A Reweighted Total Variation Algorithm with the Alternating Direction Method for Computed Tomography
Affiliation(s)
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, USA.
Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, USA.
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.
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.
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.
References
[1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] Lai, M.J. (2010) On Sparse Solutions of Underdetermined Linear Systems. Journal of Concrete & Applicable Mathematics, 8, 296-327. [10] 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 [11] 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 [12] 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 [13] 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 [14] Tao, M. and Yang, F. (2009) Alternating Direction Algorithms for Total Variation Deconvolution in Image Reconstruction. TR0918, Department of Mathematics, Nanjing University, Optimization. [15] 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 [16] 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 [17] 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 [18] 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 [19] Li, C., Yin, W. and Zhang, Y. (2010) Users Guide for TVAL3: V Minimization by Augmented Lagrangian and Alternating Direction Algorithms. CAAM Report. [20] 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
[1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] Lai, M.J. (2010) On Sparse Solutions of Underdetermined Linear Systems. Journal of Concrete & Applicable Mathematics, 8, 296-327. [10] 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 [11] 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 [12] 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 [13] 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 [14] Tao, M. and Yang, F. (2009) Alternating Direction Algorithms for Total Variation Deconvolution in Image Reconstruction. TR0918, Department of Mathematics, Nanjing University, Optimization. [15] 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 [16] 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 [17] 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 [18] 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 [19] Li, C., Yin, W. and Zhang, Y. (2010) Users Guide for TVAL3: V Minimization by Augmented Lagrangian and Alternating Direction Algorithms. CAAM Report. [20] 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