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E. Tavakkol S.M. Hosseini A.R. Hosseini

Abstract

Variational models are one of the most efficient techniques for image denoising problems. A variational method refers to the technique of optimizing a functional in order to restore appropriate solutions from observed data that best fit the original image. This paper proposes to revisit the discrete total generalized variation (TGV ) image denoising problem by redefining the operations via the inclusion of a diagonal term to reduce the staircasing effect, which is the patchy artifacts usually observed in slanted regions of the image. We propose to add an oblique scheme in discretization operators, which we claim is aware of the alleviation of the staircasing effect superior to the con ventional TGV method. Numerical experiments are carried out by using the primal-dual algorithm, and numerous real-world examples are conducted to confirm that the new proposed method achieves higher quality in terms of rel ative error and the peak signal to noise ratio compared with the conventional TGV method.

Article Details

Keywords

Image denoising;, Total variation;, Staircasing effect;, Total generalized variation;, Peak signal to noise ratio.

References
1. Aujol, J.F., Gilboa, G. and Papadakis, N. Fundamentals of non-local total variation spectral theory, In International Conference on Scale Space and Variational Methods in Computer Vision, Springer, Cham, (2015), 66–77.
2. Bauschke, H.H. and Combettes, P.L. Convex analysis and monotone operator theory in Hilbert spaces, Springer, New York, 2011.
3. Bioucas–Dias, J.M. and Figueiredo, M.A. A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration, IEEE Transactions on Image processing, 16(12) (2007), 2992–3004.
4. Blomgren, P. and Chan, T.F. Color TV: total variation methods for restoration of vector-valued images, IEEE transactions on image process ing, 7(3) (1998), 304–309.
5. Blomgren, P., Chan, T.F., Mulet, P. and Wong, C.K. Total variation image restoration: numerical methods and extensions, In Proceedings of International Conference on Image Processing, IEEE, 3 (1997), 384–387.
6. Bredies, K., Kunisch, K. and Pock, T. Total generalized variation, SIAM Journal on Imaging Sciences, 3(3) (2010), 492–526.
7. Bredies, K. and Valkonen, T. Inverse problems with second-order total generalized variation constraints, Proceedings of SampTA 201 (2011).
8. Bresson, X. and Chan, T.F. Fast dual minimization of the vectorial total variation norm and applications to color image processing, Inverse problems and imaging, 2(4) ( 2008), 455–484.
9. Burns, M., Haidacher, M., Wein, W., Viola, I. and Groeller, E. Feature emphasis and contextual cutaways for multimodal medical visualization, In EuroVis, 7 (2007), 275–282.
10. Caselles, V., Chambolle, A. and Novaga, M. The discontinuity set of solutions of the TV denoising problem and some extensions, Multiscale modeling & simulation, 6(3) (2007), 879–894.
11. Caselles, V., Chambolle, A. and Novaga, M. Regularity for solutions of the total variation denoising problem, Revista Matematica Iberoamericana, 27(1) (2011), 233–252.
12. Chambolle, A., Levine, S.E. and Lucier, B.J. An upwind finite difference method for total variation-based image smoothing, SIAM Journal on Imaging Sciences, 4(1) (2011), 277–299.
13. Chan, T.F., Osher, S. and Shen, J. The digital TV filter and nonlinear denoising, IEEE Transactions on Image processing, 10(2) (2001), 231–241.
14. Chan, T.F. and Wong, C.K. Total variation blind deconvolution, IEEE transactions on Image Processing, 7(3) (1998), 370–375.
15. Chan, T.F., Yip, A.M. and Park, F.E. Simultaneous total variation image inpainting and blind deconvolution, International Journal of ImagingSystems and Technology, 15(1) (2005), 92–102.
16. Condat, L. A primal-dual splitting method for convex optimization in volving Lipschitzian, proximable and linear composite terms, Journal of optimization theory and applications, 158(2) (2013), 460–79.
17. Elad, M. and Feuer, A. Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images, IEEE transactions on image processing, 6(12) (1997), 1646–1658.
18. Fadili, J.M. and Peyr, G. Total variation projection with first order schemes, IEEE Transactions on Image Processing, 20(3) (2011), 657–669.
19. Georgiev, T.G. and Chunev, G.N. Methods and apparatus for rendering output images with simulated artistic effects from focused plenoptic camera data, U.S. Patent 8,665,341 (2014).
20. Guichard, F. and Malgouyres, F. Total variation based interpolation, In 9th European Signal Processing Conference, IEEE, (1998), 1–4.
21. Lou, Y., Zeng, T., Osher, S. and Xin, J. A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM Journal on Imaging Sciences, 8(3) (2015), 1798–1823.
22. Nacereddine, N., Zelmat, M., Belaifa, S.S. and Tridi, M. Weld defect detection in industrial radiography based digital image processing, Trans actions on Engineering Computing and Technology 2 (2005), 145–148.
23. Richardson, W.H. Bayesian-based iterative method of image restoration, Journal of the Optical Society of America, 62(1) (1972), 55–59.
24. Ring, W. Structural properties of solutions to total variation regularization problems, ESAIM: Mathematical Modelling and Numerical Analysis, 34(4) (2000), 799–810.
25. Rudin, L.I. and Osher, S. Total variation based image restoration with free local constraints, In Proceedings of 1st International Conference on Image Processing, IEEE, 1 (1994), 31–35.
26. Rudin, L.I., Osher, S. and Fatemi, E. Nonlinear total variation based noise removal algorithms, Physica D: nonlinear phenomena, 60(1-4) (1992), 259–268.
27. Santhanam, A.P., Fidopiastis, C.M., Hamza-Lup, F.G., Rolland, J.P. and Imielinska, C.Z. Physically-based deformation of high-resolution 3D lung models for augmented reality based medical visualization, MICCAI AMI-ARCS, (2004), 21–32.
28. Svakhine, N.A., Ebert, D.S. and Andrews, W.M. Illustration-inspired depth enhanced volumetric medical visualization, IEEE Transactions on Visualization and Computer Graphics, 15(1) (2009), 77–86.
29. Vogel, C.R. and Oman, M.E. Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE transactions on image processing, 7(6) (1998), 813–824.
30. Wachs, J., Stern, H., Edan, Y., Gillam, M., Feied, C., Smith, M. and Handler, J. A real-time hand gesture interface for medical visualization applications, In Applications of Soft Computing, Springer, Berlin, Heidelberg, (2006), 153–162.
31. Zach, C., Pock, T. and Bischof, H. A duality based approach for realtime TV-L1 optical flow, In Joint Pattern Recognition Symposium, Springer, Berlin, Heidelberg, (2007), 214–223.
32. Zhang, D.D., Kong, W.K.A., You, J. and Wong, M. Online palmprint identification, IEEE Transactions on pattern analysis and machine intelligence, 25(9) (2003), 1041–1050.
33. Zhang, Z. and Blum, R.S. Region-based image fusion scheme for concealed weapon detection, In Proceedings of the 31st Annual Conference on Information Sciences and Systems, (1997), 168–173.
How to Cite
Tavakkol, E., Hosseini, S., & Hosseini, A. (2019). A new regularization term based on second order total generalized variation for image denoising problems. Iranian Journal of Numerical Analysis and Optimization, 9(2), 141-163. https://doi.org/10.22067/ijnao.v9i2.77371
Section
Research Article