Highly Undersampled 0norm Reconstruction

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Highly Undersampled 0-norm Reconstruction

• Christine Law

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Reconstruction by Optimization
Shannon sampling theory sample at Nyquist
rate. Can we take less samples? Much less than
Shannon said?
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K lt M ltlt N

• General rule M gt 4K samples

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

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Donoho, Candes 1-norm solution 0-norm solution
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96 out of 512 samples SNR37 dB
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Bypass Lin Prog Comp Sens

• Solve 0-norm directly.

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

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Example 1
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0-norm result use 4 k-space data SNR -66.2
dB 82 seconds
0-norm recon
Zero-filled Result
k-space samples used
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Example 2

• TOF image
• 360x360, 27.5 radial samples

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