1. Introduction to Compressive Sensing
Robust Compressive Sampling
Random Sensing
Conclusion
Introduction to Compressive Sensing
Mohammed Musfir
Guided By :
Mr.Edet Bijoy K
Asstistant Professor
Department of ECE
MES College of Engineering
February 20, 2012
Mohammed Musfir Introduction to Compressive Sensing

2. Introduction to Compressive Sensing
Robust Compressive Sampling
Random Sensing
Conclusion
Contents
1 Introduction to Compressive Sensing
Sensing Problem
Sparsity
Incoherence
2 Robust Compressive Sampling
Robustness
3 Random Sensing
RIP
4 Conclusion
Mohammed Musfir Introduction to Compressive Sensing

3. Introduction to Compressive Sensing
Sensing Problem
Robust Compressive Sampling
Sparsity
Random Sensing
Incoherence
Conclusion
1 Introduction to Compressive Sensing
Sensing Problem
Sparsity
Incoherence
2 Robust Compressive Sampling
Robustness
3 Random Sensing
RIP
4 Conclusion
Mohammed Musfir Introduction to Compressive Sensing

4. Introduction to Compressive Sensing
Sensing Problem
Robust Compressive Sampling
Sparsity
Random Sensing
Incoherence
Conclusion
Undersampling
m < n - undersampling, where m is the size of the
acquisition and n size of the signal f
Is reconstruction possible?
Creation of sensing matrix m << n
How to get the estimated significant f from f candidates
Mohammed Musfir Introduction to Compressive Sensing

5. Introduction to Compressive Sensing
Sensing Problem
Robust Compressive Sampling
Sparsity
Random Sensing
Incoherence
Conclusion
What is Sparsity?
Exploiting concise nature of natural signals
In sparse representation :Small coefficients discarded
without perpetual loss
Perceptual loss is hardly noticeable
Mohammed Musfir Introduction to Compressive Sensing

6. Introduction to Compressive Sensing
Sensing Problem
Robust Compressive Sampling
Sparsity
Random Sensing
Incoherence
Conclusion
Example of Compressive Sensing
a. Original image
c. Image reconstructed by discarding 97.5% coefficients
Mohammed Musfir Introduction to Compressive Sensing

7. Introduction to Compressive Sensing
Sensing Problem
Robust Compressive Sampling
Sparsity
Random Sensing
Incoherence
Conclusion
Why Incoherence?
m = C · µ2 (φ, ω) · S · log n (1)
Coherence = Covariance
Smaller the Coherence Fewer the samples required
Perceptual loss is hardly noticeable when measured set is
just m coefficients
Signal recovered from condensed set without knowledge of
the number, amplitude or position of non zero coefficients
Mohammed Musfir Introduction to Compressive Sensing

8. Introduction to Compressive Sensing
Robust Compressive Sampling
Robustness
Random Sensing
Conclusion
1 Introduction to Compressive Sensing
Sensing Problem
Sparsity
Incoherence
2 Robust Compressive Sampling
Robustness
3 Random Sensing
RIP
4 Conclusion
Mohammed Musfir Introduction to Compressive Sensing

9. Introduction to Compressive Sensing
Robust Compressive Sampling
Robustness
Random Sensing
Conclusion
Reconstruction error
Bounded by sum of two terms
Error from noiseless data
Error proportional to the noise level
Mohammed Musfir Introduction to Compressive Sensing

10. Introduction to Compressive Sensing
Robust Compressive Sampling
RIP
Random Sensing
Conclusion
1 Introduction to Compressive Sensing
Sensing Problem
Sparsity
Incoherence
2 Robust Compressive Sampling
Robustness
3 Random Sensing
RIP
4 Conclusion
Mohammed Musfir Introduction to Compressive Sensing

11. Introduction to Compressive Sensing
Robust Compressive Sampling
RIP
Random Sensing
Conclusion
Restricted Isometry Property
The subsets of S Columns from sensing matrix are nearly
orthogonal
Deterministic
Pairwise distances between S-Sparse signals well preserved
in measurement space
Mohammed Musfir Introduction to Compressive Sensing

12. Introduction to Compressive Sensing
Robust Compressive Sampling
Random Sensing
Conclusion
1 Introduction to Compressive Sensing
Sensing Problem
Sparsity
Incoherence
2 Robust Compressive Sampling
Robustness
3 Random Sensing
RIP
4 Conclusion
Mohammed Musfir Introduction to Compressive Sensing

13. Introduction to Compressive Sensing
Robust Compressive Sampling
Random Sensing
Conclusion
Compressive Sampling
Best compressed form
Only decompresssing is necessary after acquisition
Purely algebraic approach ignores the conditioning of the
information operates
Well conditioned matrices necessaryfor accurate
estimation
Mohammed Musfir Introduction to Compressive Sensing

14. Introduction to Compressive Sensing
Robust Compressive Sampling
Random Sensing
Conclusion
Applications
Compressible signals can be captured efficiently using a
number of incoherent measurements propotional to its
information leve S << n
Data compression
Channel coding
Data acquisition
Mohammed Musfir Introduction to Compressive Sensing

15. Introduction to Compressive Sensing
Robust Compressive Sampling
Random Sensing
Conclusion
mohammed.musfir@ieee.org
THANK YOU
Mohammed Musfir Introduction to Compressive Sensing