Compressive Sensing A New Approach to Signal Acquisition and Processing Richard Baraniuk Rice University

LECTURE THREE Sparsity and CS: Applications and Current Trends

Recall: CS Sensing: = random linear combinations of the entries of Recovery:Recover from via optimization measurements sparse signal nonzero entries

Gerhard Richter 4096 Farben / 4096 Colours cm X 254 cm Laquer on Canvas Catalogue Raisonné: 359 Museum Collection: Staatliche Kunstsammlungen Dresden (on loan) Sales history: 11 May 2004 Christie's New York Post-War and Contemporary Art (Evening Sale), Lot 34 US$3,703,500

“Single-Pixel” CS Camera random pattern on DMD array DMD single photon detector image reconstruction or processing w/ Kevin Kelly scene

“Single-Pixel” CS Camera random pattern on DMD array DMD single photon detector image reconstruction or processing scene Flip mirror array M times to acquire M measurements Sparsity-based (linear programming) recovery …

First Image Acquisition target pixels 1300 measurements (2%) measurements (16%)

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

Utility? DMD single photon detector Fairchild 100Mpixel CCD

Utility? DMD single photon detector Fairchild 100Mpixel CCD

CS Low-Light Imaging with PMT true color low-light imaging 256 x 256 image with 10:1 compression [Nature Photonics, April 2007]

CS Infrared Camera 20% 5%

CS Hyperspectral Imager spectrometer hyperspectral data cube nm 1M space x wavelength voxels 200k random sums

CS MRI Lustig, Pauly, Donoho et al at Stanford Goal: Speed up MRI data acquisition by reducing number of samples required for a given image reconstruction quality Approach: Design MRI sampling pattern (in frequency/k-space) to be close to random

Multi-Slice Brain Imaging [M. Lustig]

Compressive Sensing In Action A/D Converters

Analog-to-Digital Conversion Nyquist rate limits reach of today’s ADCs “Moore’s Law” for ADCs: –technology Figure of Merit incorporating sampling rate and dynamic range doubles every 6-8 years Analog-to-Information (A2I) converter –wideband signals have high Nyquist rate but are often sparse/compressible –develop new ADC technologies to exploit –new tradeoffs among Nyquist rate, sampling rate, dynamic range, … frequency hopper spectrogram time frequency

Streaming Measurements measurements streaming requires special Streaming applications: cannot fit entire signal into a processing buffer at one time

Streaming Measurements measurements streaming requires special Streaming applications: cannot fit entire signal into a processing buffer at one time

Streaming Measurements measurements streaming requires special Streaming applications: cannot fit entire signal into a processing buffer at one time RIP?

Streaming Measurements streaming requires special Many applications:Signal sparse in frequency (Fourier transform)

Random Demodulator A A B B C C D D

Sampling Rate Goal: Sample near signal’s (low) “information rate” rather than its (high) Nyquist rate A2I sampling rate number of tones / window Nyquist bandwidth

Sampling Rate Theorem [Tropp et al 2007] If the sampling rate satisfies then locally Fourier K -sparse signals can be recovered exactly with probability

Empirical Results

Example: Frequency Hopper 20x sub-Nyquist sampling spectrogram sparsogram Nyquist rate sampling

More CS In Action CS makes sense when measurements are expensive Ultrawideband A/D converters [DARPA “Analog to Information” program] Camera networks –sensing/compression/fusion Radar, sonar, array processing –exploit spatial sparsity of targets DNA microarrays –smaller, more agile arrays for bio-sensing …

Beyond Sparsity Structured Sparsity

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

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

Sparse Signals K-sparse signals comprise a particular set of K-dim subspaces

Structured-Sparse Signals A K-sparse signal model comprises a particular (reduced) set of K-dim subspaces [Blumensath and Davies] Fewer subspaces <> relaxed RIP <> stable recovery using fewer measurements M

Wavelet Sparse Typical of wavelet transforms of natural signals and images (piecewise smooth)

Tree-Sparse Model: K-sparse coefficients +significant coefficients lie on a rooted subtree Typical of wavelet transforms of natural signals and images (piecewise smooth)

Wavelet Sparse Model: K-sparse coefficients +significant coefficients lie on a rooted subtree RIP: stable embedding K-dim subspaces

Tree-Sparse Model: K-sparse coefficients +significant coefficients lie on a rooted subtree Tree-RIP: stable embedding K-dim subspaces

Recall: Iterated Thresholding update signal estimate prune signal estimate (best K-term approx) update residual

Iterated Model Thresholding update signal estimate prune signal estimate (best K-term model approx) update residual

Tree-Sparse Signal Recovery target signal CoSaMP, (RMSE=1.12) Tree-sparse CoSaMP (RMSE=0.037) N=1024 M=80 L1-minimization (RMSE=0.751) [B, Cevher, Duarte, Hegde’08]

Other Useful Models Clustered coefficients [C, Duarte, Hegde, B], [C, Indyk, Hegde, Duarte, B] Dispersed coefficients [Tropp, Gilbert, Strauss], [Stojnic, Parvaresh, Hassibi], [Eldar, Mishali], [Baron, Duarte et al], [B, C, Duarte, Hegde]

Clustered Signals targetIsing-model recovery CoSaMP recovery LP (FPC) recovery Probabilistic approach via graphical model Model clustering of significant pixels in space domain using Ising Markov Random Field Ising model approximation performed efficiently using graph cuts [Cevher, Duarte, Hegde, B’08]

Block-Sparse Model N = 4096 K = 6 active blocks J = block length = 64 M = 2.5JK = 960 msnts [Stojnic, Parvaresh, Hassibi], [Eldar, Mishali], [B, Cevher, Duarte, Hegde] target CoSaMP (MSE = 0.723) block-sparse model recovery (MSE=0.015)

Sparse Spike Trains Sequence of pulses Simple model: –sequence of Dirac pulses –refractory period between each pulse Model-based RIP if Stable recovery via iterative algorithm (exploit total unimodularity) [Hedge, Duarte, Cevher ‘09]

N=1024 K=50 =10 M=150 original model-based recovery error CoSaMP recovery error Sparse Spike Trains

Sparse Pulse Trains More realistic model: –sequence of Dirac pulses * pulse shape of length –refractory period between each pulse of length Model-based RIP if

More realistic model: –sequence of Dirac pulses * pulse shape of length –refractory period between each pulse of length Model-based RIP if N=4076 K=7 =25 =10 M=290 originalCoSaMPmodel-alg Sparse Pulse Trains

Summary Compressive sensing –randomized dimensionality reduction –integrates sensing, compression, processing –exploits signal sparsity information –enables new sensing modalities, architectures, systems –relies on large-scale optimization Why it works:preserves information in signals with concise geometric structure sparse signals | compressible signals | manifolds

Open Research Issues Links with information theory –new encoding matrix design via codes (LDPC, fountains) –new decoding algorithms (BP, etc.) –quantization and rate distortion theory Links with machine learning –Johnson-Lindenstrauss, manifold embedding, RIP Processing/inference on random projections –filtering, tracking, interference cancellation, … Multi-signal CS –array processing, localization, sensor networks, … CS hardware –ADCs, receivers, cameras, imagers, radars, …

Discussion Session 3 Discussion: –Model-based CS Computer exercises: –Manifold CS and smashed filter for classification –Model-based compressive sensing –Democracy and justice for enhanced robustness

dsp.rice.edu/cs

Open Positions open postdoc positions in sparsity / compressive sensing at Rice University dsp.rice.edu

Connexions (cnx.org) non-profit open publishing project goal: make high-quality educational content available to anyone, anywhere, anytime for free on the web and at very low cost in print open-licensed repository of Lego-block modules for authors, instructors, and learners to create, rip, mix, burn global reach: >1M users monthly from 200 countries collaborators:IEEE (IEEEcnx.org), Govt. Vietnam, TI, NI, …

Thanks! PCMI organizers and staff Chinmay Hegde, TA Rice DSP past and present –Mark Davenport, Marco Duarte, Mike Wakin, Volkan Cevher