Compressive Signal Processing

1. CompressiveSignal Processing Richard Baraniuk Rice University

2. Better, Stronger, Faster

3. Analog sensing Digital sensing

4. Sense by Sampling too much data! sample

5. Case in Point: DARPA ARGUS-IS • 1.8 Gpixel image sensor • video rate output: 444 Gbits/s • comm data rate: 274 Mbits/sfactor of 1600x way out of reach ofexisting compressiontechnology • Reconnaissancewithout conscience • too much data to transmit to a ground station • too much data to make effective real-time decisions

6. Accelerating Data Deluge • 1250 billion gigabytes generated in 2010 • # digital bits > # stars in the universe • growing by a factor of 10 every 5 years • Total data generated > total storage • Increases in generation rate >>increases in comm rate Available transmission bandwidth

8. Sense then Compress sample compress JPEG JPEG2000 … decompress

9. Sparsity largewaveletcoefficients (blue = 0) largeGabor (TF)coefficients pixels widebandsignalsamples frequency time

10. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspacesaligned w/ coordinate axes sparsesignal nonzeroentries sorted index

11. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspaces • Compressible signal: sorted coordinates decay rapidly with power-lawapproximately sparse power-lawdecay sorted index

12. Concise Signal Structure • Sparse signal: only K out of N coordinates nonzero • model: union of K-dimensional subspaces • Compressible signal: sorted coordinates decay rapidly with power-law • model: ball: power-lawdecay sorted index

13. 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 decompress

14. What’s Wrong with this Picture? • nonlinear processing • nonlinear signal model • (union of subspaces) linear processing linear signal model (bandlimited subspace) sample compress decompress

15. Compressive Sensing • Directly acquire “compressed” data via dimensionality reduction • Replace samples by more general “measurements” compressive sensing recover

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

17. Compressive Sampling • When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss through linear dimensionality reduction sparsesignal measurements nonzero entries

18. How Can It Work? • Projection not full rank…… and so loses information in general • Ex: Infinitely many ’s map to the same(null space)

19. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors columns

20. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors • is effectively MxK columns

21. How Can It Work? • Projection not full rank…… and so loses information in general • But we are only interested in sparse vectors • Design so that each of its MxKsubmatricesare full rank (ideally close to orthobasis) • Restricted Isometry Property (RIP) columns

22. RIP = Stable Embedding An information preserving projection preserves the geometry of the set of sparse signals RIP ensures that K-dim subspaces

23. How Can It Work? • Projection not full rank…… and so loses information in general • Design so that each of its MxK submatrices are full rank (RIP) • Unfortunately, a combinatorial, NP-Hard design problem columns

24. Insight from the 70’s [Kashin, Gluskin] • Draw at random • iid Gaussian • iid Bernoulli … • Then has the RIP with high probability provided columns

25. Randomized Sensing • Measurements = random linear combinations of the entries of • No information loss for sparse vectors whp sparsesignal measurements nonzero entries

26. CS Signal Recovery • Goal: Recover signal from measurements • Problem: Randomprojection not full rank(ill-posed inverse problem) • Solution: Exploit the sparse/compressiblegeometry of acquired signal

27. CS Signal Recovery • Random projection not full rank • Recovery problem:givenfind • Null space • Search in null space for the “best”according to some criterion • ex: least squares (N-M)-dim hyperplaneat random angle

28. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Closed-form solution: Wrong answer! Signal Recovery

29. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Signal Recovery “find sparsest vectorin translated nullspace”

30. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Correct! But NP-Complete alg Signal Recovery “find sparsest vectorin translated nullspace”

31. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Signal Recovery Donoho Candes Romberg Tao

32. Recovery: given(ill-posed inverse problem) find (sparse) Optimization: Convexify the optimization Correct! Polynomial time alg(linear programming) Much recent alg progress greedy, Bayesian approaches, … Signal Recovery

33. CS Hallmarks Stable acquisition/recovery process is numerically stable Asymmetrical(most processing at decoder) conventional: smart encoder, dumb decoder CS: dumb encoder, smart decoder Democratic each measurement carries the same amount of information robust to measurement loss and quantization “digital fountain” property Random measurements encrypted Universal same random projections / hardware can be used forany sparse signal class (generic)

34. Universality • Random measurements can be used for signals sparse in any basis

35. Universality • Random measurements can be used for signals sparse in any basis

36. Universality • Random measurements can be used for signals sparse in any basis sparsecoefficient vector nonzero entries

37. Analog sensing Digital sensing Computational sensing

38. Compressive SensingIn Action

39. “Single-Pixel” CS Camera scene single photon detector imagereconstructionorprocessing DMD DMD random pattern on DMD array w/ Kevin Kelly

40. “Single-Pixel” CS Camera scene single photon detector imagereconstructionorprocessing DMD DMD random pattern on DMD array … • Flip mirror array M times to acquire M measurements • Sparsity-based (linear programming) recovery

41. First Image Acquisition target 65536 pixels 1300 measurements (2%) 11000 measurements (16%)

42. Utility? Fairchild 100Mpixel CCD single photon detector DMD DMD

43. SWIR CS Camera InView “single-pixel” SWIR Camera (1024x768) Target (illuminated in SWIR only) Camera output M = 0.5 N

44. SWIR Video Recovery via CS-MUVI Naïve frame-by-frame recovery CS-MUVI • Multiscale that optimizes spatial/temporal blurs • Optical flow model for scene motion

45. CS Hyperspectral Imager (Kevin Kelly Lab, Rice U) spectrometer hyperspectral data cube 450-850nm N=1M space x wavelength voxels M=200k random measurements

46. More CS In Action • CS makes sense when measurements are expensive • Coded imagers • x-ray, gamma-ray, IR, THz, … • Camera networks • sensing/compression/fusion • Array processing • exploit spatial sparsity of targets • Ultrawideband A/D converters • exploit sparsity in frequency domain • Medical imaging • MRI, CT, ultrasound …

47. Compressive SensingThe Road Ahead

48. More Realistic Models • Beyond sparsity

49. Beyond Sparsity Sparse signal model captures simplistic primary structure pixels: background subtracted images wavelets: natural images Gabor atoms: chirps/tones

50. Structured Sparsity Sparse signal model captures simplistic primary structure Modern compression/processing algorithms capture richer secondary coefficient structure pixels: background subtracted images wavelets: natural images Gabor atoms: chirps/tones