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Richard Baraniuk Rice University Supported by NSF, ONR, DARPA TI Leadership University Program

Compressive Sensing for Networked Inference. Richard Baraniuk Rice University Supported by NSF, ONR, DARPA TI Leadership University Program. Sensor Networks. Measurement, monitoring, tracking of distributed physical phenomena (“macroscope”) using wireless embedded sensors

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Richard Baraniuk Rice University Supported by NSF, ONR, DARPA TI Leadership University Program

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  1. CompressiveSensing forNetworkedInference Richard Baraniuk Rice University Supported by NSF, ONR, DARPATI Leadership University Program

  2. Sensor Networks • Measurement, monitoring, tracking of distributed physical phenomena(“macroscope”) using wirelessembedded sensors • environmental conditions • industrial monitoring • chemicals • weather • sounds • vibrations • seismic • wildfires • pollutants…

  3. Sensor Networks • Measurement, monitoring, tracking of distributed physical phenomena(“macroscope”) using wirelessembedded sensors • environmental conditions • industrial monitoring • chemicals • weather • sounds • vibrations • seismic • wildfires • pollutants… E. Charbon, M. Vetterli, EPFL

  4. Sensor Networks • Measurement, monitoring, tracking of distributed physical phenomena(“macroscope”) using wirelessembedded sensors • environmental conditions • industrial monitoring • chemicals • weather • sounds • vibrations • seismic • wildfires • pollutants…

  5. Sensor Networks • Measurement, monitoring, tracking of distributed physical phenomena(“macroscope”) using wirelessembedded sensors • environmental conditions • industrial monitoring • chemicals • weather • sounds camera network fusioncenter light data

  6. New Hardware, Software • Hardware platforms • sensing, DSP, networking, communications, power • comm standards: 802.15.4 (Zigbee), Bluetooth, … • Crossbow motes • Berkeley motes • Smart Dust • MoteIV • Rice Gnomes • … • Operating systems • TinyOS • MagnetOS • SOS • Pumpkin • …

  7. Challenges • Computational/power asymmetry • limited compute power on each sensor node • limited (battery) power on each sensor node • Must be energy efficient • minimize communication • Hostile communication environment • multi-hop • high loss rate

  8. Pressure is on Signal Processing • Networked sensing placing increasing pressure on signal/image processing hardware and algs to support higher resolution / denser sampling • ADCs, cameras, imaging systems, … + large numbers of sensors • multi-view signal/image data bases, camera arrays and networks, pattern recognition systems, +increasing numbers of modalities • acoustic, seismic, RF, visual, IR, SAR, …

  9. Pressure is on Signal Processing • Networked sensing placing increasing pressure on signal/image processing hardware and algs to support higher resolution / denser sampling • ADCs, cameras, imaging systems, … + large numbers of sensors • multi-view target data bases, camera arrays and networks, pattern recognition systems, +increasing numbers of modalities • acoustic, seismic, RF, visual, IR, SAR, … =deluge of data • how to acquire, store, fuse, process efficiently?

  10. AntipastoSensing by Sampling

  11. Data Acquisition and Representation • Time: A/D converters, receivers, … • Space: cameras, imaging systems, … • Foundation: Shannon sampling theorem • Nyquist rate: must sample at 2x highest frequency in signal

  12. Sensing by Sampling • Long-established paradigm for digital data acquisition • sampledata (A-to-D converter, digital camera, …) • compress data (signal-dependent, nonlinear) sample compress transmit/store sparsewavelettransform receive decompress

  13. Sparsity largewaveletcoefficients largeGaborcoefficients pixels widebandsignalsamples • Many signals can be compressed in some representation/basis (Fourier, wavelets, …)

  14. Sensing by Sampling • Long-established paradigm for digital data acquisition • sampledata (A-to-D converter, digital camera, …) • compress data (signal-dependent, nonlinear) • brick wall to performance of modern acquisition systems sample compress transmit sparsewavelettransform receive decompress

  15. PastaCompressive Sensing

  16. From Samples to Measurements • Shannon was a pessimist • worst case bound for any bandlimited data • Compressive sensing (CS) principle “sparse signal statistics can be recovered from a small number of nonadaptive linear measurements” • integrates sensing, compression, processing • based on new uncertainty principlesand concept of incoherency between two bases

  17. Incoherent Bases • Spikes and sines (Fourier) (Heisenberg)

  18. Incoherent Bases • Spikes and “random basis”

  19. Incoherent Bases • Spikes and “random sequences” (codes)

  20. Incoherent Bases

  21. Sampling • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain • Samples sparsesignal measurements nonzeroentries

  22. Compressive Sensing [Candes, Romberg, Tao; Donoho] • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain • Replace samples with few linear projections sparsesignal measurements nonzeroentries

  23. Compressive Sensing [Candes, Romberg, Tao; Donoho] • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain • Replace samples with few linear projections • Random measurements will work! sparsesignal measurements nonzeroentries

  24. Compressive Sensing • Measure linear projections onto incoherentbasis where data is not sparse/compressible • Reconstruct via nonlinear processing (optimization)(using sparsity-inducing basis) project transmit/store one row of receive reconstruct

  25. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find sparsesignal measurements nonzeroentries

  26. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast

  27. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong

  28. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 number ofnonzeroentries

  29. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0correct, slowonly M=K+1 measurements required to perfectly reconstruct K-sparse signal[Bresler; Rice]

  30. CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 correct, slow • L1correct, mild oversampling[Candes et al, Donoho] linear program

  31. CS Signal Recovery original (65k pixels) 20k random projections 7k–term wavelet approximation E. J. Candès and J. Romberg, “Practical Signal Recovery from Random Projections,” 2004.

  32. Why It Works: Sparsity largewaveletcoefficients largeGaborcoefficients pixels widebandsignalsamples • Many signals can be compressed in some representation/basis (Fourier, wavelets, …)

  33. Sparse Models are Nonlinear + =

  34. Sparse Models are Nonlinear largewaveletcoefficients pixels

  35. Sparse Models are Nonlinear largewaveletcoefficients pixels

  36. Sparse Models are Nonlinear largewaveletcoefficients pixels model for all K-sparsesignals: union of subspaces(aligned with coordinate axes) K-dimhyperplanes

  37. Why L2 Doesn’t Work least squares,minimum L2 solutionis almost never sparse null space of translated to(random angle)

  38. Why L1 Works minimum L1solution= sparsest solution if

  39. Universality • Gaussian white noise basis is incoherent with any fixed orthonormal basis (with high probability) • Signal sparse in time domain:

  40. Universality • Gaussian white noise basis is incoherent with any fixed orthonormal basis (with high probability) • Signal sparse in frequency domain: • Product remains Gaussian white noise

  41. PesceCompressive Sensingin Action

  42. Single-Pixel CS Camera single photon detector random pattern on DMD array imagereconstruction w/ Kevin Kelly and students

  43. TI Digital Micromirror Device (DMD)

  44. Single Pixel Camera DMD DMD … 1 2 M

  45. Single Pixel Camera Color Filter Wheel Potential for: • new modalities beyond what can be sensed by CCD or CMOS imagers • low cost • low power DMD DMD

  46. First Image Acquisition DMD DMD image atDMD array ideal 128x128 pixels 6x sub-Nyquist

  47. Second Image Acquisition 8x sub-Nyquist

  48. World’s First Photograph • 1826, Joseph Niepce • Farm buildings and sky • 8 hour exposure • On display at UT-Austin

  49. Analog-to-Digital Conversion • Many applications – particularly in RF – have hit an A/D performance brick wall • limited bandwidth (# Hz) • limited dynamic range (# bits) • deluge of bits to process downstream • “Moore’s Law” for A/D’s: doubling in performance only every 6 years • Fresh approach: • “analog-to-information” conversion • analog CS

  50. A2I via Random Demodulation pseudo-random code • Leverage extant spread spectrum and UWB concepts and hardware • Successfully simulated at 6-20x sub-Nyquist

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