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.