Introduction to compressive sensing
1 like4,741 views
This document introduces compressive sensing. It discusses sensing problems, sparsity, and incoherence in the introduction. It then covers robust compressive sampling and its relation to reconstruction error bounds. Random sensing using the restricted isometry property is also introduced. The document concludes by discussing compressive sampling applications in data compression, channel coding, and data acquisition where signals can be captured efficiently using fewer measurements proportional to information level.
Introduction to compressive sensing
- 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