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Compressive Sensing Basics - Medical Imaging - MRI
- 1. S Compressive Sensing Stefani Thomas MD 2014
- 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. Compressive Sensing S0 Sensing Recovery Noise x ˆs Measured signal Unknown signal
- 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. Compressive Sensing S0 Sensing Recovery Noise x ˆs Sparse signal Incoherence Non-linear reconstruction
- 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. 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. Non linear reconstruction S L0 norm highly non convex and NP hard S I
- 10. Non linear reconstruction S L2 norm minimize energy not sparsity S I +
- 11. Non linear reconstruction S L1 norm is convex ! S I +
- 12. S RIP 1-eK( ) £ Ax 2 2 x 2 2 £ 1+eK( ) Restricted Isometry Property
- 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. S Gerhard Richter (1024 colours - 1974) Restricted Isometry Property
- 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. Compressive Sensing - MRI Acquisition Space = k-space Reconstructed image
- 19. Compressive Sensing x MRI Not Sparse ! Wavelet Domain it is !
- 21. Compressive Sensing Noise like pattern Noise x MRI Sparse signal
- 24. Compressive Sensing - MRI S Real life example S T2 SE matrix 512x512 : duration S T2 SE CS : duration
- 25. S DFT X=Wx Im Re Direct Fourier Transform
- 26. S DFT X=Wx Direct Fourier Transform
- 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.