A Survey of Compressive Sensing and Applications · A Survey of Compressive Sensing and...
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A Survey of Compressive Sensing and Applications
Justin Romberg
Georgia Tech, School of ECE
ENS Winter SchoolJanuary 10, 2012Lyon, France
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Signal processing trends
DSP: sample first, ask questions later
Explosion in sensor technology/ubiquity has caused two trends:
Physical capabilities of hardware are being stressed,increasing speed/resolution becoming expensive
gigahertz+ analog-to-digital conversion accelerated MRI industrial imaging
Deluge of data camera arrays and networks, multi-view target databases, streaming
video...
Compressive Sensing: sample smarter, not faster
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Classical data acquisition
Shannon-Nyquist sampling theorem (Fundamental Theorem of DSP):“if you sample at twice the bandwidth, you can perfectly reconstructthe data”
time space
Counterpart for “indirect imaging” (MRI, radar):Resolution is determined by bandwidth
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Sense, sample, process...
!"#!$%& '()!*+&,-.& /)*)&0$12%"!!3$#&
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Compressive sensing (CS)
Shannon/Nyquist theorem is pessimistic 2×bandwidth is the worst-case sampling rate —
holds uniformly for any bandlimited data
sparsity/compressibility is irrelevant
Shannon sampling based on a linear model,compression based on a nonlinear model
Compressive sensing new sampling theory that leverages compressibility
key roles played by new uncertainty principles andrandomness
Shannon
Heisenberg
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Compressive sensing
!"#$%&'(()*'+,('-(#&,
!(.#/+,012,
Essential idea:“pre-coding” the signal in analog makes it “easier” to acquire
Reduce power consumption, hardware complexity, acquisition time
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A simple underdetermined inverse problem
Observe a subset Ω of the 2D discrete Fourier plane
phantom (hidden) white star = sample locations
N := 5122 = 262, 144 pixel imageobservations on 22 radial lines, 10, 486 samples, ≈ 4% coverage
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Minimum energy reconstruction
Reconstruct g∗ with
g∗(ω1,ω2) =
f(ω1,ω2) (ω1,ω2) ∈ Ω
0 (ω1,ω2) ∈ Ω
Set unknown Fourier coeffs to zero, and inverse transform
original Fourier samples g∗
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Total-variation reconstruction
Find an image thatFourier domain: matches observationsSpatial domain: has a minimal amount of oscillation
Reconstruct g∗ by solving:
ming
i,j
|(∇g)i,j | s.t. g(ω1,ω2) = f(ω1,ω2), (ω1,ω2) ∈ Ω
original Fourier samples g∗ = originalperfect reconstruction
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Sampling a superposition of sinusoids
We take M samples of a superposition of S sinusoids:
Time domain x0(t) Frequency domain x0(ω)
Measure M samples S nonzero components(red circles = samples)
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Sampling a superposition of sinusoids
Reconstruct by solving
minx
x1 subject to x(tm) = x0(tm), m = 1, . . . ,M
original x0, S = 15 perfect recovery from 30 samples
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Numerical recovery curves
Resolutions N = 256, 512, 1024 (black, blue, red)
Signal composed of S randomly selected sinusoids
Sample at M randomly selected locations
% success
0 0.2 0.4 0.6 0.8 10
10
20
30
40
50
60
70
80
90
100
S/M
In practice, perfect recovery occurs when M ≈ 2S for N ≈ 1000
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A nonlinear sampling theorem
Exact Recovery Theorem (Candes, R, Tao, 2004):
Unknown x0 is supported on set of size S
Select M sample locations tm “at random” with
M ≥ Const · S logN
Take time-domain samples (measurements) ym = x0(tm)
Solve
minx
x1 subject to x(tm) = ym, m = 1, . . . ,M
Solution is exactly f with extremely high probability
In total-variation/phantom example, S=number of jumps
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Graphical intuition for 1
minx x2 s.t. Φx = y minx x1 s.t. Φx = y!"#$L2 %&'()*+$!&,-$
.'/(+$(01/,'(234)4313$L5 (&.1+4&)4($/.3&(+$never sparse
!"#$L1 !%&'(
)*+*),)$L1 (%-,.*%+/$L0 (01&(2(.$(%-,.*%+$*3
random orientationdimension N-M
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Acquisition as linear algebra
= resolution/ bandwidth
# samples
data
unknown signal/image
acquisition system
Small number of samples = underdetermined systemImpossible to solve in general
If x is sparse and Φ is diverse, then these systems can be “inverted”
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Sparsity/Compressibility
pixels largewaveletcoefficients
widebandsignalsamples
largeGaborcoefficients
time
frequency
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Wavelet approximation
Take 1% of largest coefficients, set the rest to zero (adaptive)
original approximated
rel. error = 0.031
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Linear measurements
Instead of samples, take linear measurements of signal/image x0
y1 = x0,φ1, y2 = x0,φ2, . . . , yM = x0,φK
y = Φx0
Equivalent to transform-domain sampling,φm = basis functions
Example: pixels
ym = ,
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Linear measurements
Instead of samples, take linear measurements of signal/image x0
y1 = x0,φ1, y2 = x0,φ2, . . . , yM = x0,φK
y = Φx0
Equivalent to transform-domain sampling,φm = basis functions
Example: line integrals (tomography)
ym = ,
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Linear measurements
Instead of samples, take linear measurements of signal/image x0
y1 = x0,φ1, y2 = x0,φ2, . . . , yM = x0,φK
y = Φx0
Equivalent to transform-domain sampling,φm = basis functions
Example: sinusoids (MRI)
ym = ,