I am not professional about compressed sensing (CS). I know CS can reconstruct the texture of the original image from highly undersampled data if some conditions are met. However, I have no clue how CS algorithms do to the noise. Is it possible that compressed sensing reconstructed image in MRI maintains or improves SNR comparing to fully sampled original image?
A good q. Just a conjecture.Intuitively it seems, that it depends, on the type of noise probability distribution in the random process. If e.g a zero mean Gaussian is assumed ,the noise content should be zero with larger samples viz regular sampling. It seems you can resolve this by considering a simulation on a simple model say vsig = A sin(x) +n(x) with n(x) generated with different probability models using for example Scilab .
Cheers
A noise reduction (or SNR improvement) due to the CS reconstruction is certainly possible - however, typically associated with loss of resolution or image details.
Compressed sensing reconstruction can be implemented using a total variation (TV) penalty in addition to an l1-norm sparsity objective. The TV penalty performs additional (edge-preserving) denoising of the resulting image - look e.g. for "TV" or "denoising" in the article reference below. (The strength/extent of denoising is controlled by the weighting factor (alpha) of the TV penalty.)
(TV denoising can also be applied to other MR images - independent of CS reconstruction; this will also improve the SNR, but potentially remove small image details.)
Article Sparse MRI: The application of compressed sensing for rapid MR imaging
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