dror baron marco duarte shriram sarvotham michael wakin ... · acknowledgements • rice dsp group...
TRANSCRIPT
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Acknowledgements
• Rice DSP Group (Slides)
– Richard Baraniuk
Mark Davenport, Marco Duarte, Chinmay Hegde, Jason Laska, Shri Sarvotham, Mona Sheikh Stephen Schnelle…
– Mike Wakin, Petros Boufounos, Dror Baron
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Outline• Introduction to Compressive Sensing (CS)
– motivation– basic concepts
• CS Theoretical Foundation
– geometry of sparse and compressible signals– coded acquisition– restricted isometry property (RIP)– structured matrices and random convolution– signal recovery algorithms– structured sparsity
• CS in Action
• Summary
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Sensing
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Digital Revolution
12MP 25fps/1080p 4KHz Multi touch
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Digital Revolution
12MP 25fps/1080p 4KHz
1977 – 5hours
<30mins
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Major Trends
higher resolution / denser sampling
12MP 25fps/1080p 4KHz
160MP 200,000fps 192,000Hz
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Major Trends
higher resolution / faster sampling
large numbers of sensors
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Major Trends
higher resolution / denser sampling large numbers of sensors
increasing # of modalities / mobility
acoustic, RF, visual, IR, UV, x-ray, gamma ray, …
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Major Trends in Sensing
higher resolution / denser sampling
xlarge numbers of sensors
xincreasing # of modalities / mobility
= ?
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Major Trends in Sensing
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Digital Data Acquisition
Foundation: Shannon/Nyquist sampling theorem
time space
“if you sample densely enough (at the Nyquist rate), you can perfectly reconstruct the original analog data”
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Sensing by Sampling
• Long-established paradigm for digital data acquisition– uniformly sample data at Nyquist rate (2x Fourier bandwidth)
sample
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Sensing by Sampling
• Long-established paradigm for digital data acquisition– uniformly sample data at Nyquist rate (2x Fourier bandwidth)
sample too much data!
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Sensing by Sampling
• Long-established paradigm for digital data acquisition– uniformly sample data at Nyquist rate (2x Fourier bandwidth)– compress data
compress transmit/store
receive decompress
sample
JPEGJPEG2000
…
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Sparsity / Compressibility
pixels largewaveletcoefficients
(blue = 0)
widebandsignalsamples
largeGabor (TF)coefficients
time
freq
uenc
y
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Sample / Compress
compress transmit/store
receive decompress
sample
sparse /compressiblewavelettransform
• Long-established paradigm for digital data acquisition– uniformly sample data at Nyquist rate – compress data
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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 transmit/store
receive decompress
sample
sparse /compressiblewavelettransform
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What’s Wrong with this Picture?linear processinglinear signal model(bandlimited subspace)
compress transmit/store
receive decompress
sample
sparse /compressiblewavelettransform
nonlinear processingnonlinear signal model(union of subspaces)
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Compressive Sensing• Directly acquire “compressed” data
• Replace samples by more general “measurements”
compressive sensing transmit/store
receive reconstruct
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Compressive Sensing
Theory IGeometrical Perspective
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• Signal is -sparse in basis/dictionary– WLOG assume sparse in space domain
Sampling
sparsesignal
nonzeroentries
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• Signal is -sparse in basis/dictionary– WLOG assume sparse in space domain
• Samples
sparsesignal
nonzeroentries
measurements
Sampling
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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
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How Can It Work?
• Projection not full rank…
… and so loses information in general
• Ex: Infinitely many ’s map to the same
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How Can It Work?
• Projection not full rank…
… and so loses information in general
• But we are only interested in sparse vectors
columns
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How Can It Work?
• Projection not full rank…
… and so loses information in general
• But we are only interested in sparse vectors
• is effectively MxK
columns
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How Can It Work?
• Projection not full rank…
… and so loses information in general
• But we are only interested in sparse vectors
• Design so that each of its MxK submatrices are full rank
columns
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How Can It Work?
• Goal: Design so that its Mx2K submatrices are full rank
– difference between two K-sparse vectors is 2K sparse in general
– preserve information in K-sparse signals
– Restricted Isometry Property (RIP) of order 2K
columns
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Unfortunately…
• Goal: Design so that its Mx2K submatrices are full rank(Restricted Isometry Property – RIP)
• Unfortunately, a combinatorial, NP-complete design problem
columns
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Insight from the 80’s [Kashin, Gluskin]
• Draw at random– iid Gaussian– iid Bernoulli
…
• Then has the RIP with high probability as long as
– Mx2K submatrices are full rank– stable embedding for sparse signals– extends to compressible signals
columns
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Compressive Data Acquisition
• Measurements = random linear combinationsof the entries of
• WHP does not distort structure of sparse signals – no information loss
measurements sparsesignal
nonzeroentries
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1. Sparse / compressible
not sufficient alone
2. Projection
information preserving (restricted isometry property - RIP)
3. Decoding algorithms
tractable
Compressive Sensing Recovery
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• Recovery: given(ill-posed inverse problem) find (sparse)
• fast
Compressive Sensing Recovery
pseudoinverse
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• Recovery: given(ill-posed inverse problem) find (sparse)
• fast, wrong
Compressive Sensing Recovery
pseudoinverse
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Why Doesn’t Work
least squares,minimum solutionis almost never sparse
null space of translated to(random angle)
for signals sparse in the space/time domain
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• Reconstruction/decoding: given(ill-posed inverse problem) find
• fast, wrong
•
Compressive Sensing Recovery
number ofnonzeroentries
“find sparsestin translated nullspace”
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• Reconstruction/decoding: given(ill-posed inverse problem) find
• fast, wrong
• correct:only M=2Kmeasurements required to reconstruct K-sparse signal
Compressive Sensing Recovery
number ofnonzeroentries
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• Reconstruction/decoding: given(ill-posed inverse problem) find
• fast, wrong
• correct:only M=2Kmeasurements required to reconstruct K-sparse signal
slow: NP-hardalgorithm
Compressive Sensing Recovery
number ofnonzeroentries
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• Recovery: given(ill-posed inverse problem) find (sparse)
• fast, wrong
• correct, slow
• correct, efficientmild oversampling[Candes, Romberg, Tao; Donoho]
number of measurements required
Compressive Sensing Recovery
linear program
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Why Works
minimum solution= sparsest solution (with high probability) if
for signals sparse in the space/time domain
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Universality
• Random measurements can be used for signals sparse in any basis
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Universality
• Random measurements can be used for signals sparse in any basis
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Universality
• Random measurements can be used for signals sparse in any basis
sparsecoefficient
vector
nonzeroentries
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Compressive Sensing• Directly acquire “compressed” data
• Replace N samples by M random projections
transmit/store
receive linear pgm
…
random measurements
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Compressive Sensing
Theory IIStable Embedding
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Johnson-Lindenstrauss Lemma
• JL Lemma: random projection stably embeds a cloud of Q points whp provided
• Proved via concentration inequality
• Same techniques link JLL to RIP [Baraniuk, Davenport, DeVore, Wakin, Constructive Approximation, 2008]
Q points
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Connecting JL to RIPConsider effect of random JL Φ on each K-plane
– construct covering of points Q on unit sphere– JL: isometry for each point with high probability– union bound isometry for all points q in Q– extend to isometry for all x in K-plane
K-plane
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Connecting JL to RIPConsider effect of random JL Φ on each K-plane
– construct covering of points Q on unit sphere– JL: isometry for each point with high probability– union bound isometry for all points q in Q– extend to isometry for all x in K-plane– union bound isometry for all K-planes
K-planes
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• Gaussian
• Bernoulli/Rademacher [Achlioptas]
• “Database-friendly” [Achlioptas]
• Random Orthoprojection to RM [Gupta, Dasgupta]
Favorable JL Distributions
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RIP as a “Stable” Embedding
• RIP of order 2K implies: for all K-sparse x1 and x2,
K-planes
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Structured Random Matrices
• There are more structured (but still random) compressed sensing matrices
• We can randomly sample in a domain whose basis vectors are incoherent with the sparsity basis
• Example: sparse in time, sample in frequency
time frequency
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Structured Random Matrices
• Signal is sparse in the wavelet domain, measured with noiselets (Coifman et al. ‘01)
2D noiselet wavelet domain noiselet domain
• Stable recovery from
measurements
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Random Convolution
• A natural way to implement compressed sensing is through random convolution
• Applications include active imaging (radar, sonar,…)
• Many recent theoretical results(R 08, Bajwa, Haupt et al 08, Rauhut 09)
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Random Convolution Theory• Convolution with a random pulse, then subsample
(each σω has unit magnitude and random phase)
• Stable recovery from
measurements
time frequency after convolution
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Coded Aperture Imaging• Allows high-levels of light and high resolution
(Marcia and Willett 08, also Brady, Portnoy and others)
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Super-resolved Imaging
(Marcia and Willet 08)
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Stability
• Recovery is robust against noise and modeling error
• Suppose we observe
• Relax the recovery algorithm, solve
• The recovery error obeys
measurement error + approximation error
best K-term approximation
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Geometrical Viewpoint, Noiseless
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Geometrical Viewpoint, Noise
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Compressive Sensing
Recovery Algorithms
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CS Recovery Algorithms• Convex optimization:
– noise-free signals Linear programming (Basis pursuit) FPC Bregman iteration, …
– noisy signals Basis Pursuit De-Noising (BPDN) Second-Order Cone Programming (SOCP) Dantzig selector GPSR, …
• Iterative greedy algorithms– Matching Pursuit (MP)– Orthogonal Matching Pursuit (OMP)– StOMP– CoSaMP– Iterative Hard Thresholding (IHT), …
software @dsp.rice.edu/cs
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L1 with equality constraints = linear programming
is equivalent to the linear program
The standard L1 recovery program
There has been a tremendous amount of progressin solving linear programs in the last 15 years
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SOCP
• Standard LP recovery
• Noisy measurements
• Second-Order Cone Program
• Convex, quadratic program
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Other Flavors of L1
• Quadratic relaxation (called LASSO in statistics)
• Dantzig selector (residual correlation constraints)
• L1 Analysis (Ψ is an overcomplete frame)
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Solving L1• “Classical” (mid-90s) interior point methods
– main building blocks due to Nemirovski– second-order, series of local quadratic approximations– boils down to a series of linear systems of equations– formulation is very general (and hence adaptable)
• Modern progress (last 5 years) has been on “first order” methods– Main building blocks due to Nesterov (mid 80s)– iterative, require applications of Φ and ΦT at each iteration– convergence in 10s-100s of iterations typically
• Many software packages available– Fixed-point continuation (Rice)– Bregman iteration-based methods (UCLA)– NESTA (Caltech)– GPSR (Wisconsin)– SPGL1 (UBC)…..
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Matching Pursuit
• Greedy algorithm
• Key ideas:(1) measurements composed of sumof K columns of
(2) identify which K columns sequentially according to size of contribution to
columns
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Matching Pursuit
• For each columncompute
• Choose largest(greedy)
• Update estimate by adding in
• Form residual measurementand iterate until convergence
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Orthogonal Matching Pursuit
• Same procedureas Matching Pursuit
• Except at each iteration:
– remove selected column
– re-orthogonalize the remaining columns of
• Converges in K iterations
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CoSAMP• Needell and Tropp, 2008• Very simple greedy algorithm, provably effective
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From Sparsity to
Model-based (structured)Sparsity
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Sparse Models
wavelets:natural images
Gabor atoms:chirps/tones
pixels:background subtracted
images
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Sparse Models • Sparse/compressible signal model captures
simplistic primary structure
sparse image
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• Sparse/compressible signal model captures simplistic primary structure
• Modern compression/processing algorithms capture richer secondary coefficient structure
Beyond Sparse Models
wavelets:natural images
Gabor atoms:chirps/tones
pixels:background subtracted
images
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• Sparse signal: only K out of N coordinates nonzero– model: union of all K-dimensional subspaces
aligned w/ coordinate axes
Signal Priors
sorted index
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• Sparse signal: only K out of N coordinates nonzero– model: union of all K-dimensional subspaces
aligned w/ coordinate axes
• Structured sparse signal: reduced set of subspaces(or model-sparse)– model: a particular union of subspaces
ex: clustered or dispersed sparse patterns
sorted index
Signal Priors
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• Model: K-sparse
• RIP: stable embedding
Restricted Isometry Property (RIP)
K-planes
Random subGaussian (iid Gaussian, Bernoulli) matrix < > RIP w.h.p.
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• Model: K-sparse + significant coefficients
lie on a rooted subtree(a known model for piecewise smooth signals)
• Tree-RIP: stable embedding
K-planes
Restricted Isometry Property (RIP)
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• Model: K-sparse
Note the difference:
• RIP: stable embedding
Restricted Isometry Property (RIP)
K-planes
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Model-Sparse Signals• Defn: A K-sparse signal model comprises a
particular (reduced) set of K-dim canonical subspaces
• Structured subspaces
<> fewer measurements
<> improved recovery perf.
<> faster recovery
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• Iterative Hard Thresholding (IHT)
CS Recovery
[Nowak, Figueiredo; Kingsbury, Reeves; Daubechies, Defrise, De Mol; Blumensath, Davies; …]
update signal estimate
prune signal estimate(best K-term approx)
update residual
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• Iterative Model Thresholding
Model-based CS Recovery
[VC, Duarte, Hegde, Baraniuk; Baraniuk, VC, Duarte, Hegde]
update signal estimate
prune signal estimate(best K-term model approx)
update residual
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Tree-Sparse Signal Recovery
target signal CoSaMP, (MSE=1.12)
L1-minimization(MSE=0.751)
Tree-sparse CoSaMP (MSE=0.037)
N=1024M=80
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Tree-Sparse Signal Recovery
• Number samples for correct recovery with noise
• Piecewise cubic signals + wavelets
• Plot the number ofsamples to reachthe noise level
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Clustered Sparsity• (K,C) sparse signals (1-D)
– K-sparse within at most C clusters
• For stable recovery
• Model approximation using dynamic programming
• Includes block sparsity as as a special case
[VC, Indyk, Hedge, Baraniuk]
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Clustered Sparsity
target Ising-modelrecovery
CoSaMPrecovery
LP (FPC)recovery
• Model clustering of significant pixels in space domain using graphical model (MRF)
• Ising model approximation via graph cuts[VC, Duarte, Hedge, Baraniuk]
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Clustered Sparsity
20% CompressionNo performance loss in tracking
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Neuronal Spike Trains
• Model the firing process of a single neuron via 1D Poisson process with spike trains
– stable recovery
• Model approximation solution:
– integer program
efficient & provable solution due to total unimodularity of linear constraint
– dynamic program [Hegde, Duarte, VC] – best paper award
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Performance of Recovery
• Using model-based IHT and CoSaMP
• Model-sparse signals
• Model-compressible signals w/restricted amplification property
CS recoveryerror
signal K-termmodel approx error
noise
[Baraniuk, VC, Duarte, Hegde]
CS recoveryerror
signal K-termmodel approx error
noise
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Compressive SensingIn Action
Cameras
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“Single-Pixel” CS Camera
randompattern onDMD array
DMD DMD
single photon detector
imagereconstruction
orprocessing
w/ Kevin Kelly
scene
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“Single-Pixel” CS Camera
randompattern onDMD array
DMD DMD
single photon detector
imagereconstruction
orprocessing
scene
• Flip mirror array M times to acquire M measurements• Sparsity-based (linear programming) recovery
…
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Single Pixel Camera
Object LED (light source)
DMD+ALP Board
Lens 1Lens 2Photodiode
circuit
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Single Pixel Camera
Object LED (light source)
DMD+ALP Board
Lens 1Lens 2Photodiode
circuit
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Single Pixel Camera
Object LED (light source)
DMD+ALP Board
Lens 1Lens 2Photodiode
circuit
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Single Pixel Camera
Object LED (light source)
DMD+ALP Board
Lens 1Lens 2Photodiode
circuit
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First Image Acquisition
target 65536 pixels
1300 measurements (2%)
11000 measurements (16%)
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Second Image Acquisition
500 random measurements
4096 pixels
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CS Low-Light Imaging with PMT
true color low-light imaging
256 x 256 image with 10:1 compression[Nature Photonics, April 2007]
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Hyperspectral Imaging
spectrometer
blue
red
near IR
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Georgia Tech Analog Imager
• Robucci and Hasler 07• Transforms image in analog,
reads out transform coefficients
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Georgia Tech Analog Imager
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Compressive Sensing Acquisition
10k DCT coefficients 10k random measurements
(Robucci et al 09)
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Compressive SensingIn Action
A/D Converters
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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
• DARPA Analog-to-Information (A2I) program– 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
freq
uenc
y
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ADC State of the Art (2005)
The bad news starts at 1 GHz…
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ADC State of the ArtFrom 2008…
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Analog-to-Information Conversion• Sample near signal’s (low) “information rate”
rather than its (high) Nyquist rate
• Practical hardware:randomized demodulator(CDMA receiver)
A2Isampling rate
number oftones /window
Nyquistbandwidth
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Example: Frequency Hopper
20x sub-Nyquist sampling
spectrogram sparsogram
Nyquist rate sampling
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Multichannel Random Demodulation
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Compressive SensingIn Action
Data Processing
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Information Scalability
• Many applications involve signal inferenceand not reconstruction
detection < classification < estimation < reconstruction
fairly computationallyintense
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Information Scalability
• Many applications involve signal inferenceand not reconstruction
detection < classification < estimation < reconstruction
• Good news: CS supports efficient learning, inference, processing directly on compressive measurements
• Random projections ~ sufficient statisticsfor signals with concise geometrical structure
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Low-dimensional signal models
pixels largewaveletcoefficients
(blue = 0)
sparsesignals
structured sparse signals
parametermanifolds
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Matched Filter• Detection/classification with K unknown
articulation parameters– Ex: position and pose of a vehicle in an image– Ex: time delay of a radar signal return
• Matched filter: joint parameter estimation and detection/classification– compute sufficient statistic for each potential target and
articulation– compare “best” statistics to detect/classify
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Matched Filter Geometry• Detection/classification with K unknown
articulation parameters
• Images are points in
• Classify by finding closesttarget template to datafor each class (AWG noise)– distance or inner product
data
target templatesfrom
generative modelor
training data (points)
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Matched Filter Geometry• Detection/classification with K unknown
articulation parameters
• Images are points in
• Classify by finding closesttarget template to data
• As template articulationparameter changes, points map out a K-dimnonlinear manifold
• Matched filter classification = closest manifold search
articulation parameter space
data
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CS for Manifolds
• Theorem:
random measurements stably embed manifoldwhp[Baraniuk, Wakin, FOCM ’08]related work:[Indyk and Naor, Agarwal et al., Dasgupta and Freund]
• Stable embedding
• Proved via concentration inequality arguments(JLL/CS relation)
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CS for Manifolds
• Theorem:
random measurements stably embed manifoldwhp
• Enables parameter estimation and MFdetection/classificationdirectly on compressivemeasurements– K very small in many
applications (# articulations)
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Example: Matched Filter
• Detection/classification with K=3 unknown articulation parameters1. horizontal translation2. vertical translation3. rotation
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Smashed Filter
• Detection/classification with K=3 unknown articulation parameters (manifold structure)
• Dimensionally reduced matched filter directly on compressive measurements
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Smashed Filter• Random shift and rotation (K=3 dim. manifold)• Noise added to measurements• Goal: identify most likely position for each image class
identify most likely class using nearest-neighbor test
number of measurements Mnumber of measurements M
avg.
shi
ft e
stim
ate
erro
r
clas
sific
atio
n ra
te (
%)
more noise
more noise
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Compressive Sensing
Summary
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CS Hallmarks
• CS changes the rules of the data acquisition game– exploits a priori signal sparsity information
• Stable– acquisition/recovery process is numerically stable
• Universal– same random projections / hardware can be used for
any compressible signal class (generic)
• Asymmetrical (most processing at decoder)– conventional: smart encoder, dumb decoder– CS: dumb encoder, smart decoder
• Random projections weakly encrypted
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CS Hallmarks
• Democratic– each measurement carries the same amount of information– robust to measurement loss and quantization simple
encoding
• Ex: wireless streaming application with data loss
– conventional: complicated (unequal) error protection of compressed data DCT/wavelet low frequency coefficients
– CS: merely stream additional measurements and reconstruct using those that arrive safely (fountain-like)