CompressiveSensingA 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 sparsesignal

nonzeroentries

Gerhard Richter 4096 Farben / 4096 Colours

1974254 cm X 254 cmLaquer on CanvasCatalogue Raisonné: 359

Museum Collection:Staatliche Kunstsammlungen Dresden (on loan)

Sales history: 11 May 2004Christie's New York Post-War and Contemporary Art (Evening Sale), Lot 34US$3,703,500  

“Single-Pixel” CS Camera

randompattern onDMD array

DMD DMD

single photon detector

imagereconstructionorprocessing

w/ Kevin Kelly

scene

“Single-Pixel” CS Camera

randompattern onDMD array

DMD DMD

single photon detector

imagereconstructionorprocessing

scene

• Flip mirror array M times to acquire M measurements• Sparsity-based (linear programming) recovery

…

First Image Acquisition

target 65536 pixels

1300 measurements (2%)

11000 measurements (16%)

World’s First Photograph

• 1826, Joseph Niepce• Farm buildings and sky • 8 hour exposure• On display at UT-Austin

Utility?

DMD DMD

single photon detector

Fairchild100Mpixel

CCD

Utility?

DMD DMD

single photon detector

Fairchild100Mpixel

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 cube450-850nm

1M space x wavelength voxels200k 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 SensingIn 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 amongNyquist rate, sampling rate,dynamic range, …

frequency hopperspectrogram

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

AB

B

C

CD

D

Random Demodulator

Random Demodulator

Sampling Rate

• Goal: Sample near signal’s (low) “information rate” rather than its (high) Nyquist rate

A2Isampling rate

number oftones /window

Nyquistbandwidth

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 transformsof natural signals and images (piecewise smooth)

Tree-Sparse

• Model: K-sparse coefficients + significant coefficients

lie on a rooted subtree

• Typical of wavelet transformsof 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=1024M=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

target Ising-modelrecovery

CoSaMPrecovery

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 = 4096K = 6 active blocksJ = block length = 64M = 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 viaiterative algorithm(exploit total unimodularity)[Hedge, Duarte, Cevher ‘09]

N=1024 K=50 =10

M=150

originalmodel-based recovery error

CoSaMPrecovery 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

original CoSaMP model-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 richb@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 monthlyfrom 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