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# Introduction to compressive sensing

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### Introduction to compressive sensing

1. 1. Introduction to Compressive Sensing Robust Compressive Sampling Random Sensing ConclusionIntroduction 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. 2. Introduction to Compressive Sensing Robust Compressive Sampling Random Sensing ConclusionContents 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. 3. Introduction to Compressive Sensing Sensing Problem Robust Compressive Sampling Sparsity Random Sensing Incoherence Conclusion1 Introduction to Compressive Sensing Sensing Problem Sparsity Incoherence2 Robust Compressive Sampling Robustness3 Random Sensing RIP4 Conclusion Mohammed Musﬁr Introduction to Compressive Sensing
4. 4. Introduction to Compressive Sensing Sensing Problem Robust Compressive Sampling Sparsity Random Sensing Incoherence ConclusionUndersampling 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. 5. Introduction to Compressive Sensing Sensing Problem Robust Compressive Sampling Sparsity Random Sensing Incoherence ConclusionWhat 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. 6. Introduction to Compressive Sensing Sensing Problem Robust Compressive Sampling Sparsity Random Sensing Incoherence ConclusionExample of Compressive Sensing a. Original image c. Image reconstructed by discarding 97.5% coeﬃcients Mohammed Musﬁr Introduction to Compressive Sensing
7. 7. Introduction to Compressive Sensing Sensing Problem Robust Compressive Sampling Sparsity Random Sensing Incoherence ConclusionWhy 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. 8. Introduction to Compressive Sensing Robust Compressive Sampling Robustness Random Sensing Conclusion1 Introduction to Compressive Sensing Sensing Problem Sparsity Incoherence2 Robust Compressive Sampling Robustness3 Random Sensing RIP4 Conclusion Mohammed Musﬁr Introduction to Compressive Sensing
9. 9. Introduction to Compressive Sensing Robust Compressive Sampling Robustness Random Sensing ConclusionReconstruction error Bounded by sum of two terms Error from noiseless data Error proportional to the noise level Mohammed Musﬁr Introduction to Compressive Sensing
10. 10. Introduction to Compressive Sensing Robust Compressive Sampling RIP Random Sensing Conclusion1 Introduction to Compressive Sensing Sensing Problem Sparsity Incoherence2 Robust Compressive Sampling Robustness3 Random Sensing RIP4 Conclusion Mohammed Musﬁr Introduction to Compressive Sensing
11. 11. Introduction to Compressive Sensing Robust Compressive Sampling RIP Random Sensing ConclusionRestricted 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. 12. Introduction to Compressive Sensing Robust Compressive Sampling Random Sensing Conclusion1 Introduction to Compressive Sensing Sensing Problem Sparsity Incoherence2 Robust Compressive Sampling Robustness3 Random Sensing RIP4 Conclusion Mohammed Musﬁr Introduction to Compressive Sensing
13. 13. Introduction to Compressive Sensing Robust Compressive Sampling Random Sensing ConclusionCompressive 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. 14. Introduction to Compressive Sensing Robust Compressive Sampling Random Sensing ConclusionApplications 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. 15. Introduction to Compressive Sensing Robust Compressive Sampling Random Sensing Conclusion mohammed.musﬁr@ieee.org THANK YOU Mohammed Musﬁr Introduction to Compressive Sensing