Prof Dr Svetlana Avramov-Zamurović
METODA SAŽETOG PRONALAŽENJA
INFORMACIJA
COMPRESSIVE SENSING
United States Naval Academy
United States Naval Academy
Annapolis
Severn River
History
1845
-
Fort Severn
Nine acres of land
50 Midshipmen
7 professors (3 civilian, 4 officers)
Current
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U.S. Naval Academy
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NASA STS 62
1994
Mission: USMP-2; OAST-2
Space Shuttle: Columbia
Launch Weight: 4,519,319 pounds
Launched: March 4, 1994; 8:53:01 a.m. EST
Landing Site: Kennedy Space Center, Florida
Landing: March 18, 1994 at 8:10 a.m. EST
Mission Duration: 13 days, 23 hours,
Orbit Altitude: 163 nautical miles
Miles Traveled: 5.8 million
The Critical Fluid Light Scattering Experiment, or ZENO, science team reported that they expect to
locate the critical temperature of xenon at "any time." Team members closely watched
computer data traces which indicate their experiment was very near the critical temperature
-- the goal of a lengthy, methodical "sensitive" search process. This is a more precise search
for the critical temperature after its location has been determined within a narrow band.
Once the temperature is located, the team will spend nearly 24 hours taking a good look at
the phenomenon they've waited years to see. They will study the properties of xenon at its
critical point, taking subtle optical measurements in the region surrounding it. A fluid's
"critical point" occurs at a condition of temperature and pressure where the fluid is
simultaneously a gas and a liquid. By understanding how matter behaves at the critical point,
scientists hope to gain a better insight into a variety of physics problems ranging from phase
changes in fluids to changes in the composition and magnetic properties of solids.
http://www.nss.org/resources/library/shuttlevideos/shuttle61.htm
People who made ZENO possible
•
Svetlana Avramov, Ratio Transformer Calibration
Matt Bieneman, Software Engineer
John Borden, Project Manager, POCC Team
Hacene Boukari, Sample Cell, POCC Team
Matt Briggs, Thermostat, POCC Team
Ed Cole, Machinist
Glenn Connor, Project Manager
Robert Gammon, Principle Investigator, POCC Team
David Gent, Sample Cell, POCC Team
Sue Landon, Project Coordinator
Meade Larson, Product Assurance
Paul Mahata, Systems Engineer
Don Martin, Technician
Lori Mora, Project Coordinator
John Pickett, Thermostat Sweathog
Jeff Shaumeyer, Project Scientist, POCC Team
Mark Sienkiewicz, Software Engineer, POCC Team
Danny Swann, Thermostat Sweathog
David Torrealba, Project Manager
Paul Wood, Machinist
Kai-Chang Zhang, Thermostat
Flight testing in Boulder Colorado
NASA STS 85
1997
Mission: CRISTA-SPAS-02
Space Shuttle: Discovery
Launched: August 7, 1997, 10:41:00 a.m. EDT
Landing Site: Kennedy Space Center, Florida
Landing: August 19, 1997, 7:07:59 a.m. EDT
Mission Duration: 11 days, 19 hours, 18 minutes, 47 seconds
Orbit Altitude: 173 statute miles
Miles Traveled: 4.7 million
CVX-2 - Critical Viscosity of Xenon - was a
science experiment to measure the viscosity
(slipperiness) of Xenon at its critical point (the
temperature and pressure where a substance is
both a liquid and a gas at the same time.) Here's
the very technical status reports the team
released during the mission.
Imaging Classical Approach
PICTURE
N samples
(ALL
measurements
are taken)
ORTHOGONALIZITION
Full set of projections
is found
SORTING
CODING
straightforward
decoding
Only K coefficients
K largest
coefficients selected are coded
N-K coef. dumped
K<<N
EXAUSTIVE SEARCH
Compressive sensing
PICTURE
CODING
Signal Reconstruction
Capture only significant components
Only M (K ≈ M) samples (measurements) are taken
(a) Measurements must be carefully designed
(b) Original signal (picture) must be sparse
(1) Underdetermined system
M<N
(2) Reconstructed signal
must have N components
(3) L1 norm is used to find
sparse representation
Scene
Image
Photodiode
Bitstream
A/D
Digital Micromirror Device
Array
Random Number Generator
Reconstruction
Major challenges
(1) Acquisition speed
(2) DSP processing speed
The images in (a) and (b) are not meant to be aligned.
(a) Conventional digital
camera image of a soccer
ball (64 × 64=4096 pixels)
(b) The same ball recovered from M = 1600 random
measurements taken by the camera
From IEEE Signal processing magazine July 2007, R. G. Baraniuk, Compressive sensing
Experimental
results
Original
65536 Pixels
3300 Measurements
(5%)
16384 Pixels
1600
Measurements
(10%)
16384 Pixels
3300 Measurements
(20%)
65536 Pixels
1300 Measurements
(2%)
Sources of noise
(1) Nonlinearities in the photodiode
(2) Non-uniform reflectance of the mirrors
through the lens focus onto the photodiode
(changing the weighting of the pattern blocks)
(3) Non-uniform mirror positions
Robustness of the CS reconstruction algorithm
Suppresses quantization noise from ADC
and photodiode circuit noise.
From D. Takhar at al.
A new Compressive Imaging Camera Architecture using Optical domain Compression
From IEEE Signal processing magazine July 2007, R. G. Baraniuk, Compressive sensing
Definitions
x target signal,
represenation of an image
in time or space domain
x is in N dimensional domain
 x1 
 11
    orthonormal basis  
 xN 
 N 1
 1N 
 
i othonormal vector

 NN 
N
 i1 
  
 iN 
Classical Approach: Find the signal projections on a given basis : x=  si i
i 1
sirepresenation of the image in  domain   iT x
x is K-sparce if only K basis vecotors have s  0 (thoeretical)
i
i
x is compressible if it has just a few large s coeficients and many small (practical)
i
Compressable signals are well represented by K-sparse representations.
Compressed sensing: Measure only significant components (M
 y1 
 11
y measurements
    measurement matrix  
y is in M dimensional domain
 yM 
M 1
y=x=s  Goal y  s
1N 


MN 
N)
COMPRESSIVE SENSING MEASUREMENT PROCESS
Φ
y
Ψ
S
=
y
Θ
=
M
x
(a)
N
K-sparse
(b)
(a) Compressive sensing measurement process with a random Gaussian measurement matrix and
discrete cosine transform (DCT) matrix . The vector of coefficients s is sparse with K = 4.
Φ (phi, measurement matrix) Ψ (psi, orthonormal basis) Θ (theta, Compressive Sensing reconstruction matrix)
(b) Measurement process with    There are four columns that correspond to nonzero si
coefficients; the measurement vector y is a linear combination of these columns.
From IEEE Signal processing magazine July 2007, R. G. Baraniuk, Compressive sensing
S
FINDING SOLUTION USING L1 NORM
a
RN
b
Ax=b underdetermined system
c
S
(a) The subspaces containing two sparse vectors in R3 lie close to the coordinate axes. (b) Visualization of the L2 minimization (5)
that finds the non-sparse point-of-contact s between the 2 ball (hyper-sphere, in red) and the translated measurement matrix null
space (in green). (c) Visualization of the L1 minimization solution that finds the sparse point-of-contact s with high probability thanks
to the pointiness of the 1 ball.
From IEEE Signal processing magazine July 2007, R. G. Baraniuk, Compressive sensing
From Imaging via
Compressive Sampling
J. Romberg
L1 AND L2 NORM COMPARISON
WHEN FINDING SOLUTION TO
COMPRESSIVE SAMPLING PROBLEM
(a) A sparse real valued signal and (b) its reconstruction
from 60 (complex valued) Fourier coefficients by l1
minimization. The reconstruction is exact. (c) The minimum
energy reconstruction obtained by substituting the 1 norm
with the l2 norm; l1 and l2 give wildly different answers. The
l2 solution does not provide a reasonable approximation to
the original signal.
From An Introduction to Compressive Sampling E. Candes M. Wakin
COMPRESSIVE SENSNING
Compressive
Imaging
Hyperspectral
Imaging
Medical Imaging
Computer Graphics
Analog-toInformation
Conversion
Geophysical Data
Analysis
Astronomy
Biosensing
Integrated Circuit
Analysis
Surface Metrology
Communications
Spectrum Analysis
Random Sampling
Remote Sensing
Multi-Sensor and
Distributed
Compressive
Sensing
Bayesian Methods
Statistical Signal
Processing
Model-based
Compressive
Sensing
Machine Learning
Compressive Radar
Foundations and
Connections
Coding and
Information
Theory
High-Dimensional
Geometry
Finite Rate of
Innovation
Ell-1 Norm
Minimization
Compressive
Sensing Recovery
Algorithms
Adaptive Sampling
Methods for
Sparse Recovery
Data Stream
Algorithms Heavy-Hitters
Dimension
Reduction and
Embeddings
Svetlana
Avramov-Zamurović
email
avramov@usna.edu
http://www.usna.edu/Users/weapsys/avramov
http://www.dsp.ece.rice.edu/cs
http://people.ee.duke.edu/~lcarin/compressive-sensing-workshop.html
Sub-Nyquist Sampling
From An Introduction to Compressive Sampling E. Candes M. Wakin
MRI Application
From Sparse MRI: The Application of Compressed Sensing for Rapid MR Imaging M. Lustig, D. Donoho and J. M. Pauly
COMPRESSIVE VIDEO SAMPLING
Vladimir Stankovic and Lina Stankovic Samuel Cheng
•
•
•
•
•
A system based on the novel concept of
compressive sampling to achieve real-time, low
complexity acquisition of video.
Each frame of the video is split into a number of
smaller non-overlapping blocks of equal size to
reduce the complexity of compressive sampling
algorithms and exploit the varying sparsity
across blocks within a frame.
Compressive sampling is performed only on
those blocks that satisfy our proposed simple
sparsity test, while the remaining blocks are
sampled fully.
Full sampled reference frames are used to
predict sparsity of the blocks within successive
frames.
Experimental results show great potential for
compressive sampling for video acquisition, with
up to 50% savings in acquisition with good
reconstruction quality.
Videos and Compressive Sensing