Compressive Signal Processing Richard Baraniuk Rice University.

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CompressiveSignal Processing

Richard Baraniuk

Rice University

Better, Stronger, Faster

Sense by Sampling

sample

Sense by Sampling

sample too much data!

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

Sense then Compress

compress

decompress

sample

JPEGJPEG2000

Sparsity

pixels largewaveletcoefficients

(blue = 0)

Sparsity

pixels largewaveletcoefficients

(blue = 0)

widebandsignalsamples

largeGabor (TF)coefficients

• Sparse signal: only K out of N coordinates nonzero

Concise Signal Structure

sorted index

sparsesignal

nonzeroentries

– model: union of K-dimensional subspacesaligned w/ coordinate axes

sorted index

sparsesignal

nonzeroentries

– model: union of K-dimensional subspaces

• Compressible signal: sorted coordinates decay rapidly with power-law

approximately sparse

sorted index

power-lawdecay

– model: union of K-dimensional subspaces

• Compressible signal: sorted coordinates decay rapidly with power-law

– model: ball:

sorted index

power-lawdecay

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?

compress

decompress

sample

What’s Wrong with this Picture?linear processinglinear signal model(bandlimited subspace)

compress

decompress

sample

nonlinear processingnonlinear signal model(union of subspaces)

Compressive Sensing• Directly acquire “compressed” data

via dimensionality reduction

• Replace samples by more general “measurements”

compressive sensing

recover

• Signal is -sparse in basis/dictionary– WLOG assume sparse in space domain

Sampling

sparsesignal

nonzeroentries

• Signal is -sparse in basis/dictionary– WLOG assume sparse in space domain

• Sampling

sparsesignal

nonzeroentries

measurements

Sampling

Compressive Sampling

• When data is sparse/compressible, can directly acquire a condensed representation with no/little information loss through linear dimensionality reduction

measurements sparsesignal

nonzeroentries

How Can It Work?

• Projection not full rank…

… and so loses information in general

• Ex: Infinitely many ’s map to the same(null space)

How Can It Work?

• But we are only interested in sparse vectors

columns

How Can It Work?

• is effectively MxK

columns

How Can It Work?

• Design so that each of its MxK submatrices are full rank (ideally close to orthobasis)– Restricted Isometry Property (RIP)– see also phase transition approach of Donoho et al.

columns

RIP = Stable Embedding• An information preserving projection preserves

the geometry of the set of sparse signals

• RIP ensures that

K-dim subspaces

RIP = Stable Embedding• An information preserving projection preserves

the geometry of the set of sparse signals

• RIP ensures that

How Can It Work?

• Design so that each of its MxK submatrices are full rank (RIP)

• Unfortunately, a combinatorial, NP-Hard design problem

columns

Insight from the 70’s [Kashin, Gluskin]

• Draw at random– iid Gaussian– iid Bernoulli

• Then has the RIP with high probability provided

columns

Randomized Sensing

• Measurements = random linear combinations of the entries of

• No information loss for sparse vectors whp

measurements sparsesignal

nonzeroentries

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

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 hyperplane

at random angle

• Recovery: given(ill-posed inverse problem) find (sparse)

• Optimization:

• Closed-form solution:

Signal Recovery

• Optimization:

• Wrong answer!

Signal Recovery

• Optimization:

• Wrong answer!

Signal Recovery

• Optimization:

Signal Recovery

“find sparsest vectorin translated nullspace”

• Optimization:

• Correct!

Signal Recovery

• Optimization:

• Correct!

• But NP-Complete alg

Signal Recovery

• Optimization:

• Convexify the optimization

Signal Recovery

Candes Romberg TaoDonoho

• Optimization:

• Convexify the optimization

• Correct!

• Polynomial time alg(linear programming)

• Much recent alg progress– greedy, Bayesian approaches, …

Signal Recovery

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 for

any sparse signal class (generic)

Universality

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

Universality

sparsecoefficient

vector

nonzeroentries

Compressive SensingIn Action

“Single-Pixel” CS Camera

randompattern onDMD array

DMD DMD

single photon detector

imagereconstructionorprocessing

w/ Kevin Kelly

“Single-Pixel” CS Camera

randompattern onDMD array

DMD DMD

imagereconstructionorprocessing

• 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%)

Utility?

DMD DMD

Fairchild100Mpixel

InView “single-pixel”SWIR Camera (1024x768)

Target (illuminated in SWIR only)

SWIR CS Camera

Camera outputM = 0.5 N

CS Hyperspectral Imager

spectrometer

hyperspectral data cube450-850nm

N=1M space x wavelength voxelsM=200k random measurements

(Kevin Kelly Lab, Rice U)

CS-MUVI for Video CS

• Pendulum speed: 2 sec/cycle

• Naïve, block-basedL1 recovery of 64x64 video frames for 3 different values of W

CS-MUVI for Video CS

Low-res preview(32x32)

Recovered video (animated)

High-res video recovery (128x128)

Effective “compression ratio” = 60:1

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

frequency

Random Demodulator

Sampling Rate

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

A2Isampling rate

number oftones /window

Nyquistbandwidth

Example: Frequency Hopper

20x sub-Nyquist sampling

spectrogram sparsogram

Nyquist rate sampling

Example: Frequency Hopper

20x sub-Nyquist sampling

spectrogram sparsogram

Nyquist rate sampling

20MHz sampling rate 1MHz sampling rate

conventional ADC CS-based AIC

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 …

Pros and Cons of Compressive Sensing

CS – Pro – Measurement Noise

• Stable recoverywith additive measurement noise

• Noise is added to

• Stability: noise only mildly amplified in recovered signal

CS – Con – Signal Noise

• Often seek recoverywith additive signal noise

• Noise is added to

• Noise folding: signal noise amplified in by 3dB for every doubling of

• Same effect seen in classical “bandpass subsampling”

CS – Con – Noise Folding

slope = -3

CS – Pro – Dynamic Range

• As amount of subsampling grows, can employan ADC with a lower sampling rate and hence higher-resolution quantizer

Dynamic Range

• Corollary: CS can significantly boost the ENOB of an ADC system for sparse

signals

conventional ADC

CS ADC w/ sparsity

CS – Pro – Dynamic Range

• As amount of subsampling grows, can employan ADC with a lower sampling rate and hence higher-resolution quantizer

• Thus dynamic range of CS ADC can far exceedNyquist ADC

• With current ADC trends, dynamic range gain is theoretically 7.9dB for each doubling in

CS – Pro – Dynamic Ranged

slope = +5 (almost 7.9)

CS – Pro vs. Con

SNR: 3dB loss for each doubling of

Dynamic Range: up to 7.9dB gain for each doubling of

Summary: CS

• Compressive sensing– randomized dimensionality reduction– exploits signal sparsity information– integrates sensing, compression, processing

• Why it works: with high probability, random projections preserve information in signals with concise geometric structures

• Enables new sensing architectures– cameras, imaging systems, ADCs, radios, arrays, …

• Important to understand noise-folding/dynamic range trade space

Open Research Issues• Links with information theory

– new encoding matrix design via codes (LDPC, fountains)– new decoding algorithms (BP, AMP, 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, arrays, radars, sonars, …– 1-bit CS and stable embeddings

dsp.rice.edu/cs

Compressive Signal Processing Richard Baraniuk Rice University.

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