S
Compressive SensingStefani Thomas MD
2014
Compressive Sensing
Exponential growth of data 48h of video uploaded/min on Youtube 571 new websites/min 100 Terabytes of dada uploaded on facebook/day
How to cope with that amount Compression Better sensing of less data
Compressive Sensing
S0
Sensing Recovery
Noise x
Measured signal
Unknown signal
Compressive Sensing
Shanon’s sampling theorem Full recovery under Nyquist sampling frequency?
Yes if fulfilling 3 criteria Sparsity Incoherence Non linear reconstruction
Compressive Sensing
Compressive Sensing
S0
Sensing Recovery
Noise x
Sparse signal
Incoherence Non-linear reconstruction
Sparsity
Desired signal has a sparse representation in some domain D x of length N x is K sparse
x has K non zeros components in D Can be reconstructed using only M measurments (K<M<N)
Wavelet transform
Incoherence
Random subsampling must show “noise-like” pattern in the transform domain Undersampling introduces noise
Randomly undersampled Fourier space is incoherent
Non linear reconstruction
L0 norm highly non convex and NP hard
SI
Non linear reconstruction
L2 norm minimize energy not sparsity
SI
+
Non linear reconstruction
L1 norm is convex !
SI
+
RIP
Restricted Isometry Property
If Φis a M x N Gaussian matrix M > O ( Klog(N))
If Ψ is a N x N sparsifying basis 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.
Gerhard Richter (1024 colours - 1974)
Restricted Isometry Property
Measurements required
How many measurements required M ≥ K+1
Only if No noise Real sparse signal
But NP hard problem (exponential numbers of subsets)
Compressive Sensing - MRI
Acquisition Space = k-space Reconstructed image
Compressive Sensing
Recovery
Noise x
MRI
Compressive Sensing
Recovery
Noise x
MRINot
Sparse !
Compressive Sensing
x
MRINot
Sparse ! Wavelet Domain it is !
Compressive Sensing
Recovery
Noise x
MRI
Sparse signal
Compressive Sensing
Noise like pattern
Noise x
MRI
Sparse signal
Compressive Sensing
Recovery
Noise x
MRI
Sparse signal
Incoherence
Compressive Sensing - MRI
Compressive Sensing - MRI
Real life example T2 SE matrix 512x512 : duration T2 SE CS : duration
DFT X=Wx
Im Re
Direct Fourier Transform
DFT X=Wx
Direct Fourier Transform
Random Fourier matrix satisfies the RIP condition: M randomly chosen columns of NxN DFT matrix 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.