1 / 20

Highly Undersampled 0-norm Reconstruction

Explore the highly undersampled realm of signal reconstruction using the 0-norm approach. Learn from Christine Law's groundbreaking work in recovering sparse signals efficiently. Discover the methods to sample and recover signals while challenging Shannon's Nyquist rate. Dive deep into linear programming to minimize nonzero entries, finding optimal solutions with fewer samples. Compare the efficiency of 0-norm and 1-norm methods, showcasing significant gains in reconstruction speed and accuracy. Explore innovative strategies to enhance sparse signal recovery and achieve higher signal-to-noise ratios.

atully
Download Presentation

Highly Undersampled 0-norm Reconstruction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related