Fast Proximal Gradient Methods for Nonsmooth Convex Optimization for Tomographic Image Reconstruction

The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a O 1 / k 2 non-asymptotic...

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Published in:	Sensing and imaging Vol. 21; no. 1
Main Authors:	Helou, Elias S., Zibetti, Marcelo V. W., Herman, Gabor T.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.12.2020 Springer Nature B.V
Subjects:	Algorithms Computational geometry Computed tomography Convergence Convex analysis Convexity Electrical Engineering Engineering Experimentation Image reconstruction Imaging Microwaves Optimization Original Paper Radiology RF and Optical Engineering Computerized tomography imaging Iterative algorithms Convex optimization Proximal gradient methods
ISSN:	1557-2064, 1557-2072
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Abstract	The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a O 1 / k 2 non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction.
AbstractList	The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a O1/k2 non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction. The Fast Proximal Gradient Method (FPGM) and the Monotone FPGM (MFPGM) for minimization of nonsmooth convex functions are introduced and applied to tomographic image reconstruction. Convergence properties of the sequence of objective function values are derived, including a O 1 / k 2 non-asymptotic bound. The presented theory broadens current knowledge and explains the convergence behavior of certain methods that are known to present good practical performance. Numerical experimentation involving computerized tomography image reconstruction shows the methods to be competitive in practical scenarios. Experimental comparison with Algebraic Reconstruction Techniques are performed uncovering certain behaviors of accelerated Proximal Gradient algorithms that apparently have not yet been noticed when these are applied to tomographic image reconstruction.
ArticleNumber	45
Author	Helou, Elias S. Herman, Gabor T. Zibetti, Marcelo V. W.
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References	Taylor, Hendrickx, cois Glineur (CR23) 2017; 27 CR18 CR17 Beck, Teboulle (CR3) 2009; 18 Garduño, Herman (CR9) 2017; 33 CR12 Zibetti, Lin, Herman (CR27) 2018; 34 CR10 Yamagishi, Yamada (CR25) 2011; 27 Herman, Meyer (CR14) 1993; 12 Erdoǧan, Fessler (CR8) 1999; 44 Nesterov (CR21) 2004 Herman (CR13) 2009 Zhang, Hager (CR26) 2004; 14 Bioucas-Dias, Figueiredo (CR4) 2007; 16 Parikh, Boyd (CR22) 2014; 1 CR5 Zibetti, Helou, Regatte, Herman (CR28) 2019; 5 Drori, Teboulle (CR6) 2014; 145 Hiriart-Urruty, Lemaréchal (CR16) 1993 Engl, Hanke, Neubauer (CR7) 2000 Grippo, Lampariello, Lucidi (CR11) 1986; 23 Nesterov (CR20) 1983; 27 Herman, Garduño, Davidi, Censor (CR15) 2012; 39 Beck, Teboulle (CR2) 2009; 2 Aharon, Elad, Bruckstein (CR1) 2006; 54 Narayan, Herman (CR19) 1999; 16 Wright, Nowak, Figueiredo (CR24) 2009; 57 H Erdoǧan (309_CR8) 1999; 44 YE Nesterov (309_CR20) 1983; 27 A Beck (309_CR3) 2009; 18 H Zhang (309_CR26) 2004; 14 JB Hiriart-Urruty (309_CR16) 1993 GT Herman (309_CR15) 2012; 39 M Aharon (309_CR1) 2006; 54 309_CR10 309_CR12 AB Taylor (309_CR23) 2017; 27 GT Herman (309_CR14) 1993; 12 309_CR18 309_CR17 E Garduño (309_CR9) 2017; 33 YE Nesterov (309_CR21) 2004 A Beck (309_CR2) 2009; 2 MVW Zibetti (309_CR27) 2018; 34 L Grippo (309_CR11) 1986; 23 M Yamagishi (309_CR25) 2011; 27 JM Bioucas-Dias (309_CR4) 2007; 16 MVW Zibetti (309_CR28) 2019; 5 SJ Wright (309_CR24) 2009; 57 N Parikh (309_CR22) 2014; 1 309_CR5 Y Drori (309_CR6) 2014; 145 GT Herman (309_CR13) 2009 TK Narayan (309_CR19) 1999; 16 HW Engl (309_CR7) 2000
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SubjectTerms	Algorithms Computational geometry Computed tomography Convergence Convex analysis Convexity Electrical Engineering Engineering Experimentation Image reconstruction Imaging Microwaves Optimization Original Paper Radiology RF and Optical Engineering
Title	Fast Proximal Gradient Methods for Nonsmooth Convex Optimization for Tomographic Image Reconstruction
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