Introduction to Compressed Sensing and its applications
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Age of Digital World
• Our life revolve around the digital world– Entertainment, communication, business, life !!
• Digital bits streams running at the background is expected to deliver “natural” performance.– Surround sound, 3D TV, sixth sense !!
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Human centric conversion process
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Nyquist Theorem
• Band limited signal with highest frequency of B Hz can be reconstructed perfectly from its samples with rate > 2B. (Nyquist Rate).
• Relation in X(t) and X(nT).– Digital operation replacing analog counter parts.– Relationship in power spectral densities of analog
and discrete random process. • Estimation and detection by DSP is possible.
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Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Spectrums of time domain signal and its samples
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Wide band signal acquisition
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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ADC
• Analog to digital converters forms heart.• Physical (analog) information streams of
numbers digital processing by software.• Intriguing task Snap shot of fast varying
signal + acquiring measurements.• Unprecedented strain on ADC’s and DSP.
– Demand is ever increasing.
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Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Ever increasing demand
• After sampling, we retain large number of bits.
• Conventional solution to storage space – Sampling Compression (exploiting redundancy)
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Ultimate question[1]
• Why so much effort is spent on acquiring (sampling) the data(redundancy) when most of it will be thrown away (compression)?
• Can’t we directly measure the part (information) which will not be thrown away?
• Why can’nt we ask such a question??[1] D.L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no.4, pp.
1289-1306, Sep. 2006.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Idea of simultaneous compression and sampling
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Common link
• Basis Function• Coefficients
• Signal
• k - sparse signal 𝑥 ≅ Φ 𝜃• Support of non zero indices for denoted as 𝜃𝑆( ) 𝜃 .
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Transform domain image representation
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Basics of Compressed Sensing
• Demo
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Whittaker–Shannon–Kotelnikov (WSK) v/s
Compressive sensing
WSK• Greater than Nyquist
rate.• Uniform / Non uniform
sampling• Non uniform sampling is
based on Lagrange interpolation.
• Theory developed for continuous time signals
CS• Sub Nyquist sampling.• Randomized
measurement matrix.• Samples are inner
product.
• Initial focus on finite dimensional signal.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Whittaker–Shannon–Kotelnikov (WSK) v/s
Compressive sensingWSK
• No underlying signal structure.
• Transform coding based compression.
• Does not consider Hardware implementation
CS• Initial sparse. Now
structure beyond sparsity is explored.
• Signal structure over and above sparsity gives higher compression.
• Structured non random measurement matrix.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Basic CS frame work
• 𝑥 is a N x 1 vector, • 𝑦 is a M x 1 vector with M << N
• is a random measurement matrix.• Sparsifying dictionary of basis
.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Recovery
• Recover• Given: and• : Class with all k sparse signal.• CS makes exhaustive search in such that
• Solution using optimization problem
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Issues with
• • optimization is computationally very
expensive: it is an non-deterministic polynomial-time (NP) hard problem.
• E.g. M = 1000, N = 5000, k = 100• search space.• Non convex
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Questions to be tackled
1. Can this problem be solved by any other mechanisms?
2. How can we get an estimated solution and what level of estimation accuracy is acceptable? (after all Engineers always looks for the workable approximate solutions !!!);
3. What kind of approximation will yield the solution closer to the desired one?
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Convex and Non convex set
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Convex relaxation - norm
• • • Set is convex, so above
problem is also convex.• A strictly posed convex problems leads to a
close form solution and guaranteed to converge at the local minima.
.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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norm
• Find Xi’s for which norm is minimal.• It is not strictly convex and it may have multiple
solution.• However, these solutions are 1. clustered around in a convex set as all optimal
solutions will have an penalty and their combination would also be convex;
2. the set is bounded; 3. among them at least one has at most k non zero
elements.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Best possible approximation
•
• approximation ℓ𝑝 norm with <1 .𝑝
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Heart: The measurement matrix
• ; : with M << N• Basis Fixed independent of signal.• Most fundamental design questions:1. how much of information about signal x is
retained in its linear measurements y ?; 2. how can the linear measurements y uniquely
represent x?; 3. how can the original signal be recovered from
its measurement?
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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The measurement matrix
• It is rank deficient with non empty Null space
• Consequence: • Unable to recover the signal from
measurements.• Design: For distinct k sparse signalsShould have a unique measurement
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Characterization of Uniqueness
• Spark = Sparse + Rank• If spark( ) > 2k uniquely
represents a k- sparse signal belonging to class
• The spark of the measurement matrix is used
to ensure stability and consistency.• Computing is search over all sub-matrices.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Coherence
• Coherence can be used to identify the sparse signal.
• It is describing the dependency between two columns.
• Supremum on sparsity k is
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Uniqueness property in presence of noise
• Modified • In CS, it is assumed that is available during
sparse signal recovery.• Modified • Measurement process should be robust to
such noise.• Sparse recovery is possible if
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Restricted Isometric property (RIP)
• A matrix satisfies (k, ) restricted iso-metry property of order k if for ,
• Isometry is a function between the two spaces which has a property to preserve distance between each pair of points.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Building Sensing Matrices
• Vandermonde matrix has spark M + 1 • : Geometric progression.• Poor conditioning. • Gabor Frame = n x n time shift matrix
• Bernoulli, Gaussian, or sub Gaussian
.
.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Research avenue
• Structured measurement matrix: Application dependent.
• Subjected to the physical constraint of the application.
M.F. Duarte & Y.C. Eldar, "Structured compressed sensing: from theory to applications," IEEE Trans. Sig. Proc., vol.59, no. 9, pp. 4053-4085, Sept. 2011.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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CS recovery
• CS recovery = L1 minimization• The choice of algorithm is based on various factors,
namely: 1. signal reconstruction timing from measurement
vector; 2. the number of measurement required for recovery
to determine the storage requirements;3. the simplicity of the implementation; 4. possible portability to the hardware for execution;5. fidelity of the signal recovery.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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BP: Basis Pursuit, BPIC: Basis Pursuit with Inequality Constraint, BPDN: Basis Pursuit De-noising, MP: Matching Pursuit, OMP: Orthogonal Matching Pursuit, IHT: Iterative Hard Thresholding
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Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Basic
•
• Its power stems from the fact that it converts the search into convex problem and provides the accurate recovery.
• Basis pursuit (BP): norm has a tendency to locate the sparse solutions if ever they exist.
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Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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2D reconstruction
(a) (b)• a) original Image N = 6400 (80 x 80); b) reconstructed
Image with M = 2400, MSE: 0.0 reconstruction time: 18.64 sec.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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BPIC
Bound based on noise. Could be user defined
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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N = 1024; k = 50; M = 220; MSE=2.02 x 10-4
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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(a) (b)• (a) reconstructed Image M = 1600, MSE: 0.25,
reconstruction time: 16.27 sec; • (b) reconstructed Image M = 800, MSE: 0.31,
reconstruction time: 31.19 sec
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Watermarking application
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Detector
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Over determined system
• with M > N• Construct matrix s.t = 0.
• Estimate using CS formulation.• s.t
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Fidelity Non malicious
Malicious manipulations
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Single Pixel Camera
• webee.technion.ac.il
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Summary
• One of the most exciting domain.• Interdisciplinary: signal processing, statistics,
probability theory, computer science, optimization, linear programming.
• Look beyond the random measurement matrix.
• Developing a better signal models: finite rate innovation (FRI), Xampling framework
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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References• D.L. Donoho, "Compressed sensing", IEEE Trans. Inf. Theory, vol. 52, no.4, pp. 1289 - 1306, Sep. 2006. • E.J. Candès & T.Tao, "Near optimal signal recovery from random projections: Universal encoding
strategies," IEEE Trans. Inf. Theory, vol.52, no. 12, pp.5406 - 5425, Dec. 2006.• E.J. Candès, J. Romberg, & T. Tao, "Robust uncertainty principles: exact signal reconstruction from
highly incomplete frequency information," IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489 - 509, Dec. 2006.
• Richard Baraniuk, "Compressive sensing," IEEE Sig. Proc. Mag., vol. 24, no. 4, pp. 118 -124, 2007.• M.F. Duarte & Y.C. Eldar, "Structured compressed sensing: from theory to applications," IEEE Trans. Sig.
Proc., vol.59, no. 9, pp. 4053-4085, Sept. 2011.• Compressed sensing, Theory and applications, (eds. Y.C. Eldar & Gitta Kutyniok), Cambridge university
press, Cambridge, UK, 2012.• E. J. Candès, J. Romberg and T. Tao, “Stable signal recovery from incomplete and inaccurate
measurements”, Comm. Pure Appl. Math., vol.59,pp. 1207–1223, 2006.• B. K. Natarajan, “Sparse approximate solutions to linear systems”, SIAM Journal on computing, vol. 24,
pp.227–234, 1995.• Nonlinear Optimization: Complexity Issues, S. A. Vavasis, Oxford University Press, New York, 1991.• E. J. Candès and T. Tao, "The power of convex relaxation: Near-optimal matrix completion," IEEE Trans.
Inform. Theory, vol. 56, no. 5, pp. 2053-2080, 2009.• Convex Optimization, Stephen Boyd & Lieven Vandenberghe, Cambridge University Press, 2004.• S. S. Chen, D. L. Donoho, & M. A. Saunders, " Atomic decomposition by basis pursuit," SIAM J. Scientific
Computing, vol. 20, no. 1, pp.33–61, 1998.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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Compressive sensing solvers
• http://users.ece.gatech.edu/~justin/l1magic/ • http://sparselab.stanford.edu/• http://www.lx.it.pt/~mtf/GPSR/• http://www.stanford.edu/~boyd/l1_ls/• http://www.personal.soton.ac.uk/tb1m08/sparsify/
sparsify.html• http://www.lx.it.pt/~mtf/SpaRSA/• https://sites.google.com/site/igorcarron2/
cs#reconstruction (Comprehensive Listing of solvers)• http://videolectures.net/bmvc09_chellappa_cscv/
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