Automatic determination of the regularization weighting for wavelet-based compressed sensing MRI reconstructions

link to paper

Automatic determination of the regularization weighting for wavelet-based compressed sensing MRI reconstructions

Gabriel Varela-Mattatall, Corey A. Baron, Ravi S. Menon



To present a method that automatically, rapidly, and in a noniterative manner determines the regularization weighting for wavelet-based compressed sensing reconstructions. This method determines level-specific regularization weighting factors from the wavelet transform of the image obtained from zero-filling in k-space.


We compare reconstruction results obtained by our method, urn:x-wiley:07403194:media:mrm28812:mrm28812-math-0001, to the ones obtained by the L-curve, urn:x-wiley:07403194:media:mrm28812:mrm28812-math-0002, and the minimum NMSE, urn:x-wiley:07403194:media:mrm28812:mrm28812-math-0003. The comparisons are done using in vivo data; then, simulations are used to analyze the impact of undersampling and noise. We use NMSE, Pearson’s correlation coefficient, high-frequency error norm, and structural similarity as reconstruction quality indices.


Our method, urn:x-wiley:07403194:media:mrm28812:mrm28812-math-0004, provides improved reconstructed image quality to that obtained by urn:x-wiley:07403194:media:mrm28812:mrm28812-math-0005 regardless of undersampling or SNR and comparable quality to urn:x-wiley:07403194:media:mrm28812:mrm28812-math-0006 at high SNR. The method determines the regularization weighting prospectively with negligible computational time.


Our main finding is an automatic, fast, noniterative, and robust procedure to determine the regularization weighting. The impact of this method is to enable prospective and tuning-free wavelet-based compressed sensing reconstructions.

Is there any way to see the full paper? The provided link seems to be broken.

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Hi @Miha - thanks for pointing this out! I’ve fixed the link, you should be able to access it now.

Indeed, I can access the paper now. Very interesting and also very relevant work.
On a detailed level (and reading sequentially), I notice the statement “The proximal operator for the L1 regularizer is [eq. (4)] where δ_λ {*} corresponds to the soft-thresholding function.”
I am somehow unsure whether this is justified in the general case. Doesn’t this depend on the expected distribution of Ψ(x) ?

In a compressed sensing reconstruction there are three practical components, not related to the consistency term of equation [2], for successful recovery of “highly” undersampled data: the regularizer, the sparsifying transform and the regularization weighting. Regarding the regularizer, we know from proximal optimization theory that the proximal operator of the L1-norm corresponds to the soft-thresholding function[1]. Maybe we are not fully understanding your question in this regard.

The regularizer is the parameter that promotes sparsity and consequently it changes the distribution Ψ(x) through the regularization weighting λ. The sparsifying transform is usually the wavelet transform as it naturally highly compresses an image. The distribution of Ψ(x) using wavelets tends to have a sharp peak at the origin (zero-coefficient), and the rest of the distribution rapidly decays as a function of the non-zero coefficients, as observed in our Fig. 1. The literature supports the fact that the levels of the wavelet transform have slightly different distributions. Higher levels lose their utility to compress the image. Finally, we used a level-specific regularization weighting per decomposition level of the wavelet transform to consider those slightly different distributions and improve image reconstruction.

We hope this clarifies what was done. And why.

  1. Parikh, N. & Boyd, S. Proximal Algorithms. Found Trends Optim 1, 127–239 (2014). "
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Thank you for your clarification. And sorry for the delay - I somehow did not receive a notification on your reply.
With your reference to proximal optimization theory, I reviewed it again to realize that my doubts (dated August 10) were unjustified.

Too hasty reading of section 3.1 leads me to the idea that a knee dataset was taken from which a brain was extracted … Oops … something did not fully fit. With somewhat more careful reading, I get the idea that (A) a 2D dataset of a knee was taken from a repository and (B) a 3D head-scan was made.
Now the following question pops up: were these the only two sets to test the methodology on? If so, is that sufficient proof of the methodology? After all, I would expect from the title (or abstract) that “automatic” refers to adaptability of the regularization factors (or wavelet thresholds) to a wide variety of circumstances, considering SNR, resolution or image content. While brain and knee images might still have somewhat similar statistics, would the method apply to imaging tasks like cardiac, diffusion or high-resolution angio? Or is the notion of automaticity particularly geared towards the ratio of the lambdas accross various wavelet-levels?