1. Introduction to Compressed Sensing and its applications

2. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

3. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Human centric conversion process
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

4. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

5. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Spectrums of time domain signal and its samples
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

6. Wide band signal acquisition
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

7. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

8. 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

9. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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 , Sep
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

10. Idea of simultaneous compression and sampling
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

11. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

12. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Common link
Basis Function
Coefficients
Signal
k - sparse signal 𝑥 ≅ Φ𝜃
Support of non zero indices for 𝜃 denoted as
𝑆(𝜃) .
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

13. Transform domain image representation
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

14. Basics of Compressed Sensing
Demo
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

15. Whittaker–Shannon–Kotelnikov (WSK) v/s Compressive sensing
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

16. Whittaker–Shannon–Kotelnikov (WSK) v/s Compressive sensing
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

17. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

18. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

19. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

20. 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

21. Convex and Non convex set
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

22. 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

23. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

24. Best possible approximation
approximation ℓ𝑝 norm with 𝑝<1 .
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

25. 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

26. 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 signals
Should have a unique measurement
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

27. 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

28. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

29. Uniqueness property in presence of noise
Modified
In CS, it is assumed that is available during sparse signal recovery.
Measurement process should be robust to such noise.
Sparse recovery is possible if
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

30. 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

31. Building Sensing Matrices.
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

32. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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 , Sept
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

33. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

34. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

35. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

36. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
2D reconstruction
(a) (b)
a) original Image N = 6400 (80 x 80); b) reconstructed Image with M = 2400, MSE: 0.0 reconstruction time: sec.
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

37. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
BPIC
Bound based on noise. Could be user defined
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

38. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
N = 1024; k = 50; M = 220; MSE=2.02 x 10-4
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

39. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
(a) (b)
(a) reconstructed Image M = 1600, MSE: 0.25, reconstruction time: sec;
(b) reconstructed Image M = 800, MSE: 0.31, reconstruction time: sec
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

40. Watermarking application
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

41. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Detector
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

42. 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

43. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Fidelity
Non malicious
Malicious manipulations
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

44. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Single Pixel Camera
webee.technion.ac.il
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

45. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
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

46. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
References
D.L. Donoho, "Compressed sensing", IEEE Trans. Inf. Theory, vol. 52, no.4, pp , Sep
E.J. Candès & T.Tao, "Near optimal signal recovery from random projections: Universal encoding strategies," IEEE Trans. Inf. Theory, vol.52, no. 12, pp , Dec
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 , Dec
Richard Baraniuk, "Compressive sensing," IEEE Sig. Proc. Mag., vol. 24, no. 4, pp , 2007.
M.F. Duarte & Y.C. Eldar, "Structured compressed sensing: from theory to applications," IEEE Trans. Sig. Proc., vol.59, no. 9, pp , Sept
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 , 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

47. Compressive sensing solvers
(Comprehensive Listing of solvers)
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015

48. Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015
Thank you
Workshop on Compressive sensing, MNIT Jaipur, 12th - 13th Sept. 2015