Journal of Applied Mathematics

A Smoothed l0-Norm and l1-Norm Regularization Algorithm for Computed Tomography

Jiehua Zhu and Xiezhang Li

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The nonmonotone alternating direction algorithm (NADA) was recently proposed for effectively solving a class of equality-constrained nonsmooth optimization problems and applied to the total variation minimization in image reconstruction, but the reconstructed images suffer from the artifacts. Though by the l0-norm regularization the edge can be effectively retained, the problem is NP hard. The smoothed l0-norm approximates the l0-norm as a limit of smooth convex functions and provides a smooth measure of sparsity in applications. The smoothed l0-norm regularization has been an attractive research topic in sparse image and signal recovery. In this paper, we present a combined smoothed l0-norm and l1-norm regularization algorithm using the NADA for image reconstruction in computed tomography. We resolve the computation challenge resulting from the smoothed l0-norm minimization. The numerical experiments demonstrate that the proposed algorithm improves the quality of the reconstructed images with the same cost of CPU time and reduces the computation time significantly while maintaining the same image quality compared with the l1-norm regularization in absence of the smoothed l0-norm.

Article information

J. Appl. Math., Volume 2019 (2019), Article ID 8398035, 8 pages.

Received: 5 November 2018
Revised: 5 April 2019
Accepted: 13 May 2019
First available in Project Euclid: 24 July 2019

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Zhu, Jiehua; Li, Xiezhang. A Smoothed ${l}_{0}$ -Norm and ${l}_{1}$ -Norm Regularization Algorithm for Computed Tomography. J. Appl. Math. 2019 (2019), Article ID 8398035, 8 pages. doi:10.1155/2019/8398035.

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  • E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principle: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489–509, 2006.
  • E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21–30, 2008.
  • D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006.
  • C. E. Shannon, “Communication in the presence of noise,” Proceedings of the IEEE, vol. 86, no. 2, pp. 447–457, 1998.
  • E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4203–4215, 2005.
  • B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM Journal on Computing, vol. 24, no. 2, pp. 227–234, 1995.
  • Y. Sun and J. Tao, “Image reconstruction from few views by $\ell _{0}$-norm optimization,” Chinese Physics B, vol. 23, no. 7, Article ID 078703, 2014.
  • H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “A fast approach for overcomplete sparse decomposition based on smoothed $\ell _{0}$ norm,” IEEE Transactions on Signal Processing, vol. 57, no. 1, pp. 289–301, 2009.
  • M. Rostami and Z. Wang, “Image super-resolution based on sparsity prior via smooth $\ell _{0}$ norm,” in Proceedings of the Symposium on Advanced Intelligent Systems, Waterloo, Canada, December 2011.
  • M. Li, C. Zhang, C. Peng et al., “Smoothed $\ell _{0}$ norm regularization for sparse-view X-ray CT reconstruction,” BioMed Research International, vol. 2016, Article ID 2180457, 12 pages, 2016.
  • L. Wang, X. Yin, H. Yue, and J. Xing, “A regularized weighted smoothed l0-norm minimization method for underdetermined blind source separation,” Sensors, vol. 18, no. 12, 2018.
  • X. Yin, L. Wang, H. Yue, and J. Xiang, “A new non-convex regularized sparse reconstruction algorithm for compressed sensing magnetic resonance image recovery,” Progress in Electromagnetics Research C, vol. 87, pp. 241–253, 2018.
  • W. Yu, C. Wang, and M. Huang, “Edge-preserving reconstruction from sparse projections of limited-angle computed tomography using $\ell _{0}$-regularized gradient prior,” Review of Scientific Instruments, vol. 88, no. 4, Article ID 043703, 2017.
  • X. Feng, C. Wu, and C. Zeng, “On the local and global minimizers of $\ell _{0}$ gradient regularized model with box constraints for image restoration,” Inverse Problems, vol. 34, no. 9, Article ID 095007, 2018.
  • L. Zhang, L. Zeng, and Y. Guo, “l0 regularization based on a prior image incorporated non-local means for limited-angle X-ray CT reconstruction,” Journal of X-Ray Science and Technology, vol. 26, no. 3, pp. 481–498, 2018.
  • C. Wang, L. Zeng, W. Yu, and L. Xu, “Existence and convergence analysis of l0 and $\ell _{2}$ regurizations for limited-angle CT reconstruction,” Inverse Problems and Imaging, vol. 12, no. 3, pp. 545–572, 2018.
  • E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted l$_{1}$ minimization,” Journal of Fourier Analysis and Applications, vol. 14, no. 5-6, pp. 877–905, 2008.
  • H. Zhang, W. Yin, and L. Cheng, “Necessary and sufficient conditions of solution uniqueness in $\ell _{1}$-norm minimization,” Journal of Optimization Theory and Applications, vol. 164, no. 1, pp. 109–122, 2015.
  • M. Tao and F. Yang, “Alternating direction algorithms for total variation deconvolution in image reconstruction,” Optimization Online TR0918, Department of Mathematics, Nanjing University, 2009.
  • J. Yang and Y. Zhang, “Alternating direction algorithms for $\ell _{1}$-problems in compressive sensing,” SIAM Journal on Scientific Computing, vol. 33, no. 1, pp. 250–278, 2011.
  • C. Li, W. Yin, H. Jiang, and Y. Zhang, “An efficient augmented Lagrangian method with applications to total variation minimization,” Computational Optimization and Applications, vol. 56, no. 3, pp. 507–530, 2013.
  • C. Li, W. Yin, and Y. Zhang, “User's guide for TVAL3: TV minimization by augmented Lagrangian and alternating direction algorithms,” CAAM Report, 2010.
  • H. Qi, Z. Chen, J. Guo, and L. Zhou, “Sparse-view computed tomography image reconstruction via a combination of L1 and SL0 regularization,” Bio-Medical Materials and Engineering, vol. 26, pp. S1389–S1398, 2015.
  • S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 33–61, 1998.
  • TEAM RADS, \endinput