Transcript Dror Baron Marco Duarte Shriram Sarvotham Michael Wakin

Compressive
Sensing
A New Approach to
Signal Acquisition
and Processing
Richard Baraniuk
Rice University
Supported by NSF, ONR, AFOSR, ARO,
DARPA, and Texas Instruments
The Digital Universe
• Size: 281 billion gigabytes generated in 2007
digital bits > stars in the universe
growing by a factor of 10 every 5 years
> Avogadro’s number (6.02x1023) in 15 years
• Growth fueled by multimedia data
audio, images, video, surveillance cameras, sensor nets, …
(2B fotos on Flickr, 4.2M security cameras in UK, …)
• In 2007 digital data generated > total storage
by 2011, ½ of digital universe will have no home
[Source: IDC Whitepaper “The Diverse and Exploding Digital Universe” March 2008]
Act I
What’s wrong with today’s
multimedia sensor systems?
why go to all the work to acquire massive
amounts of multimedia data
only to throw much/most of it away?
Act II
One way out:
dimensionality reduction
(compressive sensing)
enables the design of radically
new sensors and systems
Act III
Compressive sensing in action
new cameras, imagers, ADCs, …
Sense by Sampling
sample
Sense by Sampling
sample
too
much
data!
Sense then Compress
sample
compress
JPEG
JPEG2000
…
decompress
Sparsity
pixels
large
wavelet
coefficients
(blue = 0)
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
What’s Wrong with this Picture?
linear processing
linear signal model
(bandlimited subspace)
sample
nonlinear processing
nonlinear signal model
(union of subspaces)
compress
decompress
Compressive Sensing
• Directly acquire “compressed” data
via dimensionality reduction
• Replace samples by more general “measurements”
compressive sensing
recover
Sampling
• Signal
is
-sparse in basis/dictionary
– WLOG assume sparse in space domain
sparse
signal
nonzero
entries
Sampling
• Signal
is
-sparse in basis/dictionary
– WLOG assume sparse in space domain
• Sampling
measurements
sparse
signal
nonzero
entries
Compressive Sensing
• When data is sparse/compressible, can directly
acquire a condensed representation with
no/little information loss through
linear dimensionality reduction
measurements
sparse
signal
nonzero
entries
How Can It Work?
• Projection
not full rank…
… and so
loses information in general
• Ex: Infinitely many
(null space)
’s map to the same
How Can It Work?
• Projection
not full rank…
… and so
loses information in general
columns
• But we are only interested in sparse vectors
How Can It Work?
• Projection
not full rank…
… and so
loses information in general
columns
• But we are only interested in sparse vectors
•
is effectively MxK
How Can It Work?
• Projection
not full rank…
… and so
loses information in general
columns
• But we are only interested in sparse vectors
• Design
so that each of its MxK submatrices
are full rank (ideally close to orthobasis)
– Restricted Isometry Property (RIP)
How Can It Work?
• Projection
not full rank…
… and so
loses information in general
columns
• Design
so that each of its MxK submatrices
are full rank (RIP)
• Unfortunately, a combinatorial,
NP-complete design problem
Insight from the 70’s
• Draw
[Kashin, Gluskin]
at random
– iid Gaussian
– iid Bernoulli
…
columns
• Then
has the RIP with high probability
provided
Insight from the 70’s
• Draw
[Kashin, Gluskin]
at random
– iid Gaussian
– iid Bernoulli
…
columns
• Then
has the RIP with high probability
provided
• Ex: N = 109
K = 1000
M = c 6000
Randomized Sensing
• Measurements
= random linear combinations
of the entries of
• No information loss for sparse vectors
measurements
whp
sparse
signal
nonzero
entries
Compressive Sensing
In Action
Gerhard Richter
4096 Farben / 4096 Colours
1974
254 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
scene
single photon
detector
DMD
image
reconstruction
or
processing
DMD
random
pattern on
DMD array
w/ Kevin Kelly
and students
“Single-Pixel” CS Camera
scene
single photon
detector
image
reconstruction
or
processing
DMD
DMD
random
pattern on
DMD array
…
• Flip mirror array M times to acquire M measurements
• Sparsity-based (linear programming) recovery
First Image Acquisition
target
65536 pixels
11000 measurements
(16%)
1300 measurements
(2%)
Utility?
single photon
detector
DMD
DMD
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
450-850nm
1M space x wavelength voxels
200k random msmnts
CS THz Camera
32 x 32 PCB masks
Object mask
CS recon
300 msnts
CS recon
600 msnts
w/ Mittleman and Kelly groups
CS In Action
• CS makes sense when measurements
are expensive
• Ultrawideband A/D converters
[w/ Yehia Massoud and group]
• Medical imaging
– faster imaging/lower dosage
• Camera networks
– sensing/compression/fusion
• Radar, sonar, array processing
– exploit spatial sparsity of targets
• DNA microarrays
– smaller, more agile arrays for
bio-sensing
Application:
Decision and Inference
Information Scalability
• Many applications involve signal inference
and not reconstruction
detection < classification < estimation < reconstruction
• Good news:
CS supports efficient learning,
inference, processing directly
on compressive measurements
• Random projections ~ sufficient statistics
for signals with concise geometrical structure
Classification
• Simple object classification problem
– AWGN: nearest neighbor classifier
• Common issue:
– L unknown articulation parameters
• Common solution: matched filter
– find nearest neighbor under all articulations
CS-based Classification
• Target images form a low-dimensional
manifold as the target articulates
– random projections preserve information
in these manifolds if
CS-based Classification
• Target images form a low-dimensional
manifold as the target articulates
– random projections preserve information
in these manifolds if
CS-based Classification
• Target images form a low-dimensional
manifold as the target articulates
– random projections preserve information
in these manifolds if
• CS-based classifier: smashed filter
– find nearest neighbor under all articulations
under random projection
Smashed Filter
• Random shift and rotation (L=3 dim. manifold)
• WG noise added to measurements
• Goals:
identify most likely shift/rotation parameters
more noise
number of measurements M
classification rate (%)
avg. shift estimate error
identify most likely class
more noise
number of measurements M
Summary
• Compressive sensing
–
–
–
–
randomized dimensionality reduction
integrates sensing, compression, processing
exploits signal sparsity information
enables new sensing modalities, architectures, systems
• Why CS works: preserves information in signals
with concise geometric structure
sparse signals | compressible signals | manifolds
• Information scalability for compressive inference
– compressive measurements ~ sufficient statistics
CS Hallmarks
• Stable
– acquisition/recovery process is numerically stable
• Asymmetrical (most processing at decoder)
– conventional:
– CS:
smart encoder, dumb decoder
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)
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, …
Matrix Completion
movies
– NxN matrix
– rank R
– M entries known
users
• Given:
• Problem: Fill in rest of matrix
• Ex: “Netflix problem”
correlation matrix w/ missing data
Matrix Completion
movies
– NxN matrix
– rank R
– M entries known
users
• Given:
• Problem: Fill in rest of matrix
• Ex: “Netflix problem”
correlation matrix w/ missing data
• Result: can exactly “complete” the matrix if
dsp.rice.edu/cs
Thanks!
• Sponsors:
NSF, DARPA, ARO, AFOSR, ONR,
Texas Instruments Leadership University Program
• Group:
Volkan Cevher
Jarvis Haupt, Aswin Sankaranarayanan
Mark Davenport, Eva Dyer
Chinmay Hegde, Jason Laska
Manjari Narayan, Stephen Schnelle
Mona Sheikh
JP Slavinsky, Elizabeth Hickman
Universality
• Random measurements can be used for signals
sparse in any basis
Universality
• Random measurements can be used for signals
sparse in any basis
Universality
• Random measurements can be used for signals
sparse in any basis
sparse
coefficient
vector
nonzero
entries
Beyond Sparsity
• Sparse signal model captures
simplistic primary structure
wavelets:
natural images
Gabor atoms:
chirps/tones
pixels:
background subtracted
images
Structured Sparsity
• 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
Wavelet Tree-Sparse Recovery
target signal
CoSaMP,
(RMSE=1.12)
N=1024
M=80
Tree-sparse CoSaMP
(RMSE=0.037)
L1-minimization
(RMSE=0.751)