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# Compressive Sensing Basics - Medical Imaging - MRI

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Compressed sensing basics
Medical imaging use of compressive sensing - MRI

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### Compressive Sensing Basics - Medical Imaging - MRI

1. 1. S Compressive Sensing Stefani Thomas MD 2014
2. 2. Compressive Sensing S Exponential growth of data S 48h of video uploaded/min on Youtube S 571 new websites/min S 100 Terabytes of dada uploaded on facebook/day S How to cope with that amount S Compression S Better sensing of less data
3. 3. Compressive Sensing S0 Sensing Recovery Noise x ˆs Measured signal Unknown signal
4. 4. . . . . . . s0 æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ® N*N y . . . . . . u0 æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ® M*N F . . . x é ë ê ê ê ê ù û ú ú ú ú ® L0 . . . . . . ˆu æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ® N*N C . . . . . . ˆs æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ Compressive Sensing
5. 5. S Shanon’s sampling theorem S Full recovery under Nyquist sampling frequency? S Yes if fulfilling 3 criteria S Sparsity S Incoherence S Non linear reconstruction Compressive Sensing
6. 6. Compressive Sensing S0 Sensing Recovery Noise x ˆs Sparse signal Incoherence Non-linear reconstruction
7. 7. Sparsity S Desired signal has a sparse representation in some domain D S x of length N S x is K sparse S x has K non zeros components in D S Can be reconstructed using only M measurments (K<M<N) Wavelet transform
8. 8. Incoherence S Random subsampling must show “noise-like” pattern in the transform domain S Undersampling introduces noise S Randomly undersampled Fourier space is incoherent
9. 9. Non linear reconstruction S L0 norm highly non convex and NP hard S I
10. 10. Non linear reconstruction S L2 norm minimize energy not sparsity S I +
11. 11. Non linear reconstruction S L1 norm is convex ! S I +
12. 12. S RIP 1-eK( ) £ Ax 2 2 x 2 2 £ 1+eK( ) Restricted Isometry Property
13. 13. S If Φis a M x N Gaussian matrix S M > O ( Klog(N)) S If Ψ is a N x N sparsifying basis S  ΦΨ satisfies the RIP condition Restricted Isometry Property M.Rudelsonand, R.Vershynin, “On sparse reconstruction from Fourier and Gaussian measurements,” Commun. Pure Appl. Math., vol. 61, no. 8, pp. 1025–1045, 2008.
14. 14. S Gerhard Richter (1024 colours - 1974) Restricted Isometry Property
15. 15. Measurements required S How many measurements required S M ≥ K+1 S Only if S No noise S Real sparse signal S But S NP hard problem (exponential numbers of subsets)
16. 16. Compressive Sensing - MRI Acquisition Space = k-space Reconstructed image
17. 17. Compressive Sensing Recovery Noise x MRI
18. 18. Compressive Sensing Recovery Noise x MRI Not Sparse !
19. 19. Compressive Sensing x MRI Not Sparse ! Wavelet Domain it is !
20. 20. Compressive Sensing Recovery Noise x MRI Sparse signal
21. 21. Compressive Sensing Noise like pattern Noise x MRI Sparse signal
22. 22. Compressive Sensing Recovery Noise x MRI Sparse signal Incoherence
23. 23. . . . . . . s0 æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ® N*N y . . . . . . u0 æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ® M*N F . . . x é ë ê ê ê ê ù û ú ú ú ú ® L0 . . . . . . ˆs æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ® N*N C . . . . . . ˆu æ è ç ç ç ç ç ç ç ç ö ø ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ Compressive Sensing - MRI
24. 24. Compressive Sensing - MRI S Real life example S T2 SE matrix 512x512 : duration S T2 SE CS : duration
25. 25. S DFT X=Wx Im Re Direct Fourier Transform
26. 26. S DFT X=Wx Direct Fourier Transform
27. 27. S Random Fourier matrix satisfies the RIP condition: S M randomly chosen columns of NxN DFT matrix S M = O ( K.log (N) ) Direct Fourier Transform M.Rudelsonand, R.Vershynin, “On sparse reconstruction from Fourier and Gaussian measurements,” Commun. Pure Appl. Math., vol. 61, no. 8, pp. 1025–1045, 2008.