IntroductionReconstructing a bounded object from incomplete k-space data is a well posed problem, and it was recently shown that this incomplete spectrum approach can be used to reconstruct undersampled MRI images with similar quality to compressed sensing approaches. Here, we apply this incomplete spectrum approach to the field-to-source inverse problem encountered in quantitative magnetic susceptibility mapping (QSM). The field-to-source problem is an ill-posed problem because of conical regions in frequency space where the dipole kernel is zero or very small, which leads to the kernel's inverse being ill-defined. These “ill-posed” regions typically lead to streaking artifacts in QSM reconstructions. In contrast to compressed sensing, our approach relies on knowledge of the image-space support, more commonly referred to as the mask, of our object as well as the region in k-space with ill-defined values. In the QSM case, this mask is usually available, as it is required for most QSM background field removal and reconstruction methods.MethodsWe tuned the incomplete spectrum method (mask and band-limit) for QSM on a simulated dataset from the most recent QSM challenge and validated the QSM reconstruction results on brain images acquired in five healthy volunteers, comparing incomplete spectrum QSM to current state-of-the art-methods: FANSI, nonlinear dipole inversion, and conventional thresholded k-space division.ResultsWithout additional regularization, incomplete spectrum QSM performs slightly better than direct QSM reconstruction methods such as thresholded k-space division (PSNR of 39.9 vs. 39.4 of TKD on a simulated dataset) and provides susceptibility values in key iron-rich regions similar or slightly lower than state-of-the-art algorithms, but did not improve the PSNR in comparison to FANSI or nonlinear dipole inversion. With added (ℓ1-wavelet based) regularization the new approach produces results similar to compressed sensing based reconstructions (at sufficiently high levels of regularization).DiscussionIncomplete spectrum QSM provides a new approach to handle the “ill-posed” regions in the frequency-space data input to QSM.
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