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Highly Undersampled 0-norm Reconstruction

Highly Undersampled 0-norm Reconstruction. Christine Law. If signal is sparse (lots of zeros), then yes (2004 Donoho, Candes). How to sample? How to recover?. 1 Candes et al. IEEE Trans. Information Theory 2006 52(2):489 2 Donoho. IEEE Trans. Information Theory 2006 52(4):1289.

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Highly Undersampled 0-norm Reconstruction

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  1. Highly Undersampled 0-norm Reconstruction Christine Law

  2. If signal is sparse (lots of zeros), then yes (2004 Donoho, Candes). How to sample? How to recover? 1 Candes et al. IEEE Trans. Information Theory 2006 52(2):489 2 Donoho. IEEE Trans. Information Theory 2006 52(4):1289 Reconstruction by Optimization Shannon sampling theory: sample at Nyquist rate. Can we take less samples? Much less than Shannon said?

  3. K < M << N • General rule: M > 4K samples

  4. Use linear programming to find signal u with least nonzero entries in Yu that agrees with M observed measurements in y .

  5. Dear 0-norm god: Please find me a vector that has the least nonzero entries s.t. this equation is true.

  6. Donoho, Candes: 1-norm solution = 0-norm solution

  7. 96 out of 512 samples SNR=37 dB

  8. For p-norm, where 0 < p < 1 Chartrand (2006) proved fewer samples of y than 1-norm formulation. 3 Chartrand. IEEE Signal Processing Letters. 2007: 14(10) 707-710. Bypass Lin Prog & Comp Sens • Solve 0-norm directly.

  9. Trzasko (2007): Rewrite the problem 4 where r is tanh, laplace, log etc. such that 4 Trzasko et al. IEEE SP 14th workshop on statistical signal processing. 2007. 176-180.

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

  11. 0-norm method

  12. Demonstration • when is big (1st iteration), solving 1-norm problem. • reduce to approach 0-norm solution. • Piecewise constant image, but not sparse. • Gradient is sparse.

  13. Example 1

  14. 1-norm recon 1-norm result: use 4% k-space data SNR: -11.4 dB 542 seconds 0-norm result: use 4% k-space data SNR: -66.2 dB 82 seconds 0-norm recon Zero-filled Result k-space samples used

  15. Example 2 • TOF image • 360x360, 27.5% radial samples

  16. 0-norm method: 26.5 dB, 101 seconds 360x360 27.5% radial samples 1-norm method: 24.7 dB, 1151 seconds

  17. Summary & open problems • 0-norm minimization is fast and gives comparable results as 1-norm method. • Need better sparsifying transform. • Need 30 dB, want 50 dB

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