Toward 0-norm Reconstruction, and A Nullspace Technique for Compressive Sampling

1. Toward 0-norm Reconstruction, and A Nullspace Technique for Compressive Sampling Christine Law Gary Glover Dept. of EE, Dept. of Radiology Stanford University

2. Outline • 0-norm Magnetic Resonance Imaging (MRI) reconstruction. • Homotopic • Convex iteration • Signal separation example • 1-norm deconvolution in fMRI. • Improvement in cardinality constraint problem with nullspace technique.

3. k = 2 n Shannon vs. Sparse Sampling • Nyquist/Shannon : sampling rate ≥ 2*max freq • Sparse Sampling Theorem: (Candes/Donoho 2004) Suppose x in Rn is k-sparse and we are given m Fourier coefficients with frequencies selected uniformly at random. If m ≥ k log2 (1 + n / k) then reconstructs x exactly with overwhelming probability.

4. 1 week Rat dies 1 week after drinking poisoned wine Example by Anna Gilbert

5. x1 x2 x3 x4 x5 x6 x7 y1 y2 y3

6. k=1 n=7 m=3 x1 x2 x3 x4 x5 x6 x7 y1 y2 y3

7. Reconstruction by Optimization • Compressed Sensing theory (2004 Donoho, Candes): under certain conditions, y are measurements (rats) x are sensors (wine) Candes et al. IEEE Trans. Information Theory 2006 52(2):489 Donoho. IEEE Trans. Information Theory 2006 52(4):1289

8. For p-norm, where 0 < p < 1 Chartrand (2006) demonstrated fewer samples y required than 1-norm formulation. Chartrand. IEEE Signal Processing Letters. 2007: 14(10) 707-710. 0-norm reconstruction • Try to solve 0-norm directly.

9. method method method Homotopic Homotopic Homotopic Trzasko (2007): Rewrite the problem where r is tanh, laplace, log, etc. such that Trzasko et al. IEEE SP 14th workshop on statistical signal processing. 2007. 176-180.

10. Homotopic function in 1D Start as 1-norm problem, then reduce s slowly and approach 0-norm function.

11. Homotopic method

13. when is big (1st iteration), solving 1-norm problem. reduce to approach 0-norm solution. Demonstration x∆ original

14. Example 1 subsampled original

15. 1-norm recon 1-norm result: use 4% Fourier data error: -11.4 dB 542 seconds Homotopic result: use 4% Fourier data error: -66.2 dB 85 seconds homotopic recon Fourier sample mask Zero-filled Reconstruction

16. Example 2 • Angiography • 360x360, 27.5% radial samples original

17. reconstruction homotopic method: error: -26.5 dB, 101 seconds original 27.5% samples 360x360 reconstruction 1-norm method: error: -24.7 dB, 1151 seconds

18. Convex Iteration Chretien, An Alternating l1 approach to the compressed sensing problem, arXiv.org . Dattorro, Convex Optimization & Euclidean Distance Geometry, Meboo.

19. Convex Iteration

21. Convex Iteration demo use 4% Fourier data error: -104 dB 96 seconds Fourier sample mask reconstruction zero-filled reconstruction

22. Signal Separation by Convex Iteration

23. as convex iteration 1-norm formulation

24. = + u Signal Construction Cardinality(steps) = 7 Cardinality(cosine) = 4 u∆ uD

25. Minimum Sampling Rate m/k m measurements k cardinality n record length k/n Donoho, Tanner, 2005. Baron, Wakin et al., 2005

26. Signal Separation by Convex Iteration Cardinality(steps) = 7 Cardinality(cosine) = 4 m=28 measurements m/n= 0.1 subsample m/k= 2.5 sample rate k/n = 0.04sparsity

27. u∆ uD 1-norm -241dB reconstruction error Convex iteration TV only DCT only

28. functional Magnetic Resonance Imaging (fMRI) Haemodynamic Response Function (HRF)

29. How to conduct fMRI? Stimulus Timing Huettel, Song, Gregory. Functional Magnetic Resonance Imaging. Sinauer.

30. How to conduct fMRI? Stimulus Timing

31. Neural activity Signalling Vascular response BOLD signal Vascular tone (reactivity) Autoregulation Blood flow, oxygenation and volume Synaptic signalling arteriole B0 field glia Metabolic signalling End bouton dendrite venule What does fMRI measure?

32. rest task Oxygenated Hb Deoxygenated Hb fMRI signal origin Oxygenated Hb Deoxygenated Hb

33. Haemodynamic Response Function (HRF) Canonical HRF Stimulus Timing Predicted Data = = Time

34. Time Actual measurement Prediction Signal Intensity Time Time Which part of the brain is activated? http://www.fmrib.ox.ac.uk/

35. Variability of HRF Stimulus Timing Measurement Actual HRF = HRF calibration

36. Discrete wavelet transform Wh 1 0 (d) 10 0 -10 W: Coiflet E: monotone cone Dh y Deconvolve HRF h D(: , 1) D: convolution matrix

37. W: Coiflet wavelet E: monotone cone D: convolution matrix Dh Deconvolve HRF h smoothness D(:,1) 1 0 stimulus timing y 10 0 -10 measurement

38. Deconvolution results

39. HRF deconvolution … HRF deconvolution … … in vivo deconvolution results HRF calibration Time (s) Time (s)

42. Cardinality Constraint Problem 1 5 1 3 b A 3 x 5 A = b = A(:,2) * rand(1) + A(:,5) * rand(1)

43. = Find x with desired cardinality e.g. k = 2, want

44. From the Range perspective … In general, 1 1 n=5 3 b A x m=3 5 Check every pair for k = 2 Possible # solution:

45. 5 A 3 Particular soln. General soln. Nullspace Perspective 2 = 0 Z 5

46. Z = 0.57 0.34 -0.03 -0.26 -0.16 0.02 -0.68 -0.63 0.22 0.47 xp = 0.34 1.11 -0.30 0 0

47. How to find intersection of lines? Sum of normalized wedges

48. Znew = Z * randn(2); Znew = -0.41-0.18 -0.80-0.30 0.48 0.19 -0.26-0.07 1.00 0.37 Z = -0.59 -0.36 0.06 -0.55 0.15 0.36 0.74 -0.08 -0.27 0.66 x = 0.34 1.11 -0.30 0 0 x = 0 0.68 0 0 0.51

49. Summary • Ways to find 0-norm solutions other than 1-norm (homotopic, convex iteration) • fewer measurements • faster • In cardinality constraint problem, convex iteration and nullspace technique success more often than 1-norm.