Fast X-Ray CT image reconstruction using a linearized augmented lagrangian method with ordered subsets

IEEE Transactions on Medical Imaging

Volumn 34, Issue 2, 2015, Pages 388-399

  Nien, Hung ^a   Fessler, Jeffrey A ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Augmented Lagrangian (AL); computed tomography (CT); ordered subsets; statistical image reconstruction

Indexed keywords

ALUMINUM; CONSTRAINED OPTIMIZATION; CONVEX OPTIMIZATION; COST FUNCTIONS; IMAGE PROCESSING; IMAGE RECONSTRUCTION; LAGRANGE MULTIPLIERS; LEAST SQUARES APPROXIMATIONS; LINEARIZATION; OPTIMIZATION; PROBLEM SOLVING; SET THEORY; TOMOGRAPHY;

AUGMENTED LAGRANGIAN METHODS; AUGMENTED LAGRANGIANS; CONVEX OPTIMIZATION PROBLEMS; DOWNWARD CONTINUATION; LEAST SQUARES PROBLEMS; ORDERED SUBSETS; STATISTICAL IMAGE RECONSTRUCTION; X-RAY COMPUTED TOMOGRAPHY;

COMPUTERIZED TOMOGRAPHY;

ACCELERATION; ALGORITHM; ARTICLE; ASYMPTOTIC CONVERGENCE; COMPUTER ASSISTED TOMOGRAPHY; CONTROLLED STUDY; IMAGE RECONSTRUCTION; LINEAR SYSTEM; LINEARIZED AUGMENTED LAGRANGIAN METHOD; MATHEMATICAL PARAMETERS; NOISE; ORDERED SUBSETS ALGORITHM; OSCILLATION; SPIRAL COMPUTER ASSISTED TOMOGRAPHY; STATISTICAL ANALYSIS; HUMAN; IMAGE PROCESSING; IMAGING PHANTOM; PROCEDURES; RADIOGRAPHY; SHOULDER; STATISTICAL MODEL;

ALGORITHMS; HUMANS; IMAGE PROCESSING, COMPUTER-ASSISTED; MODELS, STATISTICAL; PHANTOMS, IMAGING; SHOULDER; TOMOGRAPHY, X-RAY COMPUTED;

EID: 84922359635     PISSN: 02780062     EISSN: 1558254X     Source Type: Journal    
DOI: 10.1109/TMI.2014.2358499     Document Type: Article

References (45)

1. M. R. Hestenes, "Multiplier and gradient methods," J. Optim. Theory Appl., vol. 4, pp. 303-320, Nov. 1969.
2. R. Glowinski and A. Marrocco, "Sur lapproximation par elements nis dordre un, et la resolution par penalisation-dualite dune classe de problemes de dirichlet nonlineaires, rev. francaise daut," Inf. Rech. Oper., vol. R-2, pp. 41-76, 1975.
3. D. Gabay and B. Mercier, "A dual algorithm for the solution of nonlinear variational problems via finite-element approximations," Comput. Math. Appl., vol. 2, no. 1, pp. 17-40, 1976.
4. J. Eckstein and D. P. Bertsekas, "On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators," Math. Program., vol. 55, pp. 293-318, Apr. 1992.
5. T. Goldstein and S.Osher, "The splitBregmanmethod for L1-regularized problems," SIAM J. Imag. Sci., vol. 2, no. 2, pp. 323-343, 2009.
6. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, "An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems," IEEE Trans. Image Process., vol. 20, no. 3, pp. 681-695, Mar. 2011.
7. S. Ramani and J. A. Fessler, "A splitting-based iterative algorithm for accelerated statistical X-ray CT reconstruction," IEEE Trans. Med. Imag., vol. 31, no. 3, pp. 677-688, Mar. 2012.
8. H. Erdogan and J.A. Fessler, "Ordered subsets algorithms for transmission tomography,"Phys.Med. Biol., vol. 44, pp. 2835-2851,Nov. 1999.
9. D. P. Bertsekas, Incremental gradient, subgradient, and proximal methods for convex optimization: A survey Rep. LIDS 2848, 2010, Report LIDS 2848.
10. H. Robbins and S. Monro, "A stochastic approximation method," Ann. Math. Stat., vol. 22, pp. 400-407, Sep. 1951.
11. D. Kim, S. Ramani, and J. A. Fessler, "Ordered subsets with momentum for accelerated X-ray CT image reconstruction," in Proc. IEEE Conf. Acoust. Speech Signal Process., 2013, pp. 920-923.
12. D. Kim, S. Ramani, and J. A. Fessler, "Accelerating X-ray CT ordered subsets image reconstruction with Nesterov's first-order methods," in Proc. Int. Meet. Fully 3-D Image Recon. Rad. Nucl. Med., 2013, pp. 22-25.
13. Y. Nesterov, "A method for unconstrained convex minimization problem with the rate of convergence ," Dokl. Akad. Nauk. USSR, vol. 269, no. 3, pp. 543-547, 1983.
14. Y. Nesterov, "Smooth minimization of non-smooth functions," Math. Program., vol. 103, pp. 127-152, May 2005.
15. A. Beck and M. Teboulle, "A fast iterative shrinkage-thresholding algorithm for linear inverse problems," SIAM J. Imag. Sci., vol. 2, no. 1, pp. 183-202, 2009.
16. O. Devolder, Stochastic first order methods in smooth convex optimization CORE Discussion Paper 2011/70, 2011.
17. M. Schmidt, N. Le Roux, and F. Bach, "Convergence rates of inexact proximal-gradient methods for convex optimization," in Adv. Neural Inf. Process. Syst., 2011, pp. 1458-1466.
18. S. Ahn and J. A. Fessler, "Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms," IEEE Trans. Med. Imag., vol. 22, no. 5, pp. 613-626, May 2003.
19. S. Ahn, J. A. Fessler, D. Blatt, and A. O. Hero, "Convergent incremental optimization transfer algorithms: Application to tomography," IEEE Trans. Med. Imag., vol. 25, no. 3, pp. 283-296, Mar. 2006.
20. H. Ouyang, N. He, L. Tran, and A. G. Gray, S. Dasgupta and D. Mcallester, Eds., "Stochastic alternating direction method of multipliers," in Proc. Int. Conf. Mach. Learn., 2013, vol. 28, pp. 80-88.
21. Z. Lin, R. Liu, and Z. Su, "Linearized alternating direction method with adaptive penalty for low-rank representation," in Adv. Neural Inf. Process. Syst., 2011, pp. 612-620.
22. E. Esser, X. Zhang, and T. Chan, "A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science," SIAM J. Imag. Sci., vol. 3, no. 4, pp. 1015-1046, 2010.
23. X. Zhang, M. Burger, X. Bresson, and S. Osher, "Bregmanized nonlocal regularization for deconvolution and sparse reconstruction," SIAM J. Imag. Sci., vol. 3, no. 3, pp. 253-276, 2010.
24. X. Zhang, M. Burger, and S. Osher, "A unified primal-dual algorithm framework based on Bregman iteration," J. Sci. Comput., vol. 46, no. 1, pp. 20-46, 2011.
25. D. Kim, D. Pal, J.-B. Thibault, and J. A. Fessler, "Accelerating ordered subsets image reconstruction for X-ray CT using spatially non-uniform optimization transfer," IEEE Trans. Med. Imag., vol. 32, no. 11, pp. 1965-1978, Nov. 2013.
26. A. Chambolle and T. Pock, "A first-order primal-dual algorithm for convex problems with applications to imaging," J. Math. Imag. Vis., vol. 40, no. 1, pp. 120-145, 2011.
27. R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, 1970.
28. I. Daubechies,M. Defrise, and C. D.Mol, "An iterative thresholding algorithm for linear inverse problems with a sparsity constraint," Comm. Pure Appl. Math., vol. 57, pp. 1413-1457, Nov. 2004.
29. D. P. Bertsekas, Nonlinear Programming,2 ed. Nashua, NH: Athena Scientific, 1999.
30. S. Boyd,N. Parikh, E. Chu, B. Peleato, and J. Eckstein, "Distributed optimization and statistical learning via the alternating direction method of multipliers," Found. Trends Mach. Learn., vol. 3, no. 1, pp. 1-122, 2010.
31. Y. Nesterov, "On an approach to the construction of optimal methods of minimization of smooth convex functions," (in Russian) Ekonomika i Mateaticheskie Metody, vol. 24, pp. 509-517, 1988.
32. B. O'Donoghue and E. Candès, "Adaptive restart for accelerated gradient schemes," Found. Comput. Math., vol. 13, Jul. 2013.
33. S. Kontogiorgis and R. R. Meyer, "A variable-penalty alternating directions method for convex optimization," Math. Program., vol. 83, no. 1-3, pp. 29-53, 1998.
34. J. A. Fessler and W. L. Rogers, "Spatial resolution properties of penalized-likelihood image reconstruction methods: Space-invariant tomographs," IEEE Trans. Image Process., vol. 5, no. 9, pp. 1346-1358, Sep. 1996.
35. G. T. Herman and L. B. Meyer, "Algebraic reconstruction techniques can be made computationally efficient," IEEE Trans. Med. Imag., vol. 12, no. 3, pp. 600-609, Sep. 1993.
36. J. Barzilai and J. Borwein, "Two-point step size gradient methods," IMA J. Num. Anal., vol. 8, no. 1, pp. 141-148, 1988.
37. P. J. Huber, Robust Statistics. New York: Wiley, 1981.
38. J.-B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, "A three-dimensional statistical approach to improved image quality for multi-slice helical CT," Med. Phys., vol. 34, pp. 4526-4244, Nov. 2007.
39. W. P. Shuman, D. E. Green, J. M. Busey, O. Kolokythas, L. M. Mitsumori, K. M. Koprowicz, J.-B. Thibault, J. Hsieh, A. M. Alessio, E. Choi, and P. E. Kinahan, "Model based iterative reconstruction versus adaptive statistical iterative reconstruction and filtered back projection in 64-MDCT: Focal lesion detection, lesion conspicuity, and image noise," Am. J. Roentgenol., vol. 200, pp. 1071-1076, May 2013.
40. D. Kim and J. A. Fessler, "Ordered subsets acceleration using relaxed momentum for X-ray CT image reconstruction," in Proc. IEEE Nucl. Sci. Symp. Med. Imag. Conf., 2013, pp. 1-5.
41. N. Le Roux, M. Schmidt, and F. Bach, "A stochastic gradient method with an exponential convergence rate for strongly-convex optimization with finite training sets," in Adv. Neural Inf. Proc. Syst., 2012, pp. 2672-2680.
42. M. Schmidt, N. Le Roux, and F. Bach,Minimizing finite sums with the stochastic average gradient arXiv:1309.2388, 2013.
43. H. Nien and J. A. Fessler, "Fast splitting-based ordered-subsets X-ray CT image reconstruction," in Proc. 3rd Int.Meet. ImageFormat. X-ray CT, 2014, pp. 291-294.
44. M. G. McGaffin and J. A. Fessler, "Fast edge-preserving image denoising via group coordinate descent on the GPU," in Proc. SPIE 9020 Comput. Imag. XII, 2014, p. 90200.
45. M. G. McGaffin and J. A. Fessler, "Edge-preserving image denoising via group coordinate descent on the GPU," IEEE Trans. Image Process., 2014, submitted for publication.