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Primal‐dual and forward gradient implementation for quantitative susceptibility mapping
Author(s) -
Kee Youngwook,
Deh Kofi,
Dimov Alexey,
Spincemaille Pascal,
Wang Yi
Publication year - 2017
Publication title -
magnetic resonance in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.696
H-Index - 225
eISSN - 1522-2594
pISSN - 0740-3194
DOI - 10.1002/mrm.26627
Subject(s) - imaging phantom , algorithm , mathematics , rate of convergence , biconjugate gradient stabilized method , convergence (economics) , conjugate gradient method , computer science , nuclear medicine , medicine , key (lock) , computer security , economic growth , economics
Purpose To investigate the computational aspects of the prior term in quantitative susceptibility mapping (QSM) by (i) comparing the Gauss‐Newton conjugate gradient (GNCG) algorithm that uses numerical conditioning (ie, modifies the prior term) with a primal‐dual (PD) formulation that avoids this, and (ii) carrying out a comparison between a central and forward difference scheme for the discretization of the prior term. Theory and Methods A spatially continuous formulation of the regularized QSM inversion problem and its PD formulation were derived. The Chambolle‐Pock algorithm for PD was implemented and its convergence behavior was compared with that of GNCG for the original QSM. Forward and central difference schemes were compared in terms of the presence of checkerboard artifacts. All methods were tested and validated on a gadolinium phantom, ex vivo brain blocks, and in vivo brain MRI data with respect to COSMOS. Results The PD approach provided a faster convergence rate than GNCG. The GNCG convergence rate slowed considerably with smaller (more accurate) values of the conditioning parameter. Using a forward difference suppressed the checkerboard artifacts in QSM, as compared with the central difference. The accuracy of PD and GNCG were validated based on excellent correlation with COSMOS. Conclusions The PD approach with forward difference for the gradient showed improved convergence and accuracy over the GNCG method using central difference. Magn Reson Med 78:2416–2427, 2017. © 2017 International Society for Magnetic Resonance in Medicine.