signal exactly from msmntsvia linear program or greedy algorithm signal sparsein basis signal

13. CS Hallmarks • CS changes the rules of the data acquisition game • enables the design of new hardware and algorithms • sub-Nyquist A/D converters, cameras, imaging algorithms, … • Universal • same random projections / hardware can be used for anycompressible signal class (generic hardware) • Democratic • each measurement carries the same amount of information • simple encoding • robust to measurement loss and quantization • Asymmetrical • most processing at decoder • Random projections weakly encrypted

14. DistributedCompressed Sensing

15. Network of Sensors Transmitting raw datatypically inefficient destination rawdata

16. Can we exploit intra-sensor and inter-sensorstructure to jointly compress? Signal Structure

17. Collaborative Sensing Collaboration introduces inter-sensor communication overhead complexity at sensors destination compressed data

18. Take incoherent (random) measurements at each sensor Reconstruct individuallyat destination Exploit intra-sensor structure(sparsity/compressibility) IndependentCompressive Sensing destination compressed data

19. Distributed Compressed Sensing (DCS) destination compressed data • Take incoherent (random) measurements at each sensor • Reconstruct/process jointlyat destination • Exploit intra/inter-sensorstructure [D. Baron, M. Wakin, M. Duarte, S. Sarvotham, R. Baraniuk, 2005]

20. Example Applications • Distributed compression/reconstruction • exploit commonality in signal structure • Distributed processing for target localization • exploit spatial sparsity • Distributed multiview imaging • Exploit commonality in background and sparsity in foreground

21. Signal Compression/Reconstruction viaCommon SparseSupport

22. Common Sparse Support Model Joint sparsity model Observe J signals, each K-sparse in some basis Signals share sparse component locations, but have different coefficients …

23. Common Sparse Support Model Ex: audio signals • sparse in Fourier domain • same frequencies received by each microphone • different attenuations and delays (magnitudes and phases) …

24. Common Sparse Support Model Theorem As the number of sensors J, the number of measurements required per sensor for perfect reconstruction  K+1 …

25. Common Support Recovery K=5 N=50 J= Independent Joint

26. Real Data Example Environmental Sensing in Intel Berkeley Lab J = 49 sensors, N =1024 samples each Compare: transform coding approx K largest terms per sensor independent CS 4K measurements per sensor DCS (JSM-2) 4K measurements per sensor

27. Light Intensity - Wavelets, K = 100

28. Temperature - Wavelets, K = 20

29. Decentralized Localization via Spatial Sparsity

30. Localization as Sparse Approximation Number of targets is sparse over the space Sparse approximation Create a sparsity basis for target locations Sample at K log N rate!

31. Localization via Spatial Sparsity • Spatial localization problem = sparse approximation problem • Use observed signal at each sensor to predict signals at other sensors • Compressive (random) measurements of signal predictions compared with actual compressive measurements TO DO WHAT? • Slide is still vague

32. Localization via Spatial Sparsity Synthetic Example: Decentralized consensus

33. Localization via Spatial Sparsity Field Example: Field example: 5 vehicle convoy, 2 HMV’s and 3 commercial SUV’s.

34. Multiview Imaging viaForeground Sparsity

35. Compressive Multiview Tracking

36. Compressive 3D Reconstruction Comparison with state-of-the-art Normalized computation time

37. Conclusions • Compressive sensing naturally suited to sensor network applications • sub-Nyquist sampling at the signals’ joint sparsity rate • communication bandwidth usage scales logarithmically with the number of sensors and/or desired resolution • democratic compressive measurements robust to quantization, noise, and packet loss • universality of compressive measurements enables design/deployment of inexpensive generic sensing hardware • Example applications • distributed compression/reconstruction • one-bit decentralized localization • multiview camera network processing

38. Conclusions • Compressive sensing naturally suited to sensor network applications • sub-Nyquist sampling at the signals’ joint sparsity rate • communication bandwidth usage scales logarithmically with the number of sensors and/or desired resolution • democratic compressive measurements robust to quantization, noise, and packet loss • universality of compressive measurements enables design/deployment of inexpensive generic sensing hardware • Open research problems • new joint sparsity models • efficient reconstruction/processing algorithms • relationship with information theory (Slepian-Wolf coding)