EE123 Digital Signal Processing

Transcription

1 EE123 Digital Signal Processing Lecture 24 Compressed Sensing III M. Lustig, EECS UC Berkeley

2 RADIOS sp15/radio.html Interfaces and radios on Wednesday -- please come to pick up Midterm II this Friday -- same deal - open everything covers everything including 2D M. Lustig, EECS UC Berkeley

3 Traditional Sensing x R N is a signal Make N linear measurements y Φ Arbitrary sensing x = A good sensing matrix is orthogonal N Φ* Φ = I NxN N sensing matrix M. Lustig, EECS UC Berkeley

4 Compressed Sensing!! (Candes, Romber, Tao 2006; Donoho 2006) x R N is a K-sparse signal (K<<N) Make M (K<M<<N) incoherent linear projections y Φ x M = MxN A good compressed sensing matrix is incoherent i.e, approximately orthogonal Φ* Φ I sensing matrix K Incoherency can preserve information M. Lustig, EECS UC Berkeley

5 CS recovery Given y = Φx find x But there s hope, x is sparse! Under-determined y Φ x = M. Lustig, EECS UC Berkeley

6 CS recovery Given y = Φx find x But there s hope, x is sparse! Under-determined M. Lustig, EECS UC Berkeley

7 CS recovery Given y = Φx find x But there s hope, x is sparse! Under-determined minimize x 2 s.t. y = Φx WRONG! M. Lustig, EECS UC Berkeley

8 CS recovery Given y = Φx find x But there s hope, x is sparse! Under-determined minimize x 0 s.t. y = Φx HARD! M. Lustig, EECS UC Berkeley

9 CS recovery Given y = Φx find x But there s hope, x is sparse! Under-determined minimize x 1 s.t. y = Φx need M K log(n) <<N Solved by linear-programming M. Lustig, EECS UC Berkeley

10 Geometric Interpretation domain of sparse signals minimum x 1 minimum x 2 M. Lustig, EECS UC Berkeley

11 A non-linear sampling theorem f C N supported on a set Ω in Fourier Shannon: Ω is known connected set, size B Exact recovery from B equispaced time samples Linear reconstruction by sinc interpolation Non-linear sampling theorem Ω is an arbitrary, unknown set of size B Exact recovery from ~ B logn (almost) arbitrary placed samples Nonlinear reconstruction by convex programming M. Lustig, EECS UC Berkeley

12 Practicality of CS Can such sensing system exist in practice? Fourier matrix M. Lustig, EECS UC Berkeley

13 Practicality of CS Can such sensing system exist in practice? Fourier matrix M. Lustig, EECS UC Berkeley

14 Practicality of CS Can such sensing system exist in practice? Fourier matrix M. Lustig, EECS UC Berkeley

15 Practicality of CS Can such sensing system exist in practice? Randomly undersampled Fourier is incoherent MRI samples in the Fourier domain! Φ* Φ I = M. Lustig, EECS UC Berkeley

16 M. Lustig, EECS UC Berkeley

17 Intuitive example of CS

18 Intuitive example of CS FFT sampling Nyquist

19 Intuitive example of CS FFT equispaced sub-nyquist

20 Intuitive example of CS FFT sub-nyquist Ambiguity

21 Intuitive example of CS FFT random sub-nyquist

22 M. Lustig, EECS UC Berkeley

23 Intuitive example of CS FFT sub-nyquist Looks like random noise

24 Intuitive example of CS FFT sub-nyquist But it s not noise!

25 M. Lustig, EECS UC Berkeley

26 M. Lustig, EECS UC Berkeley

27 Intuitive example of CS FFT Recovery Example inspired by Donoho et. Al, 2007

28 M. Lustig, EECS UC Berkeley

29 Question! What if this was the signal? Would CS still work? random sub-nyquist

30 Domains in Compressed Sensing Not Sparse! Signal Sampling Domain Sparse! incoherent Sparse Domain

31 MRI Signal Sampling Domain Not Sparse! Signal Sparse! incoherent Sparse Domain

32 Acquired Data Compressed Sensing Reconstruction Sparse denoising

33 Acquired Data Compressed Sensing Reconstruction Sparse denoising

34 Acquired Data Compressed Sensing Reconstruction Sparse denoising Undersampled Final Image Tutorial & code available at

35 6 year old male abdomen. Fine structures (arrows) are buried in noise (artifactual + noise amplification) and are recovered by CS with L1-wavelets. x8 acceleration Linear Reconstruction Compressed sensing portal vein liver hepatic vein

36 6 year old male abdomen. Fine structures (arrows) are buried in noise (artifactual + noise amplification) and are recovered by CS with L1-wavelets. Zoom Linear Reconstruction Zoom Compressed sensing not seen portal vein Hepatic vein liver bile duct pancreatic duct

37 Back to Results 6 year old 8- fold acceleration 16 second scan mm in- plane 1.6 slice thickness

38 Principles of Magnetic Resonance Imaging EE c225e / BIOE c265 Spring 2016 Shameless Promotion

39 Other Applications Compressive Imaging Medical Imaging Analog to information conversion Biosensing Geophysical Data Analysis Compressive Radar Astronomy Communications More...

40 Resources CS + parallel imaging matlab code, examples Rice University CS page: papers, tutorials, codes,. IEEE Signal Processing Magazine, special issue on compressive sampling 2008;25(2) March 2010 Issue Wired Magazine: Filling the Blanks Igor Caron Blog: Thank you! ת ו ד ה רבה