Highly Undersampled 0-norm Reconstruction Christine Law

Reconstruction by Optimization Shannon sampling theory: sample at Nyquist rate. Can we take less samples? Much less than Shannon said? 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

General rule: M > 4K samples Different from sparsifying transform matrix K < M << N General rule: M > 4K samples

Use linear programming to find signal u with least nonzero entries in Yu that agrees with M observed measurements in y . Want little correlation between rows and columns of phi and psi

Psi here is a DCT matrix but it can be anything Psi here is a DCT matrix but it can be anything. Doesn’t have to be orthogronal ***Questions *** Phi can be random matrix, Fourier matrix as in MRI *** in MRI, y is k-space data and u is image

Dear 0-norm god: Please find me a vector that has the least nonzero entries s.t. this equation is true. Psi not required to be orthnormal or invertible. ***Dantzig

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

96 out of 512 samples SNR=37 dB

Bypass Lin Prog & Comp Sens Solve 0-norm directly. 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. L1: solve a large linear program. Want to know if there is a way to solve L0 problem directly and bypass compressed sensing theory and bypass linear program. Solve faster.

Trzasko (2007): Rewrite the problem 4 where r is tanh, laplace, log etc. such that Y is incomplete k-space data in MRI *** Phi is incomplete Fourier matrix *** approach 0-norm function *** sequence of sigma *** not continuous limit but discretized sigma 4 Trzasko et al. IEEE SP 14th workshop on statistical signal processing. 2007. 176-180.

1D Example of Start as 1-norm problem, then reduce s slowly 1D vector *** rho is any function that goes from 1-norm to 0-norm as the parameter goes from infinity to 0 Start as 1-norm problem, then reduce s slowly and approach 0-norm function.

0-norm method Finding zero in gradient

Demonstration Piecewise constant image, but not sparse. when is big (1st iteration), solving 1-norm problem. reduce to approach 0-norm solution. Demo: 10 times faster than other known technique including L1 magic. Picture refreshs when inverse matrix (to Ax=b) in CG is updated. Inverse is solved in multiple passes in CG. Until a fixed point solution is found. At that point, the inverse is exact. Piecewise constant image, but not sparse. Gradient is sparse.

Example 1

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

Example 2 TOF image 360x360, 27.5% radial samples

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

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