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Compressive Sensing: A New Framework for Computational Signal and Image Processing. Richard Baraniuk Rice University dsp.rice.edu/cs. Pressure is on Digital Signal Processing. Shannon/Nyquist sampling theorem no information loss if we sample at 2x signal bandwidth
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CompressiveSensing: A New Framework for Computational Signaland Image Processing Richard Baraniuk Rice University dsp.rice.edu/cs
Pressure is on Digital Signal Processing • Shannon/Nyquist sampling theorem • no information loss if we sample at 2x signal bandwidth • DSP revolution: sample first and ask questions later • Increasing pressure on DSP hardware, algorithms • ever faster sampling and processing rates • ever larger dynamic range • ever larger, higher-dimensional data • ever lower energy consumption • ever smaller form factors • multi-node, distributed, networked operation • radically new sensing modalities • communication over ever more difficult channels
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
Sparsity largewaveletcoefficients largeGaborcoefficients pixels widebandsignalsamples • Many signals can be compressed in some representation/basis (Fourier, wavelets, …)
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
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
Sampling • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain sparsesignal nonzeroentries
Sampling • Signal is -sparse in basis/dictionary • WLOG assume sparse in space domain • Samples sparsesignal measurements nonzeroentries
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
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
Compressive Sensing • Measure linear projections onto randombasis where data is not sparse/compressible • Reconstruct via nonlinear processing (optimization)(using sparsity-inducing basis) project transmit/store one row of receive reconstruct
CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find sparsesignal measurements nonzeroentries
CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast
CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong
CS Signal Recovery • Reconstruction/decoding: given(ill-posed inverse problem) find • L2 fast, wrong • L0 number ofnonzeroentries
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] number ofnonzeroentries
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
CS Signal Recovery original (65k pixels) 20k random projections 7k–term wavelet approximation
Universality • Gaussian white noise basis is incoherent with any fixed orthonormal basis (with high probability) • Signal sparse in time domain:
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
Why It Works: Sparsity largewaveletcoefficients largeGaborcoefficients pixels widebandsignalsamples • Many signals can be compressed in some representation/basis (Fourier, wavelets, …)
Sparse Models are Nonlinear largewaveletcoefficients pixels
Sparse Models are Nonlinear largewaveletcoefficients pixels model for all K-sparsesignals: union of subspaces(aligned with coordinate axes) K-dimhyperplanes
Why L2 Doesn’t Work least squares,minimum L2 solutionis almost never sparse null space of translated to(random angle)
Why L1 Works minimum L1solution= sparsest solution if
New Acquisition Hardware • CS changes the rules of the data acquisition game • exploits a priori signal/image sparsity information • Same random projections / hardware can be used for any compressible signal class (generic) • Simplifies hardware and algorithm design • Random projections automatically encrypted • Very simple encoding • Robust to measurement loss and quantization • Asymmetrical processing (most at decoder)
Rice Single-Pixel Camera single photon detector DMD DMD random pattern on DMD array imagereconstructionorprocessing
TI Digital Micromirror Device (DMD) DLP 1080p --> 1920 x 1080 resolution
Miniturized DMD • TI DLP “picoprojector” destined for cell phones
(Pseudo) Random Optical Projections • Binary patterns are loaded into mirror array: • light reflected towards the lens/photodiode (1) • light reflected elsewhere (0) • pixel-wise products summed by lens • Pseudorandom number generator outputs measurement basis vectors • Mersenne Twister [Matsumoto/Nishimura, 1997] • Binary sequence (0/1) • Period 219937-1 …
Rice Single-Pixel Camera Object Light DMD+ALP Board Lens 1 Lens 2 Photodiode circuit
First Image Acquisition DMD DMD target128x128 pixels TV reconstruction6x sub-Nyquist
Another Image Acquisition 8x sub-Nyquist
World’s First Photograph • 1826, Joseph Niepce • Farm buildings and sky • 8 hour exposure • On display at UT-Austin
Rice Single-Pixel Camera single photon detector DMD DMD random pattern on DMD array imagereconstruction imagereconstructionorprocessing
More Complex Photodetectors http://micro.magnet.fsu.edu/primer/digitalimaging/concepts/photomultipliers.html
Low-Light Color Imaging w/ PMT Mandrill 256x256 Mandrill 10x sub-Nyquist
Information Scalability • Random projections ~ sufficient statistics • Same random projections / hardware can be used for a range of different signal processing tasks reconstruction > estimation > recognition > detection • Many fewer measurements may be required to detect/classify/recognize than to reconstruct • Example applications: • matched filtering for target detection/classification • manifold learning and processing for camera networks • pattern recognition …
Hyperspectral Image Classification • AVIRIS image with 224 frequency channels • each pixel • Random projection over frequency • each pixel • Trained Neyman-Pearson SVM to classify land/water
Analog-to-Information 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 • DARPA A2I program
A2I via Random Demodulation pseudo-random code • Leverage extant spread spectrum and UWB concepts and hardware • Measurement system must be “real time”
Conclusions • Compressive sensing • new approach to signal and image acquisition • exploit a priori image sparsity information • integrates sensing, compression, processing • information scalable • Proof of concept: single-pixel CS camera • flexible, single sensor element • universal, simple, robust imaging • enables imaging beyond the visible + new geometries • Current research • fast measurement and reconstruction (LPDC/BP) • distributed CS for sensor nets (Slepian-Wolf) • video and volumetric data acquisition • CS meets Johnson-Lindenstrauss • A2I converters (analog CS) • R/D analysis dsp.rice.edu/cs