ABSTRACT A variety of alternating direction methods have been proposed for solving a class of optimization problems. The applications in computed tomography (CT) perform well in image reconstruction. The reweighted schemes were applied in l1-norm and total variation minimization for signal and image recovery to improve the convergence of algorithms. In this paper, we present a reweighted total variation algorithm using the alternating direction method (ADM) for image reconstruction in CT. The numerical experiments for ADM demonstrate that adding reweighted strategy reduces the computation time effectively and improves the quality of reconstructed images as well.
Cite this paper
Li, X. and Zhu, J. (2019) A Reweighted Total Variation Algorithm with the Alternating Direction Method for Computed Tomography. Advances in Computed Tomography, 8, 1-9. doi: 10.4236/act.2019.81001.
 Chen, S., Donoho, D. and Saunders M. (1998) Atomic Decomposition by Basis Pursuit. SIAM Journal on Scienti_c Computing, 20, 33-61. https://doi.org/10.1137/S1064827596304010
 Donoho, D. and Logan, B.F. (1992) Signal Recovery and the Large Sieve. SIAM Journal on Applied Mathematics, 52, 577-591. https://doi.org/10.1137/0152031
 Donoho, D. and Stark, P.B. (1989) Uncertainty Principles and Signal Recovery. SIAM Journal on Applied Mathematics, 49, 906-931. https://doi.org/10.1137/0149053
 Santosa, F. and Symes, W. (1986) Linear Inversion of Band-Limited Reection Seismograms. SIAM Journal on Scienti_c and Statistical Computing, 7, 1307-1330. https://doi.org/10.1137/0907087
 Tibshirani, R. (1996) Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society: Series B, 58, 267-288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x
 Zhang, H., Yin, W. and Cheng, L. (2015) Necessary and Sufficient Conditions of Solution Uniqueness in l1-Norm Minimization. Journal of Optimization Theory and Applications, 154, 09-122. https://doi.org/10.1007/s10957-014-0581-z
 Candes, E., Wakins, W. and Boyd, S. (2008) Enhancing Sparsity by Reweighted l1-Minimization. Journal of Fourier Analysis Application, 14, 877-905. https://doi.org/10.1007/s00041-008-9045-x
 Foucart, S. and Lai, M. (2009) Sparsest Solutions of Underdetermined Linear Systems via lq-Minimization for 0 < q _ 1. Applied and Computational Harmonical Analysis, 26, 395-407. https://doi.org/10.1016/j.acha.2008.09.001
 Lai, M.J. (2010) On Sparse Solutions of Underdetermined Linear Systems. Journal of Concrete & Applicable Mathematics, 8, 296-327.
 Zhu, J. and Li, X. (2013) A Generalized l1 Greedy Algorithm for Image Reconstruction in CT. Applied Mathematics and Computation, 219, 5487-5494. https://doi.org/10.1016/j.amc.2012.11.052
 Kozlov, I. and Petukhov, A. (2010) Sparse Solutions for Underdetermined Linear System. In: Freeden, W., Nashed, M.Z. and Sonar, T., Eds., Handbook of Geomathematics, Vol. 1, Springer, New York, 1243-1260. https://doi.org/10.1007/978-3-642-01546-5 42
 Zhu, J., Li, X., Arroyo, F. and Arroyo, E. (2015) Error Analysis of Reweighted l1 Greedy Algorithm for Noisy Reconstruction. Journal of Computational and Applied Mathematics, 286, 93-101. https://doi.org/10.1016/j.cam.2015.02.038
 Kim, H., Chen, J., Wang, A., Chuang, C., Held, M. and Poullot, J. (2016) Non-Local Total-Variation (NLTV) Minimization Combined with Reweighted l1-Norm for Compressed Sensing CT Reconstruction. Physics in Medicine & Biology, 61, 6878-6891. https://doi.org/10.1088/0031-9155/61/18/6878
 Tao, M. and Yang, F. (2009) Alternating Direction Algorithms for Total Variation Deconvolution in Image Reconstruction. TR0918, Department of Mathematics, Nanjing University, Optimization.
 Yang, J. and Zhang, Y. (2011) Alternating Direction Algorithms for l1-Problems in Compressive Sensing. SIAM Journal on Scientific Computing, 33, 250-278. https://doi.org/10.1137/090777761
 Goldstein, T., O'Donoghue, B., Setzer, S. and Baraniuk, R. (2014) Fast Alternating Direction Optimization Methods. SIAM Journal on Imaging Sciences, 7, 1588-1623. https://doi.org/10.1137/120896219
 Lu, C., Feng, J., Yan, S. and Liu, Z. (2018) A Uni_ed Alternating Direction Method of Multipliers by Majorization Minimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 40, 527-541. https://doi.org/10.1109/TPAMI.2017.2689021
 Li, C., Yin, W., Jiang, H. and Zhang, T. (2013) An E_cient Augmented Lagrangian Method with Applications to Total Variation inimization. Computational Optimization and Applications, 56, 07-530. ttps://doi.org/10.1007/s10589-013-9576-1
 Li, C., Yin, W. and Zhang, Y. (2010) Users Guide for TVAL3: V Minimization by Augmented Lagrangian and Alternating Direction Algorithms. CAAM Report.
 Kak, A.C. and Slaney, M. (2001) Principles of Computerized Tomographic Imaging. Society of Industrial and Applied Mathematics, Philadelphia. ttps://doi.org/10.1137/1.9780898719277