A four directions variational method for solving image processing problems

Iranian Journal of Numerical Analysis and Optimization

مقاله 5، دوره 10، شماره 2 - شماره پیاپی 18، آذر 2020، صفحه 87-104   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v10i2.84900

نویسندگان

Alireza H.^* ؛ E.E. Esfahani

School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Iran.

چکیده

In this paper, based on a discrete total variation model, a modified discretization of total variation (TV) is introduced for image processing problems. Two optimization problems corresponding to compressed sensing magnetic resonance imaging (MRI) data reconstruction problem and image denoising are proposed. In the proposed method, instead of applying isotropic TV whose gradient field is a two directions vector, a four directions discretization with some modification is applied for the inverse problems. A dual formulation for the proposed TV is explained and an efficient primal dual algorithm is employed to solve the problem. Some important image test problems in MRI and image denoising problems are considered in the numerical experiments. We compare our model with the state of the art methods.

کلیدواژه‌ها

Total variation؛ Magnetic resonance imaging؛ Primal-dual optimization method؛ Regularization؛ Image denoising

مراجع

1. Abergel, R. and Moisan, L. The Shannon total variation, J. Math. Imaging Vision 59 (2017), 341–370.

2. Alter, F., Caselles, V. and Chambolle, A. Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow, Interfaces Free Bound. 7 (2005), 29–53 .

3. Beck, A. First-order methods in optimization, SIAM, Philadelphia, 2017.

4. Bredies, K. Recovering piecewise smooth multichannel images by minimization of convex functional with total generalized variation penal, A. Bruhn et al. eds., Global Optimization Methods, Lecture Notes in Computer Science 8293, (2014), 44–77.

5. Bredies, K., Kunisch, K. and Pock, T. Total generalized variation, SIAM J. Imaging Sci. 3 (2010), 492–526.

6. Buades, A., Coll, B. and Morel, J.M. Image denoising methods. A new nonlocal principle, SIAM Rev. 52 (2010), 113–147.

7. Chambolle, A., Caselles, V., Novaga, M., Cremers, D. and Pock, T. An introduction to total variation for image analysis, Theoretical Foundations and Numerical Methods for Sparse Recovery, De Gruyter, Radon Series Comp. Appl. Math. 9 (2010), 263–340.

8. Chambolle, A., Levine, S.E. and Lucier, B.J. An upwind finite-difference method for total variation based image smoothing, SIAM J. Imaging Sci., 4 (2011), 277–299.

9. Chambolle, A. and Pock, T. A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis. 40 (2010), 120–145.

10. Condat, L. A primal-dual splitting method for convex optimization in volving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl. 158, 460–479 (2013).

11. Ghazel, M., Freeman, G.H. and Vrscay, E.R. Fractal image denoising. IEEE Trans. Image Process. 12 (2003), 1560–1578.

12. Guo, W., Kin, J. and Yin, W. A new detail-preserving regularization scheme, SIAM J. Imaging Sci. 7 (2014), 1309–1334.

13. Hosseini, A. New discretization of total variation functional for image processing tasks, Signal Process. Image Commun. 78 (2019), 62–76.

14. Huang, J., Zhang, S. and Metaxas, D. Efficient MR image reconstruction for compressed MR imaging, Med. Image Anal. 15 (2011), 670–679.

15. Knoll, F., Bredies, K., Pock, T. and Stollberger, R. Second order total generalized variation (TGV) for MRI, Magn. Reson. Med. 65 (2011), 480–491.

16. Kou, G., Zhao, Y., Peng, Y. and Shi, Y. A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data, IEEE J-STSP. 4 (2010), 288–297.

17. Lore, K.G., Akintayo, A. and Sarkar, S. LLNEt: A deep autoencoder approach to natural low-light image enhancement, Pattern Recognit, 61 (2015), 650–662.

18. Lustig, M. Michael Lustig homepage, https://people.eecs.berkeley. edu/~mlustig/index.html, Online, accessed February 2019.

19. Lustig, M., Donoho, D. and Pauly, J. Sparse MRI: The application of compressed sensing for rapid MR imaging, Magn. Reson. Med. 58 (2007), 1182–1195.

20. Ma, S., Yin, W., Zhang, Y. and Chakraborty, A. An efficient algorithm for compressed MR imaging using total variation and wavelets, IEEE Conference on Computer Vision and Pattern Recognition, CVPR (2008), 1–8.

21. Perona, P. and Malik, J. Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990), 629–639.

22. Ravishankar, S. and Bresler, Y. MR image reconstruction from highly undersampled k-space data by dictionary learning, IEEE Trans. Med. Imag ing, 30 (2011), 1029–1041.

23. Ravishankar, S. and Bresler, Y. Efficient blind compressed sensing using sparsifying transforms with convergence guarantees and application to magnetic resonance imaging, SIAM J. Imaging Sci. 8 (2015), 2519–2557.

24. Rudin, L., Osher, S. and Fatemi, E. Nonlinear total variation based noise removal algorithms, Physica D. 60 (1992), 259–268.

25. Wang, N., Tao, D., Gao, X., Li, X. and Li, J. A comprehensive survey to face hallucination, Int. J. Comput. Vis. 106 (2014), 9–30.

26. Weickert, J. Anisotropic diffusion in image processing, Teubner, Stuttgart, 1998.