A primal–dual hybrid gradient method for nonlinear operators with applications to MRI

Handle URI:
http://hdl.handle.net/10754/597381
Title:
A primal–dual hybrid gradient method for nonlinear operators with applications to MRI
Authors:
Valkonen, Tuomo ( 0000-0001-6683-3572 )
Abstract:
We study the solution of minimax problems min xmax yG(x) + K(x), y - F*(y) in finite-dimensional Hilbert spaces. The functionals G and F* we assume to be convex, but the operator K we allow to be nonlinear. We formulate a natural extension of the modified primal-dual hybrid gradient method, originally for linear K, due to Chambolle and Pock. We prove the local convergence of the method, provided various technical conditions are satisfied. These include in particular the Aubin property of the inverse of a monotone operator at the solution. Of particular interest to us is the case arising from Tikhonov type regularization of inverse problems with nonlinear forward operators. Mainly we are interested in total variation and second-order total generalized variation priors. For such problems, we show that our general local convergence result holds when the noise level of the data f is low, and the regularization parameter α is correspondingly small. We verify the numerical performance of the method by applying it to problems from magnetic resonance imaging (MRI) in chemical engineering and medicine. The specific applications are in diffusion tensor imaging and MR velocity imaging. These numerical studies show very promising performance. © 2014 IOP Publishing Ltd.
Citation:
Valkonen T (2014) A primal–dual hybrid gradient method for nonlinear operators with applications to MRI. Inverse Problems 30: 055012. Available: http://dx.doi.org/10.1088/0266-5611/30/5/055012.
Publisher:
IOP Publishing
Journal:
Inverse Problems
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
1-May-2014
DOI:
10.1088/0266-5611/30/5/055012
Type:
Article
ISSN:
0266-5611; 1361-6420
Sponsors:
This work has been financially supported by the King Abdullah University of Science and Technology (KAUST) Award no. KUK-I1-007-43 as well as the EPSRC / Isaac Newton Trust Small Grant 'Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors' and the EPSRC first grant no. EP/J009539/1 'Sparse & Higher-order Image Restoration'. The author is grateful to Florian Knoll for providing the in vivo DTI data set.
Appears in Collections:
Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorValkonen, Tuomoen
dc.date.accessioned2016-02-25T12:32:02Zen
dc.date.available2016-02-25T12:32:02Zen
dc.date.issued2014-05-01en
dc.identifier.citationValkonen T (2014) A primal–dual hybrid gradient method for nonlinear operators with applications to MRI. Inverse Problems 30: 055012. Available: http://dx.doi.org/10.1088/0266-5611/30/5/055012.en
dc.identifier.issn0266-5611en
dc.identifier.issn1361-6420en
dc.identifier.doi10.1088/0266-5611/30/5/055012en
dc.identifier.urihttp://hdl.handle.net/10754/597381en
dc.description.abstractWe study the solution of minimax problems min xmax yG(x) + K(x), y - F*(y) in finite-dimensional Hilbert spaces. The functionals G and F* we assume to be convex, but the operator K we allow to be nonlinear. We formulate a natural extension of the modified primal-dual hybrid gradient method, originally for linear K, due to Chambolle and Pock. We prove the local convergence of the method, provided various technical conditions are satisfied. These include in particular the Aubin property of the inverse of a monotone operator at the solution. Of particular interest to us is the case arising from Tikhonov type regularization of inverse problems with nonlinear forward operators. Mainly we are interested in total variation and second-order total generalized variation priors. For such problems, we show that our general local convergence result holds when the noise level of the data f is low, and the regularization parameter α is correspondingly small. We verify the numerical performance of the method by applying it to problems from magnetic resonance imaging (MRI) in chemical engineering and medicine. The specific applications are in diffusion tensor imaging and MR velocity imaging. These numerical studies show very promising performance. © 2014 IOP Publishing Ltd.en
dc.description.sponsorshipThis work has been financially supported by the King Abdullah University of Science and Technology (KAUST) Award no. KUK-I1-007-43 as well as the EPSRC / Isaac Newton Trust Small Grant 'Non-smooth geometric reconstruction for high resolution MRI imaging of fluid transport in bed reactors' and the EPSRC first grant no. EP/J009539/1 'Sparse & Higher-order Image Restoration'. The author is grateful to Florian Knoll for providing the in vivo DTI data set.en
dc.publisherIOP Publishingen
dc.subjectconvergenceen
dc.subjectMRIen
dc.subjectnon-convexen
dc.subjectnonlinearen
dc.subjectprimal-dualen
dc.titleA primal–dual hybrid gradient method for nonlinear operators with applications to MRIen
dc.typeArticleen
dc.identifier.journalInverse Problemsen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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