Abstract
We consider the MRI physics in a low-field MRI scanner, in which permanent magnets are used to generate a magnetic field in the millitesla range. A model describing the relationship between measured signal and image is derived, resulting in an ill-posed inverse problem. In order to solve it, a regularization penalty is added to the least-squares minimization problem. We generalize the conjugate gradient minimal error (CGME) algorithm to the weighted and regularized least-squares problem. Analysis of the convergence of generalized CGME (GCGME) and the classical generalized conjugate gradient least squares (GCGLS) shows that GCGME can be expected to converge faster for ill-conditioned regularization matrices. The {ell}_{p}-regularized problem is solved using iterative reweighted least squares for p=1 and p=frac{1}{2}, with both cases leading to an increasingly ill-conditioned regularization matrix. Numerical results show that GCGME needs a significantly lower number of iterations to converge than GCGLS.
Highlights
In low-field magnetic resonance imaging (MRI), magnetic field strengths in the millitesla range are used to visualize the internal structure of the human body
This research is part of a project that aims toward creating an inexpensive low-field MRI scanner using a Halbach cylinder that can be used for medical purposes
We focus on a low-field MRI setting, this algorithm is generally applicable to p-regularized least-squares problems
Summary
In low-field magnetic resonance imaging (MRI), magnetic field strengths in the millitesla (mT) range are used to visualize the internal structure of the human body. The resulting reconstruction problem is very ill-posed This is due to the nonlinearity of the magnetic field inside the Halbach cylinder that we consider. In the center of the cylinder, there is very little variation in the field, limiting the spatial resolution in that area. Another complication we face is low signal-to-noise ratios. In a similar project, Cooley et al [6] have shown that it is possible to reconstruct magnetic resonance images given signals obtained with a device based on a Halbach cylinder, using a simplified signal model in which similar assumptions are made as in high-field MRI. We revisit the underlying physics and formulate the general signal model for MRI without making these assumptions
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